Research Article Propagation of Water Waves over Uneven …downloads.hindawi.com/journals/ijde/2015/805625.pdf · 2019-07-31 · Research Article Propagation of Water Waves over Uneven

Research ArticlePropagation of Water Waves over Uneven Bottom underthe Effect of Surface Tension

Juan Carlos Muntildeoz Grajales

Departamento de Matematicas Universidad del Valle Calle 13 Nro 100-00 Cali Colombia

Correspondence should be addressed to Juan Carlos Munoz Grajales jcarlmzyahoocom

Received 29 July 2015 Accepted 8 September 2015

Academic Editor Mayer Humi

Copyright copy 2015 Juan Carlos Munoz Grajales This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We establish existence and uniqueness of solutions to the Cauchy problem associated with a new one-dimensional weakly-nonlinear weakly-dispersive system which arises as an asymptotical approximation of the full potential theory equations formodelling propagation of small amplitude water waves on the surface of a shallow channel with variable depth taking into accountthe effect of surface tension Furthermore numerical schemes of spectral type are introduced for approximating the evolution intime of solutions of this system and its travelling wave solutions in both the periodic and nonperiodic case

1 Introduction

In this paper we study the propagation of water waves on thesurface of a shallow channel with variable depth consideringthe effect of surface tension To describe this phenomenonwewill derive a new water wave model from Eulerrsquos equations(in dimensionless variables) for an inviscid incompressibleliquid bounded above by a free surface and bounded belowby an impermeable bottom topography [1]

120573120601119909119909+ 120601

119910119910= 0

for minus 119867(119909120574) lt 119910 lt 120572120578 (119909 119905) minusinfin lt 119909 lt infin

(1)

with the nonlinear free surface conditions

120578119905+ 120572120601

119909120578119909minus1

120573120601119910= 0

120578 + 120601119905+120572

2(1206012119909+1

1205731206012119910) minus 120573120590120597

119909(

120578119909

radic1 + 1205731205722 (120578119909)2

)

= 0

(2)

at 119910 = 120572120578(119909 119905) Here 120601(119909 119910 119905) denotes the potential velocityand 120578(119909 119905) the wave elevation measured with respect to

the undisturbed free surface 119910 = 0 The dimensionlessparameters 120572 and 120573 are small positive real numbers whichmeasure the strength of nonlinear and dispersive effectsrespectively The parameter 120574 measures the ratio inhomo-geneitieswavelength and the parameter 120590 is associated withthe surface tensionTheNeumann condition at the imperme-able bottom is

120601119910+120573

1205741198671015840 (

119909

120574)120601

119909= 0 (3)

The bottom topography is described by 119910 = minus119867(119909120574) where

119867(119909

120574) =

1 + 119899(119909

120574) when 0 lt 119909 lt 119871

1 when 119909 le 0 or 119909 ge 119871(4)

We will see in Section 3 that system (1)ndash(3) is equivalent toleading order provided that 120572 ≪ 1 120573 ≪ 1 to the weakly-nonlinear weakly-dispersive system

(119868 minus120573

21205972120585) V119905=2120573

31205974120585Φ minus 120597

120585((1 +

120572V1198722)Φ

120585)

(119868 minus120573

21205972120585)Φ

119905

Hindawi Publishing CorporationInternational Journal of Differential EquationsVolume 2015 Article ID 805625 21 pageshttpdxdoiorg1011552015805625

2 International Journal of Differential Equations

= 120573120590(V120585120585minus 31198721015840

1198724V120585+ 3(1198721015840)

2

1198725V minus11987210158401015840

1198724V) minus

1

119872V

minus120572

21198722Φ2120585 V (120585 0) = V

0(120585) Φ (120585 0) = Φ

0(120585)

(5)

The variables 120585 and 119905 are the space and time coordinatesrespectively V = 119872(120585)120578(120585 119905) with 120578(120585 119905) denoting theelevation of the free surface and Φ(120585 119905) represents thepotential fluid velocitymeasured at the channelrsquos bottomThemetric coefficient 119872(120585) is related to the channelrsquos bottom119910 = minus119867(119909120574) and it is defined as

119872(120585) =120587

4radic120573intinfin

minusinfin

119867(119909 (1205850 minusradic120573) 120574)

cosh2 ((1205872radic120573) (1205850minus 120585))

1198891205850 (6)

where (119909 119910) rarr (120585 120577) is the coordinate transformation usedin the derivation of system (5) to map the original physicalchannel with variable depth onto a strip in the complexplane This strategy of change of variable was introduced byHamilton [2] and later used successfully by Nachbin in [3] tostudy wave propagation over a channel with a highly variabletopography Observe from (6) that the coefficient 119872(120585) isinfinitely differentiable although the function describing thechannelrsquos bottom is not smooth In case of constant depth thecoefficient119872(120585) is identically one and system (5) reduces toa system derived by Quintero and Montes in [4]

Wave-topography interaction has been the subject ofconsiderable mathematical research [5ndash18] The physicalapplications range from coastal surface waves [19] to atmo-spheric flows over mountain ranges [20 21] In particular theinteraction of waves with fine features of the topography is ofgreat interest As pointed out in the introduction to theOrog-raphy proceedings [21] of the European Centre for Medium-Range Weather Forecasts (ECMWF) ldquothe representation of subgrid-scale orographic processes is recognized as crucialto numerical weather prediction at all time rangesrdquo In theatmospheric literature orography implies mountain ranges[20]

In previous works some weakly-nonlinear weakly-dispersive models have been developed to describe the inter-action of a long pulse with small amplitude that propagateson the surface of a channel with a variable bottom [2 3 5 7ndash9 22ndash24] However these models either neglect the effect ofsurface tension on the free surface where the wave propagatesor are not applicable to bottoms described by a discon-tinuousnondifferentiable function or the wave elevation isremoved in their physical derivation For instance Milewski[23] derived a bidirectional scalar Benney-Luke type equa-tion in terms of the potential velocity which includes thesurface tension effect and the influence of a variable bottomHowever the asymptotical derivation of this model implieseliminating the wave elevation and consequently neglectingseveral second-order terms in the parameters 120572 and 120573

Some of the features of new formulation (5) are thefollowing

(i) It can be applied to study wave propagation over ashallow channel with a discontinuous or nondifferen-tiable bottom provided that the bottomrsquos fluctuationssatisfy |119899| lt 1 This is a consequence of introducingthe conformal mapping (119909 119910) rarr (120585 120577) mentionedabove Observe that all coefficients of the reducedequations (5) in the new coordinate system result inbeing infinitely differentiable since they depend onthe smooth function119872(120585)

(ii) One-dimensional system (5) models bidirectionalwaves and it incorporates the simultaneous effects ofsurface tension and variable depth upon the shapeof a water wave that propagates on the surface of anirregular shallow channel

(iii) Furthermore in the derivation of system (5) we donot eliminate the wave elevation (which prevents usfrom neglecting additional terms of order 119874(1205722 1205732))as done for example in [23] Therefore the newformulation (5) is expected to be a more accurateapproximation of the full potential theory equations

In the present paper we establish existence and unique-ness of a solution to Cauchy problem (5) using classicalsemigroup theory and Banachrsquos fixed point principle In [4]the well-posedness of system (5) was analyzed but only inthe case that 119872(120585) equiv 1 In second place we formulate aGalerkin-spectral numerical scheme of spectral accuracy inspace to approximate the solutions of system (5) based on theFourier basis and using an implicit-explicit (IMEX) second-order strategy for time stepping This type of temporaldiscretization is described for instance in [25] and it hasbeen used in conjunction with spectral methods [26 27] forthe time integration of spatially discretized PDEs of diffusion-convection type IMEX schemes have also been successfullyapplied to the incompressible Navier-Stokes equations [28]and in environmental modelling studies [29] On the otherhand we develop a numerical solver to compute travellingwave solutions of system (5) by using a Fourier-collocationstrategy combined with a Newton-type iterative procedureWe also indicate how to determine appropriate starting pointsfor this iterative process in order to achieve convergenceExistence results on travelling wave solutions (in the periodicand nonperiodic case) of system (5) with 119872 equiv 1 have alsobeen established in [4] Travelling wave solutions exist as aconsequence of a balance between nonlinear and dispersiveeffects present in a system these waves travel with a constantspeed without any temporal evolution in shape or size whenthe frame of referencemoveswith the same speed of thewaveIn the last decades the study of travelling waves has grownenormously because they appear in several and varied fieldsof application such as fluid mechanics optics acousticsoceanography and astronomy Thus to determine existenceand properties of such type of solutions is a fundamentalproblem in the theory of ordinary and partial differentialequations of great interest for both pure and applied math-ematicians

International Journal of Differential Equations 3

The rest of this paper is organized as follows In Section 2we introduce the functional spaces and notation to beemployed in the paper In Section 3 we present in detail thederivation of model (5) starting from the potential theoryequations including the surface tension effect In Section 4we discuss existence and uniqueness of a solution of theCauchy problem associated with system (5) by using Banachrsquosfixed point principle and semigroup theory In Section 5we introduce the numerical schemes for approximating theevolution of a solution of system (5) and computing theirtravelling wave solutions (periodic and nonperiodic cases)Section 6 presents a set of numerical simulations to check theaccuracy of the numerical schemes developed in the paperFurthermore some numerical experiments are included toillustrate the interaction between thewave-topography effectsand surface tension Finally in Section 7 are the conclusionsof the paper

2 Preliminaries

To analyze existence of solutions of problem (5) we will usethe standard notation For 1 le 119901 le infin we will denoteby 119871119901(R) (or simply 119871119901) the Banach space of measurablefunctions inR such that int

R|119891(119909)|119901119889119909 lt infin if 1 le 119901 lt infin and

ess supR|119891| lt infin if 119901 = infin We define the norm in 119871119901(R) for1 le 119901 lt infin by

10038171003817100381710038171198911003817100381710038171003817119871119901 = (int

R

1003816100381610038161003816119891 (119909)1003816100381610038161003816119901

)1119901

(7)

and in119871infin(R) by 119891infin= ess supR|119891|119871

2(R) is aHilbert spacefor the scalar product

⟨119891 119892⟩ = intR

119891 (119909) 119892 (119909) 119889119909 (8)

We set 119891 = 1198911198712 For function 119891 isin 1198711(R) the Fourier

transform is defined as

F (119891) (119910) = 119891 (119910) = intR

119891 (119909) 119890minus119894119909119910119889119909 119910 isin R (9)

and the inverse Fourier transform is defined by

Fminus1 (119891) (119910) = (119910) =

1

2120587intR

119891 (119909) 119890119894119909119910119889119909 119910 isin R (10)

We will also denote by F(119891) (or 119891) and Fminus1(119891) (or ) theextensions of these operators to 1198712(R) The convolution oftwo functions 119891 119892 isin 1198712(R) is defined as

119891 lowast 119892 (119909) = intR

119891 (119909 minus 119910) 119892 (119910) 119889119910 (11)

We recall that 119891 lowast 119892(119910) = 119891(119910)119892(119910) For 119904 isin R we definethe Sobolev space119867119904(R) (sometimes written for simplicity as119867119904) as the completion of the Schwartz space (rapidly decayingfunctions) defined as

119878 (R) = 119891 isin 119862infin

(R) 10038171003817100381710038171003817119909

]12059712057311990911989110038171003817100381710038171003817infin ltinfin for any ] 120573

isin Z+ (12)

with respect to the norm

10038171003817100381710038171198911003817100381710038171003817119867119904 = (int

R

(1 + 1199102)119904 10038161003816100381610038161003816119891 (119910)

100381610038161003816100381610038162

119889119910)12

(13)

For simplicity we also denote this norm by 119891119904 The inner

product in119867119904(R) is defined as

⟨119891 119892⟩119904= int

R

(1 + 1199102)119904

119891 (119910) 119892 (119910) 119889119910 (14)

The product norm in the space119867119904(R) times 119867119904(R) is defined by

1003817100381710038171003817(120577 119906)1003817100381710038171003817119867119904times119867119904 =

10038171003817100381710038171205771003817100381710038171003817119867119904 + 119906119867119904 (15)

for (120577 119906) isin 119867119904(R) times 119867119904(R) Sometimes we will also use theequivalent product norm

1003817100381710038171003817(120577 119906)1003817100381710038171003817119867119904times119867119904 = (

100381710038171003817100381712057710038171003817100381710038172

119867119904 + 119906

2

119867119904)12

(16)

For 119879 gt 0 we will denote by 119862([0 119879]119867119904(R)) the space ofcontinuous functions 119891 [0 119879] rarr 119867119904(R) that is the spaceof continuous functions 119905 rarr 119891(119905 sdot) isin 119867119904(R) 119905 isin [0 119879] withthe supremum norm

10038171003817100381710038171198911003817100381710038171003817119862(0119879) = sup

119905isin[0119879]

1003817100381710038171003817119891 (119905 sdot)1003817100381710038171003817119867119904 (17)

and the product norm

1003817100381710038171003817(120577 119906)1003817100381710038171003817119862(0119879)2 =

10038171003817100381710038171205771003817100381710038171003817119862(0119879) + 119906119862(0119879) (18)

for (120577 119906) isin 119862([0 119879] 119867119904(R) times 119867119904(R))

3 Governing Equations

We start by presenting the potential theory formulation forEulerrsquos equations (in dimensionless variables) for an inviscidincompressible liquid bounded above by a free surface andbounded below by an impermeable bottom topography andincluding the effect of surface tension [1]

120573120601119909119909+ 120601

119910119910= 0


(19)


120578119905+ 120572120601

119909120578119909minus1

120573120601119910= 0

120578 + 120601119905+120572

2(1206012119909+1

1205731206012119910) minus 120573120590120597

119909(

120578119909

radic1 + 1205731205722 (120578119909)2

)

= 0

(20)


2

Symmetric flow region

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

minus4 minus3 minus2 minus1 0 1 4 5minus5 3

Figure 1 The symmetric domain in the complex 119911-plane where119911 = 119909(120585 120577) + 119894119910(120585 120577) The lower half (119909 isin [minus5 5] 119910 isin [minus3 0]) isthe physical channel with 119910 = 120577 = 0 indicating the undisturbedfree surface Superimposed in this complex 119911-plane domain arethe (curvilinear) coordinate level curves from the 119908-plane system120585120577 The polygonal line at the bottom of the figure is a schematicrepresentation of the topography (where 120577 = plusmnradic120573) This figure wasgenerated using SC Toolbox [30]

at 119910 = 120572120578(119909 119905) Here 120601(119909 119910 119905) denotes the potential velocityand 120578(119909 119905) the wave elevation measured with respect tothe undisturbed free surface 119910 = 0 The dimensionlessparameters 120572 and 120573 measure the strength of nonlinear anddispersive effects respectively The parameter 120574 measuresthe ratio inhomogeneitieswavelength and 120590 is related tothe surface tension effects The Neumann condition at theimpermeable bottom is

120601119910+120573

1205741198671015840 (

119909

120574)120601

119909= 0 (21)


119867(119909

120574) =

1 + 119899(119909

120574) when 0 lt 119909 lt 119871

1 when 119909 le 0 or 119909 ge 119871(22)

The bottom profile is described by the (possibly rapidlyvarying) function minus119899(119909120574)The topography is rapidly varyingwhen 120574 ≪ 1 The scale 119871 represents the total length of theirregular section of the coast The undisturbed depth is givenby 119910 = minus1 and the topography can be of large amplitudeprovided that |119899| lt 1 The fluctuations 119899 are not assumed tobe small nor continuous nor slowly varying

Let us consider a symmetric flow domain by reflecting theoriginal one about the undisturbed free surface (cf Figure 1)This domain is denoted by Ω

119911where 119911 = 119909 + 119894radic120573119910 and can

be considered as the conformal image of the strip Ω119908where

119908 = 120585 + 119894120577 with |120577| le radic120573 Then 119911 = 119909(120585 120577) + 119894radic120573119910(120585 120577) =119909(120585 120577)+119894119910(120585 120577)with 119909 and 119910 a pair of harmonic functions onΩ119908 Following the strategy suggested by Hamilton in [2] and

Nachbin [3] within the weakly-nonlinear weakly-dispersiveregime (120572 ≪ 1 120573 ≪ 1) and using the relationships

120601119909=1

|119869|[119910120577120601120585minus 119910

120585120601120577]

120601119910=1

|119869|[minus119909

120577120601120585+ 119909

120585120601120577]

1206012119909+ 1206012

119910=1

|119869|(1206012120585+ 1206012

120577

)

120597119909=1

|119869|(119910120577120597120585minus 119910

120585120597120577)

(23)

where

|119869| = 119909120585119910120577minus 119910

120585119909120577= 1199102

120577

+ 1199102120585 (24)

and the variable free surface coefficient119872(120585) defined as

119872(120585) = 119910120577(120585 0)

= 1 +120587


minusinfin

119899 (119909 (1205850 minusradic120573) 120574)

cosh2 (1205872radic120573) (1205850minus 120585)1198891205850

(25)

the potential theory equations can be approximated to order119874(120572 120573) in the orthogonal curvilinear coordinates (120585 120577) with120577 = 120577radic120573 + 1 by the equation

120573120601120585120585+ 120601

120577120577= 0 at 0 lt 120577 lt 1 + 120572119873 (120585 119905) (26)

with conditions at the free surface 120577 = 1 + 120572119873(120585 119905)

|119869|119873119905+ 120572120601

120585119873120585minus1

120573120601120577= 0

120578 + 120601119905+120572

2 |119869|(1206012120585+1

1205731206012120577) minus 120573120590

1

|119869|119872120597

120585(1

119872120578120585)

= 119874 (1205722 120572120573 1205732)

(27)

and condition at the channelrsquos bottom

120601120577= 0 at 120577 = 0 (28)

Observe that the change of variables 120577 rarr 120577 lets the originof the curvilinear coordinate system at the bottom TheJacobian for the (120585 120577) rarr (119909 119910) coordinate transformationis represented by |119869| and 120577 = 1 + 120572119873(120585 119905) corresponds tothe position of the free surface in the curvilinear coordinatesystem By performing an asymptotic simplification as in [1](page 464) through a power series expansion in terms of


the dispersion parameter 120573 near the bottom of the channelin the form

120601 (120585 120577 119905) =119896

sum119895=0

(minus1)119895 1205731198951205772119895

(2119895)1205972119895

120585Φ (120585 119905) + 119874 (120573

119896+1) (29)

into (26)ndash(28) for 120572 120573 being small we find that free surfaceconditions (27) can be approximated to order 119874(120572) 119874(120573) bythe equations

119872(120585) 120578119905+ [(1 +

120572

119872(120585)120578)Φ

120585]120585

minus120573

6Φ120585120585120585120585

= 119874 (1205722 120572120573 1205732)

(30)

120578 + Φ119905minus120573

2Φ120585120585119905+

120572

21198722 (120585)Φ2120585minus 120573120590

1

119872120597120585(1

119872120578120585)

= 119874 (120572120573 1205732)

(31)

Here Φ(120585 119905) = 120601(120585 0 119905) denotes the potential fluid velocityat the channelrsquos bottom 120577 = 0 We point out that (30) and(31) with 120590 = 0 (no surface tension) correspond to thosederived in [3] (bottom of page 915 and top of page 916) Inthe derivation of (30)-(31) we used the relationship

|119869| (120585 119905) = 119872 (120585)2

+ 119874 (1205722) (32)

which means that at leading order the Jacobian of theconformal coordinate transformation is time independent

Let us introduce the new variable V(120585 119905) = 119872(120585)120578(120585 119905)Thus observe that from (30)

V119905= 119872120578

119905= minus1205972

120585Φ + 119874 (120572 120573) (33)

This relationship allows us to change the form of dispersiveterms in the equations above In particular by using thedecomposition

minus120573

61205974120585Φ =

120573

21205974120585Φ minus2120573

31205974120585Φ

= minus120573

21205972120585V119905minus2120573

31205974120585Φ + 119874(120572120573 1205732)

(34)

in (30) we obtain that system (30)-(31) can be approximatedto order 119874(120572 120573) by the following equations

(119868 minus120573

21205972120585) V119905=2120573

31205974120585Φ minus 120597

120585((1 +

120572V1198722)Φ

120585)

(119868 minus120573

21205972120585)Φ

119905

= 120573120590(1

1198723V120585120585minus 31198721015840

1198724V120585+ 3(1198721015840)

2

1198725V minus11987210158401015840

1198724V)

minus1

119872V minus

120572

21198722Φ2120585

(35)

We point out that the system above is applicable to modellingof propagation of water waves over an arbitrary rapidly vary-ing depth In case of a slowly varying channelrsquos topographyand 119899(119909) = radic120573119899

0(119909) with 119899

0being a stationary random

process with standard deviation and correlation length oforder one the metric coefficient 119872(120585) can be expanded as(see [9])

119872(120585) = 1 + radic1205731198990(120585) + 119874 (120573) (36)

Thus

1205731205901

1198723V120585120585= 120573120590(1 minus 3radic120573119899

0(120585) + 119874 (120573)) V

120585120585 (37)

and system (35) can be approximated to order 119874(120572 120573) by

(119868 minus120573

21205972120585) V119905=2120573

31205974120585Φ minus 120597

120585((1 +

120572V1198722)Φ

120585) (38)

(119868 minus120573

21205972120585)Φ

119905

= 120573120590(V120585120585minus 31198721015840

1198724V120585+ 3(1198721015840)

2

1198725V minus11987210158401015840

1198724V) minus

1

119872V

minus120572

21198722Φ2120585

(39)

Note that the coefficient119872(120585) is smooth even when the func-tion describing the bottom 119910 = minus119867(119909120574) is discontinuousor nondifferentiable The function119872(120585) is time independentand becomes identically one for a channel with constantdepth In this case system (38)-(39) reduces to that studiedin [4] Moreover in applications this coefficient is boundedand infR119872(120585) gt 0 These properties will be important toobtain existence and uniqueness results for system (38)-(39)We also point out that the function119872(120585) actually depends onthe dispersion parameter 120573 For this reason it will be denotedby119872(120585 120573) whenever we need to emphasize this dependence

4 Existence and Uniqueness

System (38)-(39) can be written as

(V

Φ)119905

= A(V

Φ) +G(

V

Φ) (40)

where


A(V

Φ) = (

0 (119868 minus120573

21205972120585)minus1

(119868 minus2120573

31205972120585) (minus1205972

120585)

(119868 minus120573

21205972120585)minus1

1205731205901205972120585

0

)(V

Φ)

G(V

Φ) = (

minus(119868 minus120573

21205972120585)minus1

(120572

1198722VΦ120585)120585

(119868 minus120573

21205972120585)minus1

(minus31205731205901198721015840

1198724V120585+ 3120573120590

(1198721015840)2

1198725V minus 120573120590

11987210158401015840

1198724V minus1

119872V minus

120572

21198722Φ2120585)

)

(41)

System (40) is supplemented with the initial conditions

V (120585 0) = V0(120585)

Φ (120585 0) = Φ0(120585)

(42)

Taking Fourier transform in the spatial variable 120585 in system(40) we have

119905= 119860 (119910) + G (119880)

(119910 0) = 0(119910)

(43)

where

119880 = (V

Φ)

1198800= (

V0

Φ0

)

119860 (119910) = (0

1199102 (1 + (21205733) 1199102)

1 + (1205732) 1199102

minus1205731205901199102

1 + (1205732) 11991020

)

(44)

41 Analysis of the Linear Semigroup Consider system (40)withG equiv 0

119905= 119860 (119910)

(119910 0) = 0(119910)

(45)

which has unique solution in the form

(119910 119905) = 119890119860(119910)1199050(119910) (46)

where

119890119860(119910)119905 = (11988611(119910 119905) 119886

12(119910 119905)

11988621(119910 119905) 119886

22(119910 119905)

)

= (cos (120582 (119910) 119905) Γ (119910) sin (120582 (119910) 119905)

Ψ (119910) sin (120582 (119910) 119905) cos (120582 (119910) 119905))

120582 (119910) =21199102radic120573120590 (3 + 21205731199102)

radic3 (2 + 1205731199102)

Γ (119910) =radic3 + 21205731199102

radic3120573120590

Ψ (119910) = minusradic3120573120590

radic3 + 21205731199102

(47)

By using the inverse Fourier transform in (46) we have that

119880 (119909 119905) = 119878 (119905) 1198800(119909) (48)

where

119878 (119905) 119891 = Fminus1 (119890119860(sdot)119905119891 (sdot)) (49)

with

119891 = (1198911

1198912

) (50)

Theorem 1 The family of linear operators (119878(119905))119905ge0

is a 1198620-

semigroup in 119867119904minus2 times 119867119904minus1 Furthermore A 119867119904minus1 times 119867119904 rarr119867119904minus2 times 119867119904minus1 is its infinitesimal generator in119867119904minus2 times 119867119904minus1


Proof Let 119891 = (1198911 1198912)119879 Then

1003817100381710038171003817119878 (119905) 11989110038171003817100381710038172

119867119904minus2times119867119904minus1

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

(Fminus1 (119886

111198911+ 119886

121198912)

Fminus1 (119886211198911+ 119886

221198912))

10038171003817100381710038171003817100381710038171003817100381710038171003817

2


=10038171003817100381710038171003817F

minus1 (119886111198911) +Fminus1 (119886

121198912)100381710038171003817100381710038172

119867119904minus2

+10038171003817100381710038171003817F

minus1 (119886211198911) +Fminus1 (119886

221198912)100381710038171003817100381710038172

119867119904minus1

le 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381611988611

10038161003816100381610038162 100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381611988612

10038161003816100381610038162 100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381611988621

10038161003816100381610038162 100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381611988622

10038161003816100381610038162 100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

le 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381610038161198911

100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus2

(3 + 21205731199102)100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 1

3 + 21205731199102100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381610038161198912

100381610038161003816100381610038162

119889119910

le 119862 (1003817100381710038171003817119891110038171003817100381710038172

119867119904minus2 +1003817100381710038171003817119891210038171003817100381710038172

119867119904minus1)

(51)

Hereafter 119862 will denote a generic constant and we recall that0 lt 120573 lt 1 Therefore 119878(119905) 119905 ge 0 is a family of continuouslinear operators in119867119904minus2 times 119867119904minus1 On the other hand it is easyto see that 119878(0) = 119868 119878(119905 + 119904) = 119878(119905)119878(119904) = 119878(119904)119878(119905) 119905 119904 ge 0Finally

1003817100381710038171003817119878 (119905) 119891 minus 11989110038171003817100381710038172


le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(Fminus1 (119886

111198911+ 119886

121198912minus 119891

1)

Fminus1 (119886211198911+ 119886

221198912minus 119891

2))

10038171003817100381710038171003817100381710038171003817100381710038171003817

2


le 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381611988611 minus 1

10038161003816100381610038162 100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381611988612

10038161003816100381610038162 100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381611988621

10038161003816100381610038162 100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381611988622 minus 1

10038161003816100381610038162 100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

(52)

Observe that 11988612(119910 119905) rarr 0 119886

21(119910 119905) rarr 0 as 119905 rarr 0+ and

11988611(119910 119905) rarr 1 119886

22(119910 119905) rarr 1 as 119905 rarr 0+ By using Lebesguersquos

dominated convergence theorem we can conclude that

1003817100381710038171003817119878 (119905) 119891 minus 1198911003817100381710038171003817119867119904minus2times119867119904minus1 997888rarr 0 (53)

as 119905 rarr 0+ We conclude that 119878(119905) 119905 ge 0 is a stronglycontinuous semigroup in119867119904minus2 times 119867119904minus1

On the other hand

1003817100381710038171003817A11989110038171003817100381710038172

119867119904minus2times119867119904minus1 le int

R

(1 + 1199102)119904minus2

sdot100381610038161003816100381610038161003816100381610038161003816F((119868 minus

120573

21205972120585)minus1

(119868 minus2120573

31205972120585) 12059721205851198912)100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ intR

(1 + 1199102)119904minus1

sdot100381610038161003816100381610038161003816100381610038161003816F((119868 minus

120573

21205972120585)minus1

12057312059012059721205851198911)100381610038161003816100381610038161003816100381610038161003816

2

119889119910

le intR

(1 + 1199102)119904minus2(1 + (21205733) 1199102)

2

1199104

(1 + (1205732) 1199102)2

100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

+ intR

(1 + 1199102)119904minus1 120573212059021199104

(1 + (1205732) 1199102)2

100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

le 119862 (1003817100381710038171003817119891110038171003817100381710038172

119904minus1+1003817100381710038171003817119891210038171003817100381710038172

119904) = 119862

100381710038171003817100381711989110038171003817100381710038172

119867119904minus1times119867119904

(54)

To see thatA is the infinitesimal generator of the semigroup119878(119905) 119905 ge 0 observe that for 119891 isin 119867119904minus1 times 119867119904

10038171003817100381710038171003817100381710038171003817

119878 (ℎ) 119891 minus 119891

ℎminusA119891

10038171003817100381710038171003817100381710038171003817

2


=100381710038171003817100381710038171003817100381710038171003817Fminus1 (119890119860ℎ minus 119868

ℎ119891

minus 119860119891)100381710038171003817100381710038171003817100381710038171003817

2


=

10038171003817100381710038171003817100381710038171003817100381710038171003817

Fminus1 (11988611(sdot ℎ) 119891

1+ 119886

12(sdot ℎ) 119891

2minus 119891

1)

ℎminus (119868

minus120573

21205972120585)minus1

(119868 minus2120573

31205972120585) (minus1205972

1205851198912)

10038171003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+

10038171003817100381710038171003817100381710038171003817100381710038171003817

Fminus1 (11988621(sdot ℎ) 119891

1+ 119886

22(sdot ℎ) 119891

2minus 119891

2)

ℎminus (119868

minus120573

21205972120585)minus1

12057312059012059721205851198911

10038171003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus1

= intR

(1 + 1199102)119904minus2


sdot

10038161003816100381610038161003816100381610038161003816100381610038161003816

(cos (120582ℎ) minus 1) 1198911+ Γ sin (120582ℎ) 119891

2

ℎ

minus1199102 (1 + (21205733) 1199102) 119891

2

1 + (1205732) 1199102

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910 + intR

(1 + 1199102)119904minus1

sdot

1003816100381610038161003816100381610038161003816100381610038161003816

Ψ sin (120582ℎ) 1198911+ (cos (120582ℎ) minus 1) 119891

2

ℎ

minusminus1205731205901199102119891

1

1 + (1205732) 1199102

1003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

(55)

But by virtue of

limℎrarr0

cos (120582ℎ) minus 1ℎ

= 0

limℎrarr0

Γsin (120582ℎ)ℎ

= Γ120582 =1199102 (1 + (21205733) 1199102)

1 + (1205732) 1199102

limℎrarr0

Ψsin (120582ℎ)ℎ

= Ψ120582 = minus1205731205901199102

1 + (1205732) 1199102

(56)

and using again Lebesguersquos dominated convergence theoremwe get that

limℎrarr0

10038171003817100381710038171003817100381710038171003817

119878 (ℎ) 119891 minus 119891

ℎminusA119891

10038171003817100381710038171003817100381710038171003817

2


= 0 (57)

42 Analysis of the Nonlinear Term

Theorem 2 Let 119904 gt 32 The application G maps 119867119904minus1 times 119867119904on itself and

G (119880)119867119904minus1times119867119904 le 119862 119880

119867119904minus1times119867119904 (58)

Furthermore

1003817100381710038171003817G (1198801) minusG (1198802)10038171003817100381710038172

119867119904minus1times119867119904

le 119862 [(1 +1003817100381710038171003817Φ110038171003817100381710038172

119867119904)1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1

+ (1003817100381710038171003817V210038171003817100381710038172

119867119904minus1 +1003817100381710038171003817Φ1 + Φ2

10038171003817100381710038172

119867119904)1003817100381710038171003817Φ1 minus Φ2

10038171003817100381710038172

119867119904]

(59)

where 119862 gt 0 is a constant and 119880119894= (

V119894Φ119894) 119894 = 1 2

Proof In the first place let 119880 = (V Φ)119879

G (119880)2

119867119904minus1times119867119904

le 119862intR

(1 + 1199102)119904minus1

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

2 100381610038161003816100381610038161003816100381610038161003816

(

V1198722Φ120585)120585

100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

210038161003816100381610038161003816100381610038161003816100381610038161003816

1015840

1198724V120585

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

2

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(1198721015840)

2

1198725V

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

210038161003816100381610038161003816100381610038161003816100381610038161003816

10158401015840

1198724V10038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

2 10038161003816100381610038161003816100381610038161003816

V119872

10038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

2100381610038161003816100381610038161003816100381610038161003816100381610038161003816

Φ2120585

1198722

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

(60)

Taking into account the inequalities

1003817100381710038171003817100381710038171003817

V1198722Φ120585

1003817100381710038171003817100381710038171003817119867119904minus2le 1198621003817100381710038171003817100381710038171003817

V1198722

1003817100381710038171003817100381710038171003817119867119904minus210038171003817100381710038171003817Φ12058510038171003817100381710038171003817119867119904minus1

le 1198621003817100381710038171003817100381710038171003817

1

1198722

1003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1 Φ

119867119904

100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724V120585

100381710038171003817100381710038171003817100381710038171003817119867119904minus2le 119862100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724

100381710038171003817100381710038171003817100381710038171003817119867119904minus1

10038171003817100381710038171003817V12058510038171003817100381710038171003817119867119904minus2

le 119862100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724

100381710038171003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725V100381710038171003817100381710038171003817100381710038171003817100381710038171003817119867119904minus2

le 119862

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725

100381710038171003817100381710038171003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus2

le 119862

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725

100381710038171003817100381710038171003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1

100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724V100381710038171003817100381710038171003817100381710038171003817119867119904minus2

le 119862100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724

100381710038171003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1

1003817100381710038171003817100381710038171003817

V119872

1003817100381710038171003817100381710038171003817119867119904minus2le 1198621003817100381710038171003817100381710038171003817

1

119872

1003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus2

le 1198621003817100381710038171003817100381710038171003817

1

119872

1003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1


1003817100381710038171003817100381710038171003817100381710038171003817

Φ2120585

1198722

1003817100381710038171003817100381710038171003817100381710038171003817119867119904minus2le 11986210038171003817100381710038171003817100381710038171003817

Φ120585

1198722Φ120585

10038171003817100381710038171003817100381710038171003817119867119904minus2

le 11986210038171003817100381710038171003817Φ12058510038171003817100381710038171003817119867119904minus1

10038171003817100381710038171003817100381710038171003817

Φ120585

1198722

10038171003817100381710038171003817100381710038171003817119867119904minus2

le 119862 Φ119904

1003817100381710038171003817100381710038171003817

1

1198722

1003817100381710038171003817100381710038171003817119867119904minus110038171003817100381710038171003817Φ12058510038171003817100381710038171003817119867119904minus2

le 119862 Φ119904

1003817100381710038171003817100381710038171003817

1

1198722

1003817100381710038171003817100381710038171003817119867119904minus1Φ

119867119904

(61)

valid for 119904 gt 32 (applying Corollary 316 in [31]) and usingthe fact that the function119872 is bounded we get that

G (119880)2

119867119904minus1times119867119904 le 119862(

1003817100381710038171003817100381710038171003817

V1198722Φ120585

1003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724V120585

100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725V

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724V100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+1003817100381710038171003817100381710038171003817

V119872

1003817100381710038171003817100381710038171003817

2

119867119904minus2

+

1003817100381710038171003817100381710038171003817100381710038171003817

Φ2120585

1198722

1003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

) le 119862(V2119867119904minus1 + Φ

2

119867119904)

(62)

On the other hand for 119880119894= (V

119894 Φ119894)119879 isin 119867119904minus1 times 119867119904 we have

that

1003817100381710038171003817100381710038171003817100381710038171003817G(

V1

Φ1

) minusG(V2

Φ2

)

1003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus1times119867119904

le 119862(1003817100381710038171003817100381710038171003817

V1

1198722Φ1120585

minusV2

1198722Φ2120585

1003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724(V1minus V

2)120585

100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725(V1minus V

2)

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724(V1minus V

2)100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+1003817100381710038171003817100381710038171003817

1

119872(V1minus V

2)1003817100381710038171003817100381710038171003817

2

119867119904minus2

+1003817100381710038171003817100381710038171003817

1

1198722(Φ21120585minus Φ2

2120585)1003817100381710038171003817100381710038171003817

2

119867119904minus2

)

le 119862(10038171003817100381710038171003817V1Φ1120585 minus V2Φ2120585

100381710038171003817100381710038172

119867119904minus2+1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1 +10038171003817100381710038171003817Φ2

1120585

minus Φ22120585

100381710038171003817100381710038172

119867119904minus2) le 119862 [(

10038171003817100381710038171003817(V1 minus V2)Φ112058510038171003817100381710038171003817119867119904minus2

+10038171003817100381710038171003817V2 (Φ1120585 minus Φ2120585)

10038171003817100381710038171003817119867119904minus2)2

+1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1 +10038171003817100381710038171003817Φ1120585

+ Φ2120585

100381710038171003817100381710038172

119867119904minus1

10038171003817100381710038171003817Φ1120585 minus Φ2120585100381710038171003817100381710038172

119867119904minus1] le 119862 [

1003817100381710038171003817V1 minus V210038171003817100381710038172

119867119904minus1

sdot10038171003817100381710038171003817Φ1120585100381710038171003817100381710038172

119867119904minus2+1003817100381710038171003817V210038171003817100381710038172

119867119904minus1

10038171003817100381710038171003817Φ1120585 minus Φ2120585100381710038171003817100381710038172

119904minus2

+1003817100381710038171003817V1

minus V2

10038171003817100381710038172

119867119904minus1 +1003817100381710038171003817Φ1 + Φ2

10038171003817100381710038172

119867119904

1003817100381710038171003817Φ1 minus Φ210038171003817100381710038172

119867119904] le 119862 [(1

+1003817100381710038171003817Φ110038171003817100381710038172

119867119904)1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1 + (

1003817100381710038171003817V210038171003817100381710038172

119867119904minus1

+1003817100381710038171003817Φ1 + Φ2

10038171003817100381710038172

119867119904)1003817100381710038171003817Φ1 minus Φ2

10038171003817100381710038172

119867119904]

(63)

43 Local Existence and Uniqueness

Theorem 3 Let 119904 gt 32 and 120601 = (V0 Φ0)119879 isin 119867119904minus1 times 119867119904

Then there exists 119879(119904 120601) gt 0 and unique 119880 = (V Φ)119879 isin119862([0 119879]119867119904minus1 times 119867119904) which satisfies the integral equation

119880 (119905) = 119878 (119905) 120601 + int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840 (64)

Proof Let 119872119879 gt 0 be fixed constants and consider thenonlinear operator

(Ψ119880) (119905) = 119878 (119905) 120601 + int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840 (65)

defined in the complete metric space

X119872(119879) = 119880 isin 119862 ([0 119879] 119867

119904minus1 times 119867119904)

sup119905isin[0119879]

1003817100381710038171003817119880 (119905) minus 119878 (119905) 1206011003817100381710038171003817119867119904minus1times119867119904 le119872

(66)

Let us prove that if 119880 isin X119872(119879) then Ψ119880 isin 119862([0 119879]119867119904minus1 times

119867119904) Indeed if 119880 isin X119872(119879) then

(Ψ119880) (119905) minus (Ψ119880) (120591)119867119904minus1times119867119904

le1003817100381710038171003817(119878 (119905) minus 119878 (120591)) 120601

1003817100381710038171003817119867119904minus1times119867119904

+10038171003817100381710038171003817100381710038171003817int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840

minus 119878 (120591 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840

10038171003817100381710038171003817100381710038171003817119867119904minus1times119867119904

(67)

We remark that if 119880 = (V Φ)119879 isin X119872(119879) then

119880 (119905)119867119904minus1times119867119904 le1003817100381710038171003817119880 (119905) minus 119878 (119905) 120601

1003817100381710038171003817119867119904minus1times119867119904

+1003817100381710038171003817119878 (119905) 120601

1003817100381710038171003817119867119904minus1times119867119904

le 119872 +10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904

(68)


Given that (119878(119905))119905ge0

is a 1198620-semigroup then (119878(119905) minus

119878(120591))120601119867119904minus1times119867119904 rarr 0 as 120591 rarr 119905 To analyze the second

expression let us suppose that 120591 gt 119905 gt 0 We obtain that

10038171003817100381710038171003817100381710038171003817int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840 minus int

120591

0

119878 (120591 minus 1199051015840)G (119880)

sdot (1199051015840) 119889119905101584010038171003817100381710038171003817100381710038171003817119867119904minus1times119867119904

le int119905

0

10038171003817100381710038171003817(119878 (119905 minus 1199051015840) minus 119878 (120591 minus 1199051015840))

sdotG (119880) (1199051015840)10038171003817100381710038171003817119867119904minus1times119867119904 119889119905

1015840 + int120591

119905

10038171003817100381710038171003817119878 (120591 minus 1199051015840)G (119880)

sdot (1199051015840)10038171003817100381710038171003817119867119904minus1times119867119904 119889119905

1015840 = 1198681+ 1198682

(69)

Observe that since 119886119894119895(119910 120591minus1199051015840) rarr 119886

119894119895(119910 119905minus1199051015840) as 120591 rarr 119905 119894 119895 =

1 2 and using Lebesguersquos dominated convergence theoremwe get that 119868

1rarr 0 as 120591 rarr 119905

On the other hand due toTheorems 1 and 2 we arrive at

1198682= int

120591

119905

10038171003817100381710038171003817119878 (120591 minus 1199051015840)G (119880) (119905

1015840)10038171003817100381710038171003817119867119904minus1times119867119904 119889119905

1015840

le 119862int120591

119905

10038171003817100381710038171003817G (119880) (1199051015840)10038171003817100381710038171003817119867119904minus1times119867119904 119889119905

1015840

le 119862int120591

119905

119880119867119904minus1times119867119904 1198891199051015840

le 119862 (120591 minus 119905) (119872 +10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904)

(70)

Therefore we get that 1198682rarr 0 as 120591 rarr 119905 This means that

Ψ119880 isin 119862([0 119879]119867119904minus1 times 119867119904)Let us prove existence of small time 119879 gt 0 such that

Ψ(X119872(119879)) sub X

119872(119879) Let 119880 = (V Φ)119879 isin X

119872(119879) For any

119905 isin [0 119879]

1003817100381710038171003817(Ψ119880) (119905) minus 119878 (119905) 1206011003817100381710038171003817119867119904minus1times119867119904

le int119905

0

10038171003817100381710038171003817119878 (119905 minus 1199051015840)G (119880) (119905

1015840)10038171003817100381710038171003817119867119904minus1times119867119904

le 119862int119905

0

10038171003817100381710038171003817G (119880) (1199051015840)10038171003817100381710038171003817119867119904minus1times119867119904

le 119862119879 sup119905isin[0119879]

119880 (119905)119867119904minus1times119867119904 le 119862119879 (119872 +

10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904)

le 119872

(71)

for 119879 being small enough Therefore Ψ(119880) isin X119872(119879)

Finally let us show that time isin (0 119879] exists so that theoperator Ψ is a contraction on X

119872() Let 119880

1= (V

1 Φ1)119879

1198802= (V

2 Φ2)119879 isin X

119872(119879) Then usingTheorem 2 we arrive at

1003817100381710038171003817(Ψ1198801) (119905) minus (Ψ1198802) (119905)1003817100381710038171003817119867119904minus1times119867119904 le int

119905

0

10038171003817100381710038171003817119878 (119905 minus 1199051015840)

sdot (G (1198801) (1199051015840) minusG (119880

2) (1199051015840))

10038171003817100381710038171003817119867119904minus1times119867119904 1198891199051015840

le 119862int119905

0

1003817100381710038171003817G (1198801) minusG (1198802)1003817100381710038171003817119867119904minus1times119867119904 119889119905

1015840 le int119905

0

(1

+1003817100381710038171003817Φ11003817100381710038171003817119867119904)1003817100381710038171003817V1 minus V2

1003817100381710038171003817119867119904minus1 + (1003817100381710038171003817V21003817100381710038171003817119867119904minus1

+1003817100381710038171003817Φ1 + Φ2

1003817100381710038171003817119867119904)1003817100381710038171003817Φ1 minus Φ2

1003817100381710038171003817119867119904 1198891199051015840 le 119862119879 [(1 +119872

+10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904)

1003817100381710038171003817V1 minus V21003817100381710038171003817119867119904minus1 + (3119872 + 3

10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904)

sdot1003817100381710038171003817Φ1 minus Φ2

1003817100381710038171003817119867119904] le 119862119879 [3119872 + 310038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904]

sdot sup119905isin[0119879]

10038171003817100381710038171198801 minus 11988021003817100381710038171003817119867119904minus1times119867119904

(72)

We see that it is possible to select time isin (0 119879] such that

119862 [3119872 + 310038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904] lt 119872 (73)

so that the operator Ψ is a contraction on the closed ballX119872() sub 119862([0 ]119867119904minus1 times 119867119904) The fixed point principle

guarantees the existence of unique solution

119880 isin 119862 ([0 ] 119867119904minus1 times 119867119904) (74)

of integral equation (64) Finally uniqueness of this solutionin themetric space119862([0 ]119867119904minus1times119867119904) can be established viaGronwallrsquos lemma

5 Numerical Schemes

In this section we describe the numerical scheme we proposeto approximate solutions of system (38)-(39) In the firstplace it is convenient to rewrite it as

(119868 minus120573

21205972120585) V119905=2120573

31205973120585119906 minus 120597

120585((1 +

120572V1198722) 119906)

(119868 minus120573

21205972120585)119906

119905

= 1205731205901205973120585V + 120573120590(minus

31198721015840

1198724V120585+ 3(1198721015840)

2

1198725V minus11987210158401015840

1198724V)

minus 120597120585(1

119872V) minus

120572

2120597120585(1

11987221199062)

(75)

subject to the initial conditions V(120585 0) = V0(120585) 119906(120585 0) =

1199060(120585) where we introduced the new variable 119906 = Φ

120585 In

the numerical solver to be introduced the computationaldomain [0 119871] is discretized by 119873 equidistant points withspacing Δ120585 = 119871119873 and the unknowns V and 119906 are expanded


as truncated Fourier series in space with time-dependentcoefficients

V (120585 119905) = sum119895

V119895(119905) 119890

119894119908119895120585

119906 (120585 119905) = sum119895

119895(119905) 119890

119894119908119895120585(76)

with

119908119895=2120587119895

119871 119895 = minus

119873

2+ 1 0

119873

2 (77)

The time-dependent coefficients V119895(119905) 119895 = minus1198732 +

1 0 1198732 are calculated by means of the equation

V119895(119905) =

1

119871int119871

0

V (120585 119905) 119890minus119894119908119895120585119889120585 (78)

and analogously for 119895(119905) Substituting these expressions into

(75) and projecting the resulting equations with respect to the1198712-orthonormal basis 120601

119895= 119871minus12119890119894119908119895120585 and the inner product

⟨119891 119892⟩ = int119871

0

119891 (120585) 119892 (120585) 119889120585 (79)

it follows that

V1015840119895(119905) = 120574

1119895+ 119873

1

1015840119895(119905) = 120574

2V119895+ 119873

2

(80)

where

1205741=minus (21205733) 1198941199083

119895

1 + (1205732)1199082119895

1205742=minus1205731205901198941199083

119895

1 + (1205732)1199082119895

1198731= minus119894119908119895119875119895[(1 + 120572V1198722) 119906]1 + (1205732)1199082

119895

1198732=119894119908119895119875119895[120573120590 (minus (311987210158401198724) V

120585+ (3 (1198721015840)

2

1198725) V minus (119872101584010158401198724) V) minus (1119872) V minus (12057221198722) 1199062]

1 + (1205732)1199082119895

(81)

and 119875119895[sdot] denotes the operator

119875119895[119892] =

1

119871int119871

0

119892 (120585) 119890minus119894119908119895120585119889120585 (82)

When the period 119871 is taken large enough this numericalscheme can be applied to approximate the rapidly decayingsolutions of system (75) on the entire real line R Thistechnique was used successfully in [9]

51 Temporal Discretization Note that (80) can be seenas a system of ordinary differential equations where theunknowns are the time-dependent Fourier coefficients ofthe solutions To solve it we use an implicit-explicit scheme(IMEX) in the form

V119899+1119895minus V119899

119895

Δ119905=1205741(119899+1119895+ 119899

119895)

2+3

21198731198991minus1

2119873119899minus11

119899+1119895minus 119899

119895

Δ119905=1205742(V119899+1119895+ V119899

119895)

2+3

21198731198992minus1

2119873119899minus12

(83)

Here V119899 119906119899 denote the approximations of the unknownsV(120585 119905) 119906(120585 119905) respectively at time 119905 = 119899Δ119905 where Δ119905 is thetime step of the method and119873119899

1 1198731198992denote the approxima-

tions of the functions1198731 1198732evaluated at time 119899Δ119905 Similarly

V119899119895 119899119895denote the approximations to the Fourier transforms of

the functions V and 119906 respectively with respect to the variable120585 evaluated at time 119899Δ119905The numerical approach adopted forsolving system (75) ensures the scheme results to be linearlyunconditionally stable which can be easily verified Furtherobserve that the linear dispersive terms are approximated byusing an implicit strategy in contrast to the nonlinear termsand terms where the variable coefficient119872 is present whichare treated in explicit form Implicit-explicit schemes (IMEX)were already applied in [25] for scalar dispersive evolutionequations The main advantage of the numerical schemedescribed here is that at each time step we can solve explicitlythe approximations V119899+1

119895 119899+1119895

119895 = minus1198732 + 1 1198732 ofthe Fourier coefficients of the unknowns V(120585 119905) and 119906(120585 119905)from (83) without using implicit Newton-type iterationsThus the scheme results in being cheap and its computerimplementation is easier


In the scheme the spatial derivatives 119906120585and V

120585are

computed by exact differentiation of the truncated Fourierseries For instance

V120585(120585 119905) = sum

119895

119894119908119895119875119895[V (sdot 119905)] 119890119894119908119895120585 (84)

The numerical calculations presented in this paper werecarried out in double precision by usingMATLAB R2012b ona Mac platform The Fourier-type integral appearing in theoperator 119875

119895[sdot] (see (82)) is approximated through the well-

known Fast Fourier Transform (FFT) routine

52 Approximating Travelling Wave Solutions In this sectionwe are interested in computing solutions of system (75) overa channel with flat bottom (ie when the metric coefficient119872 is equivalent to 1) of the form

V (120585 119905) = V (120585 minus 119888119905)

119906 (120585 119905) = (120585 minus 119888119905) (85)

which are named as travelling wave solutions The parameter119888 is called the wave velocity

After dropping the tildes travelling wave solution (V 119906)with speed 119888 of system (75) must satisfy the followingequations

(V10158401015840

11990610158401015840)

=1

(1205734) 1198882 minus 21205731205903(

119888

2

2

3

120590119888

2

)(119888V minus ((1 + 120572V) 119906)

119888119906 minus V minus120572

21199062)

(86)

In first place we are interested in finding approximations toeven periodic travelling wave solutions (V 119906) with period 2119897119897 gt 0 of system (75) Thus let us introduce truncated cosineexpansions for V and 119906

V (119909) asymp V0+1198732

sum119899=1

V119899cos(119899120587

119897119909)

119906 (119909) asymp 1199060+1198732

sum119899=1

119906119899cos(119899120587

119897119909)

(87)

where

V0=1

119897int119897

0

V (119909) 119889119909 =1

2119897int2119897

0

V (119909) 119889119909

V119899=2

119897int119897

0

V (119909) cos(119899120587119909119897) 119889119909

=2

2119897int2119897

0

V (119909) cos(119899120587119909119897) 119889119909

(88)

and analogous expressions for 119906 By substituting expressions(87) into (86) evaluating them at the 1198732 + 1 collocationpoints

119909119895=2119897 (119895 minus 1)

119873 119895 = 1

119873

2+ 1 (89)

we obtain a system of119873 + 2 nonlinear equations in the form

119865 (V0 V1 V

1198732 1199060 1199061 119906

1198732) = 0 (90)

where the 119873 + 2 coefficients V119899 119906119899are the unknowns

Nonlinear system (90) can be solved by Newtonrsquos iterationComputation of the cosine series in (87) and the integralsin (88) is performed using the FFT (Fast Fourier Transform)algorithmThe Jacobian of the vector field119865 R119873+2 rarr R119873+2

is approximated by the second-order accurate formula

119869119894119895119865 (119909) asymp

119865119894(119909 + ℎ119890

119895) minus 119865

119894(119909 minus ℎ119890

119895)

2ℎ

119895 = 1 119873 + 2

(91)

where 119890119895= (0 1 0) and ℎ = 001 We stop Newtonrsquos

iteration when the relative error between two successiveapproximations and the value of the vector field 119865 are smallerthan 10minus12

The starting point for Newtonrsquos procedure in the periodicframe is taken as

1199060(119909) = cos(120587119909

119897)

V0(119909) = cos(120587119909

119897)

(92)

In second place we are also interested in approximatingsolitary wave solutions of system (75) that is when thefunctions V 119906 and their derivatives decay to zero at plusmninfin Inthis case we can also approximate V 119906 by a truncated Fourierseries as in (87) using a large enough length 119871 To derive anappropriate initial point for Newtonrsquos iteration we computean approximate solitary wave solution of system (75) with119872 equiv 1 and 120572 120573 120590 being small To accomplish this we willuse original system (38)-(39) in the variables (V Φ) Let usobserve that from (39)

V = minus(119868 minus120573

21205972120585)Φ

119905+ 120573120590V

120585120585minus120572

2Φ2120585

= minusΦ119905+ 119874 (120572 120573 120590)

(93)

Substituting this into (38) and neglecting second-order termsin 120572 120573 120590 we derive the Benney-Luke type equation for thepotentialΦ

minus Φ119905119905+ Φ

120585120585+ 120573 (1 minus 120590)Φ

120585120585119905119905minus2120573

3Φ120585120585120585120585minus 2120572Φ

120585Φ120585119905

minus 120572Φ119905Φ120585120585= 0

(94)


We look for a solitary wave solution of (94) in the formΦ (120585 119905) = Φ (120585 minus 119888119905) (95)

Therefore abandoning the tildes the functionΦmust satisfythe following equation

(1 minus 1198882)Φ10158401015840 + (120573 (1 minus 120590) 1198882 minus2120573

3)Φ1015840101584010158401015840

+3120572119888

2(Φ10158402)

1015840

= 0

(96)

Integrating the equation above and using the fact that Φ andtheir derivatives decay to zero at plusmninfin we arrive at

(1 minus 1198882)Φ1015840 + 120573((1 minus 120590) 1198882 minus2

3)Φ101584010158401015840 +

3120572119888

2(Φ1015840)

2

= 0

(97)

Multiplying the previous equation by 2Φ10158401015840 and integratingagain we obtain that

(1 minus 1198882) (Φ1015840)2

+ 120573((1 minus 120590) 1198882 minus2

3) (Φ10158401015840)

2

+ 120572119888 (Φ1015840)3

= 0

(98)

Furthermore letting 119906 = Φ1015840 we find from (93)

V = 119888119906 minus120573119888

211990610158401015840 + 12057312059011988811990610158401015840 minus

120572

21199062 (99)

Assume the solution form ofΦ1015840 to be119906 = Φ1015840 (119909) = 119860 sech2 (119861119909) (100)

where119860 119861 are constantsThen by replacing this into (98) weget that the constants 119860 119861 are given by

119860 =1198882 minus 1

120572119888

119861 = radic1198882 minus 1

4120573 ((1 minus 120590) 1198882 minus 23)

(101)

Now substituting into (99) we arrive at the following approx-imation of the wave elevation

V = 119888119860 sech2 (119861119909) minus 12057221198602sech4 (119861119909) + 120573119888 (120590 minus 1

2)

sdot [41198601198612sech2 (119861119909) tanh2 (119861119909)

minus 21198601198612sech4 (119861119909)]

(102)

6 Description of the Numerical Experiments

In this section we compute some approximations to solutionsof system (75) using the numerical schemes described in theprevious section In first place some travelling wave solutionsare approximated by using the numerical scheme discussedin Section 52 and then these solutions are checked throughcomputer simulations conducted with numerical solver (83)Finally we apply it to compute the evolution of a gaussianpulse over a channel with variable depth taking into accountthe surface tension effect

61 Periodic TravellingWave Solutions Using Newtonrsquos itera-tion as explained in Section 52 we develop some numericalsimulations to compute periodic travelling wave solutions ofsystem (75)The results are shown in Figures 2 3 4 and 5 fordifferent values of the modelling parameters 120572 120573 120590 and wavespeed 119888 The initial step for Newtonrsquos scheme is taken as in(92) The number of FFT point is 27 and we recall that herethe channel has constant depth that is119872 equiv 1 In Figure 6 wecheck the accuracy of the periodic travelling wave solutionscomputed by comparing the prediction of numerical scheme(83) at time 119905 = 10 (using 119873 = 210 and Δ119905 = 001) with theprofile in Figure 2 shifted at a distance of 119888119905 = 10 times 12 = 12to the right Observe that the profiles coincide with goodaccuracy and they propagate with the expected wave velocityThe difference between the two profiles is of order 10minus3Similar results were obtained for the other periodic travellingwave solutions shown in Figures 3 4 and 5

62 Solitary Wave Solutions Numerical experiments in thecase of solitary wave solutions are shown in Figures 7 8and 9 for different values of the modelling parameters 120572 120573 120590and wave speed 119888 The initial step for Newtonrsquos scheme istaken as in (100)-(102) centered at the position 120585 = 50The number of FFT point is 210 and we recall that herethe channel has constant depth that is 119872 equiv 1 In thisnonperiodic scenario the spatial computational domain isthe interval [0 100] which is large enough so that the pulsesdo not reach the computational boundaries within the timeinterval modeled In Figure 10 we check the accuracy of theperiodic travelling wave solutions computed by comparingthe prediction of the numerical scheme (83) at time 119905 = 10(using 119873 = 210 and Δ119905 = 001) with the profile in Figure 2shifted at a distance of 119888119905 = 10 times 11 = 11 to the right Asin the experiments in the previous section we see that theprofiles coincide with good accuracy and they propagate withthe expected wave velocity The difference between the twoprofiles is of order 10minus3 Similar results were obtained for theother periodic travelling wave solutions shown in Figures 8and 9

63 Variable Depth Finally in this section we include somenumerical simulations to illustrate the dynamics of incomingGaussian pulses in the form

119906 (120585 0) = V (120585 0) = 119890minus20(119909minus28)2

(103)

governed by (75) under the simultaneous effects of surfacetension (120590) dispersion (120573) and channelrsquos topography (coef-ficient 119872(120585)) We further remark that the incoming pulsesare located at two units to the left of the irregular part of thechannel which covers the interval [30 40] In order to focuson these three phenomena we only consider the linear casethat is 120572 = 0

In Figure 11 is displayed the output of numerical scheme(83) for the value of the dispersion parameter 120573 = 001 andwithout surface tension effect that is 120590 = 0 for a constantdepth channel The numerical parameters are 119873 = 211 andΔ119905 = 001 and the computational domain is the interval


v u

2 4 6 8 100x

0

02

04

06

08

1

12

14

2 4 6 8 100x

0

05

1

15

Figure 2 Periodic travelling wave solution of system (75) for 120572 = 120573 = 03 120590 = 0 wave speed 119888 = 12 and period 119871 = 10 obtained after 7Newtonrsquos iterations

v u

minus1

minus05

0

05

2 4 6 8 100x

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

02

2 4 6 8 100x


[0 60] which again is large enough so that the pulses do notreach the computational boundaries We point out that asa consequence of dispersion (which produces a differencebetween the phase velocity of the Fourier modes containedin the pulse) the incident pulses progressively develop anoscillatory tail that propagates to the left while the leadingwave propagates to the right We contrast this experimentto the dynamics of a pulse under the influence of surfacetensionwhich is presented in Figure 12 Note that in this caseboth the oscillatory tail and the wave front propagate to theright side of the computational domainWe conclude that thesurface tension effect alters the dispersive characteristics ofmodel (75)

We can anticipate the influence of the additional third-order term 1205731205901205972

120585V (due to the surface tension effects) on the

solutions of system (75) (with 120572 = 0 and flat channel119872 equiv 1)by analyzing its linear dispersion relation given by

1198622120590=1199082

1198962

=1 + (21205733) 1198962

(1 + (1205732) 1198962)2+1205731205901198962 (1 + (21205733) 1198962)

(1 + (1205732) 1198962)2

gt 11986220

(104)

We recall that this expression arises by studying the propaga-tion of solutions of system (75) in the form 119906 = 119906

0119890119894(119896120585minus119908119905)

V = V0119890119894(119896120585minus119908119905) where 119906

0 V0are constants


v u

minus02

0

02

04

06

08

1

12

14

16

2 4 6 8 100

x

0

02

04

06

08

1

12

14

2 4 6 8 100

x


v

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

2 4 6 8 100

x

u

minus25

minus2

minus15

minus1

minus05

0

2 4 6 8 100

x


In Figure 13 the phase velocities 119862120590(with 120590 = 06)

and 1198620(no surface tension) are compared as functions of

radic120573119896 (119896 is called the wavenumber) Observe that in thepresence of surface tension the phase speed 119862

120590is larger

than 1 and it approximates to the value 81205903 as radic120573119896 tendsto infinity In contrast without surface tension effects thephase speed 119862

0is lower than 1 and Fourier components

with high wavenumber travel more slowly than the Fouriermodes with small wavenumber These observations are inaccordance with the results presented in Figures 11 and 12

We further point out that the fact that the phase velocity insystem (75) is bounded by 81205903 for allradic120573119896 has an advantagein improving the stability of numerical solvers formulatedto approximate solutions of this system In particular theproperties of numerical solver (83) employed in the presentpaper such as the convergence stability and error estimateswill be studied in a future research

After analyzing the flat bottom case we consider a numer-ical simulation in Figure 15 where the channelrsquos topographyis variable The topography considered is shown in Figure 14


at t = 10

0

05

1

15u at t = 10

0

02

04

06

08

1

12

14

2 4 6 8 100

120585

2 4 6 8 100

120585

Figure 6 Evolution of a periodic travelling wave solution of system (75) for 120572 = 120573 = 03 120590 = 0 wave speed 119888 = 12 and period 119871 = 10 usingthe numerical scheme (83)

v

minus01

0

01

03

02

04

05

06

07

45 50 55 6040

x

u

45 50 55 6040

x

minus01

0

01

03

02

04

05

06

07

Figure 7 Solitary wave solution of system (75) for 120572 = 120573 = 03 120590 = 03 and wave speed 119888 = 11 obtained after 5 Newtonrsquos iterations

together with the metric coefficient119872(120585 120573) computed usingthe Schwarz-Christoffel Toolbox designed for MATLAB byDriscoll [30] Note that the irregular part covers the interval[30 40] (10 units long) A nonnegligible value of surfacetension 120590 = 06 is incorporated into model equations(75) Again the level of dispersion is taken as 120573 = 001The numerical parameters are the same as in the previouscomputer simulation In addition to the effect of the surfacetension on the direction of propagation of the oscillatorytail we now observe irregular fluctuations (coda) developingbehind the wave front and propagating to the left end ofthe computational domain This tail signal is due to themultiple reflections of the incoming pulses produced by every

variation of the channelrsquos depthThese phenomena have beenstudied in previous works using other water wave modelsthat neglect the surface tension effect [9ndash11] In the presentpaper our purpose was to initiate the study of these wavephenomena and numerical scheme (83) and system (75) haveproved to be valuable tools in developing this research in afuture work

To see in more detail the effect of surface tension onthe wave-topography interaction in Figure 16 we show theevolution of the Gaussian pulse

119906 (120585 0) = V (120585 0) = 119890minus20(119909minus198)2 (105)


v

0

005

01

015

02

025

03

035

04

045

45 50 55 6040

x

u

0

005

01

015

02

025

03

035

04

45 50 55 6040

x


v

minus05

0

05

1

15

2

25

45 50 55 6040

x

u

minus05

0

05

1

15

2

45 50 55 6040

x


on the surface of a channel with a large irregular bottomcovering the interval [200 300] (10 times larger than in theprevious numerical simulation) in the case of 120590 = 0 (nosurface tension) and 120590 = 06 As in the previous experimentsthe physical depth is described by a linear piecewise functionand the coefficient119872(120585 120573) is computed again using Driscollrsquospackage for MATLAB The numerical parameters for thiscomputer simulation are 119873 = 214 Δ119905 = 00025 120572 =0 120573 = 001 and the computational domain is the interval[0 400] Observe that the pulse without the presence ofsurface tension appears to be delayedwith respect to the pulsepropagating under the influence of a nonnegligible value ofsurface tension 120590 = 06 and the long channelrsquos topographyintroducedThese phenomena and further research using full

model equations (35) for the case of a rapidly varying bottom(ie 120574 ≪ 1) will be studied in detail in a future work Weremark that numerical scheme (83) could be adapted to solvefull system (35)

7 Conclusions

In this paper we introduced a new model for describingthe propagation of a small amplitude water wave over thesurface of a shallow channel with a possibly varying depthIt was derived formally as an asymptotical approximation ofthe full potential theory equations in the weakly-nonlinearweakly-dispersive regime Existence and uniqueness of asolution to the initial value problem associated with (5) were


at t = 10 u at t = 10

minus01

0

01

02

03

04

05

06

07

minus01

0

01

02

03

04

05

06

07

60 65 7055

120585

60 65 7055

120585

Figure 10 Evolution of a solitary wave solution of system (75) for 120572 = 120573 = 03 120590 = 03 and wave speed 119888 = 11 using numerical scheme(83)

u at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

20 30 40 5010

120585

20 30 40 5010

120585

Figure 11 Evolution of a Gaussian pulse over a flat channel without surface tension effect Parameters 120572 = 0 120573 = 001 and 120590 = 0

also studied using Banachrsquos fixed point argument and theclassical semigroup theory in appropriate Sobolev spacesThemain features of model (5) considered in this work are thefollowing

(i) It can be applied to study wave propagation over ashallow channel with a discontinuous or nondifferen-tiable bottom provided that the bottomrsquos fluctuationssatisfy |119899| lt 1 Observe that in the derivation ofsystem (5) the function 119899was not assumed to be small

nor continuous since the conformal mapping has asmoothing effect on the topography Note that theeffective variable coefficient119872(120585) that appears in finalsystem (5) is defined as a convolution between thefunction describing the bottomrsquos fluctuations 119899 andthe sech2 function (see (25)) This fact implies that119872(120585) and all variable coefficients in system (5) areinfinitely differentiable even when the function 119899 isdiscontinuous or nondifferentiable In Figure 14 the


minus04

minus03

minus02

minus01

0

01

02

03

04

05u at t = 20

35 40 45 50 55 6030120585

v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 55 6030120585

Figure 12 Evolution of a Gaussian pulse over a flat channel with surface tension effect Parameters 120572 = 0 120573 = 001 and 120590 = 06

0 5 10 15 20 25 30 35 40 45 50

C2120590

C20

81205903

radic120573k

0

02

04

06

08

1

12

14

16

Figure 13 The phase velocity for system (75)

coefficient119872(120585) is displayed for a (nondifferentiable)piecewise linear topography We see that the coeffi-cient 119872(120585) is a regularized version of the physicaltopography


(iii) Furthermore in the derivation of system (5) wedo not eliminate the wave elevation 120578 (which pre-vents us from neglecting additional terms of order

05060708091

1112131415

30 32 34 36 38 40 4228

x

Figure 14 The metric coefficient 119872(120585 120573 = 001) (in solid line)comparedwith a (nondifferentiable) piecewise linear topography (indashed line) Observe that the coefficient119872 is smooth even thoughthe physical topography is not differentiable The irregular part ofthe bottom is located over the interval [30 40] and the channelrsquosbottom is flat outside this interval

119874(1205722 1205732)) as done for example in themodel derivedby Milewski [23] This means that we did not reduce(30)-(31) to only one equation for the potentialvelocity Φ where 120578 does not appear Therefore newformulation (5) is expected to be a more accurateapproximation of the original potential theory equa-tions

In second place we introduced a numerical Galerkin-spectral discretization of second order in time and spectralaccuracy in space to approximate solutions of system (5) anda Fourier-collocation strategy combined with a Newton-typeiterative procedure to compute its travelling wave solutionsIn all computer simulations these solutions were observedto propagate approximately without changing their shapewithin a large time interval


u at t = 20

35 40 45 50 5530120585

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 5530120585

Figure 15 Evolution of a Gaussian pulse over a channel with the variable depth shown in Figure 14 and taking into account surface tensioneffect Parameters 120572 = 0 120573 = 001 and 120590 = 06

v at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

290 300 310 320280

120585

u at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

280 300 310 320290

120585

Figure 16 Propagation of a Gaussian pulse over a channel with variable depth Observe the effect of surface tension on the dynamics of thepulse Parameters 120572 = 0 120573 = 001 Solid line 120590 = 06 Dashed line 120590 = 0

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This researchwas supported byUniversidad del Valle Calle 13No 100-00 Cali Colombia under the Research Project CI988

References

[1] G B Whitham Linear and Nonlinear Waves John Wiley ampSons New York NY USA 1974

[2] J Hamilton ldquoDifferential equations for long-period gravitywaves on fluid of rapidly varying depthrdquo Journal of FluidMechanics vol 83 no 2 pp 289ndash310 1977

[3] A Nachbin ldquoA terrain-following Boussinesq systemrdquoThe SIAMJournal on Applied Mathematics vol 63 no 3 pp 905ndash9222003


[4] J R Quintero and A M Montes ldquoExistence physical senseand analyticity of solitons for a 2D Boussinesq-Benney-Lukesystemrdquo Dynamics of Partial Differential Equations vol 10 no4 pp 313ndash342 2013

[5] O Nwogu ldquoAlternative form of Boussinesq equations fornearshorewave propagationrdquo Journal ofWaterway Port Coastalamp Ocean Engineering vol 119 no 6 pp 618ndash638 1993

[6] P L-F Liu S B Yoon and J T Kirby ldquoNonlinear refraction-diffraction of waves in shallow waterrdquo Journal of Fluid Mechan-ics vol 153 pp 185ndash201 1985

[7] P A Madsen R Murray and O R Soslashrensen ldquoA new formof the Boussinesq equations with improved linear dispersioncharacteristicsrdquo Coastal Engineering vol 15 no 4 pp 371ndash3881991

[8] P A Madsen and H A Schaffer ldquoHigher-order Boussinesq-type equations for surface gravity waves derivation and anal-ysisrdquo Philosophical Transactions of the Royal Society A Mathe-matical Physical and Engineering Sciences vol 356 no 1749 pp3123ndash3184 1998

[9] J Garnier J CM Grajales and A Nachbin ldquoEffective behaviorof solitary waves over random topographyrdquoMultiscaleModelingand Simulation vol 6 no 3 pp 995ndash1025 2007

[10] J-P Fouque J Garnier and A Nachbin ldquoTime reversal fordispersive waves in random mediardquo SIAM Journal on AppliedMathematics vol 64 no 5 pp 1810ndash1838 2004

[11] J-P Fouque J Garnier and A Nachbin ldquoShock structure dueto stochastic forcing and the time reversal of nonlinear wavesrdquoPhysica D Nonlinear Phenomena vol 195 no 3-4 pp 324ndash3462004

[12] CCMei ldquoResonant reection of surfacewaterwaves by periodicsandbarsrdquo Journal of FluidMechanics vol 152 pp 315ndash335 1985

[13] C C Mei and Y Li ldquoEvolution of solitons over a randomlyrough seabedrdquo Physical Review E vol 70 no 1 Article ID016302 2004

[14] A Nachbin and K Soslashlna ldquoApparent diffusion due to topo-graphic microstructure in shallow watersrdquo Physics of Fluids vol15 no 1 pp 66ndash77 2003

[15] R Snieder ldquoThe influence of topography on the propagationand scattering of surface wavesrdquo Physics of the Earth andPlanetary Interiors vol 44 no 3 pp 226ndash241 1986

[16] P L-F Liu ldquoWave-current interactions on a slowly varyingtopographyrdquo Journal of Geophysical Research vol 88 no 7 pp4421ndash4426 1983

[17] Y-Z Liu and J Z Shi ldquoA theoretical formulation for wavepropagations over uneven bottomrdquo Ocean Engineering vol 35no 3-4 pp 426ndash432 2008

[18] G Gottwald and R Grimshaw ldquoThe effect of topography onthe dynamics of interacting solitary waves in the context ofatmospheric blockingrdquo Journal of the Atmospheric Sciences vol56 no 21 pp 3663ndash3678 1999

[19] C C MeiThe Applied Dynamics of Ocean Surface Waves JohnWiley amp Sons New York NY USA 1983

[20] P G Baines Topographic Effects in Stratified Flows CambridgeUniversity Press Cambridge UK 1995

[21] European Centre for Medium-Range Weather Forecasts(ECMWF) Proceedings of a Workshop Held at ECMWF onOrography 10 to 12 November 1997 European Centre forMedium-Range Weather Forecasts (ECMWF) Reading UK1998

[22] H Peregrine ldquoLong waves on a beachrdquo Journal of FluidMechanics vol 27 pp 815ndash827 1967

[23] P A Milewski ldquoA formulation for water waves over topogra-phyrdquo Studies in Applied Mathematics vol 100 no 1 pp 95ndash1061998

[24] M Chen ldquoEquations for bidirectional waves over an unevenbottomrdquo Mathematics and Computers in Simulation vol 62no 1-2 pp 3ndash9 2003 Proceedings of the Nonlinear WavesComputation andTheory II

[25] U M Asher S J Ruuth and B T R Wetton ldquoImplicit-explicitmethods for time-dependent partial differential equationsrdquoSIAM Journal on Numerical Analysis vol 32 no 3 pp 797ndash8231995

[26] J Kim and P Moin ldquoApplication of a fractional-step method toincompressible Navier-Stokes equationsrdquo Journal of Computa-tional Physics vol 59 no 2 pp 308ndash323 1985

[27] C Canuto M Y Hussaini and A Quarteroni Spectral Methodsin Fluid Dynamics Series in Computational Physics SpringerBerlin Germany 1988

[28] G E Karniadakis M Israeli and S A Orszag ldquoHigh-ordersplitting methods for the incompressible Navier-Stokes equa-tionsrdquo Journal of Computational Physics vol 97 no 2 pp 414ndash443 1991

[29] J G Verwer J G Blom and W Hundsdorfer ldquoAn implicit-explicit approach for atmospheric transport-chemistry prob-lemsrdquo Applied Numerical Mathematics vol 20 no 1-2 pp 191ndash209 1996

[30] T Driscoll ldquoSchwarz-Christoffel Toolbox for MATLABrdquo httpwwwmathudeledusimdriscollSC

[31] T Tao ldquoMultilinear weighted convolution of 1198712 functionsand applications to nonlinear dispersive equationsrdquo AmericanJournal of Mathematics vol 123 no 5 pp 839ndash908 2001

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Algebra

Discrete Dynamics in Nature and Society



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Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


= 120573120590(V120585120585minus 31198721015840

1198724V120585+ 3(1198721015840)

2

1198725V minus11987210158401015840

1198724V) minus

1

119872V

minus120572

21198722Φ2120585 V (120585 0) = V

0(120585) Φ (120585 0) = Φ

0(120585)

(5)

The variables 120585 and 119905 are the space and time coordinatesrespectively V = 119872(120585)120578(120585 119905) with 120578(120585 119905) denoting theelevation of the free surface and Φ(120585 119905) represents thepotential fluid velocitymeasured at the channelrsquos bottomThemetric coefficient 119872(120585) is related to the channelrsquos bottom119910 = minus119867(119909120574) and it is defined as

119872(120585) =120587


minusinfin

119867(119909 (1205850 minusradic120573) 120574)

cosh2 ((1205872radic120573) (1205850minus 120585))

1198891205850 (6)

where (119909 119910) rarr (120585 120577) is the coordinate transformation usedin the derivation of system (5) to map the original physicalchannel with variable depth onto a strip in the complexplane This strategy of change of variable was introduced byHamilton [2] and later used successfully by Nachbin in [3] tostudy wave propagation over a channel with a highly variabletopography Observe from (6) that the coefficient 119872(120585) isinfinitely differentiable although the function describing thechannelrsquos bottom is not smooth In case of constant depth thecoefficient119872(120585) is identically one and system (5) reduces toa system derived by Quintero and Montes in [4]

Wave-topography interaction has been the subject ofconsiderable mathematical research [5ndash18] The physicalapplications range from coastal surface waves [19] to atmo-spheric flows over mountain ranges [20 21] In particular theinteraction of waves with fine features of the topography is ofgreat interest As pointed out in the introduction to theOrog-raphy proceedings [21] of the European Centre for Medium-Range Weather Forecasts (ECMWF) ldquothe representation of subgrid-scale orographic processes is recognized as crucialto numerical weather prediction at all time rangesrdquo In theatmospheric literature orography implies mountain ranges[20]

In previous works some weakly-nonlinear weakly-dispersive models have been developed to describe the inter-action of a long pulse with small amplitude that propagateson the surface of a channel with a variable bottom [2 3 5 7ndash9 22ndash24] However these models either neglect the effect ofsurface tension on the free surface where the wave propagatesor are not applicable to bottoms described by a discon-tinuousnondifferentiable function or the wave elevation isremoved in their physical derivation For instance Milewski[23] derived a bidirectional scalar Benney-Luke type equa-tion in terms of the potential velocity which includes thesurface tension effect and the influence of a variable bottomHowever the asymptotical derivation of this model implieseliminating the wave elevation and consequently neglectingseveral second-order terms in the parameters 120572 and 120573

Some of the features of new formulation (5) are thefollowing

(i) It can be applied to study wave propagation over ashallow channel with a discontinuous or nondifferen-tiable bottom provided that the bottomrsquos fluctuationssatisfy |119899| lt 1 This is a consequence of introducingthe conformal mapping (119909 119910) rarr (120585 120577) mentionedabove Observe that all coefficients of the reducedequations (5) in the new coordinate system result inbeing infinitely differentiable since they depend onthe smooth function119872(120585)


(iii) Furthermore in the derivation of system (5) we donot eliminate the wave elevation (which prevents usfrom neglecting additional terms of order 119874(1205722 1205732))as done for example in [23] Therefore the newformulation (5) is expected to be a more accurateapproximation of the full potential theory equations

In the present paper we establish existence and unique-ness of a solution to Cauchy problem (5) using classicalsemigroup theory and Banachrsquos fixed point principle In [4]the well-posedness of system (5) was analyzed but only inthe case that 119872(120585) equiv 1 In second place we formulate aGalerkin-spectral numerical scheme of spectral accuracy inspace to approximate the solutions of system (5) based on theFourier basis and using an implicit-explicit (IMEX) second-order strategy for time stepping This type of temporaldiscretization is described for instance in [25] and it hasbeen used in conjunction with spectral methods [26 27] forthe time integration of spatially discretized PDEs of diffusion-convection type IMEX schemes have also been successfullyapplied to the incompressible Navier-Stokes equations [28]and in environmental modelling studies [29] On the otherhand we develop a numerical solver to compute travellingwave solutions of system (5) by using a Fourier-collocationstrategy combined with a Newton-type iterative procedureWe also indicate how to determine appropriate starting pointsfor this iterative process in order to achieve convergenceExistence results on travelling wave solutions (in the periodicand nonperiodic case) of system (5) with 119872 equiv 1 have alsobeen established in [4] Travelling wave solutions exist as aconsequence of a balance between nonlinear and dispersiveeffects present in a system these waves travel with a constantspeed without any temporal evolution in shape or size whenthe frame of referencemoveswith the same speed of thewaveIn the last decades the study of travelling waves has grownenormously because they appear in several and varied fieldsof application such as fluid mechanics optics acousticsoceanography and astronomy Thus to determine existenceand properties of such type of solutions is a fundamentalproblem in the theory of ordinary and partial differentialequations of great interest for both pure and applied math-ematicians



2 Preliminaries




10038171003817100381710038171198911003817100381710038171003817119871119901 = (int

R

1003816100381610038161003816119891 (119909)1003816100381610038161003816119901

)1119901

(7)



⟨119891 119892⟩ = intR

119891 (119909) 119892 (119909) 119889119909 (8)



F (119891) (119910) = 119891 (119910) = intR

119891 (119909) 119890minus119894119909119910119889119909 119910 isin R (9)


Fminus1 (119891) (119910) = (119910) =

1

2120587intR

119891 (119909) 119890119894119909119910119889119909 119910 isin R (10)


119891 lowast 119892 (119909) = intR

119891 (119909 minus 119910) 119892 (119910) 119889119910 (11)


119878 (R) = 119891 isin 119862infin

(R) 10038171003817100381710038171003817119909


isin Z+ (12)


10038171003817100381710038171198911003817100381710038171003817119867119904 = (int

R

(1 + 1199102)119904 10038161003816100381610038161003816119891 (119910)

100381610038161003816100381610038162

119889119910)12

(13)



⟨119891 119892⟩119904= int

R

(1 + 1199102)119904

119891 (119910) 119892 (119910) 119889119910 (14)


1003817100381710038171003817(120577 119906)1003817100381710038171003817119867119904times119867119904 =

10038171003817100381710038171205771003817100381710038171003817119867119904 + 119906119867119904 (15)


1003817100381710038171003817(120577 119906)1003817100381710038171003817119867119904times119867119904 = (

100381710038171003817100381712057710038171003817100381710038172

119867119904 + 119906

2

119867119904)12

(16)


10038171003817100381710038171198911003817100381710038171003817119862(0119879) = sup

119905isin[0119879]

1003817100381710038171003817119891 (119905 sdot)1003817100381710038171003817119867119904 (17)


1003817100381710038171003817(120577 119906)1003817100381710038171003817119862(0119879)2 =

10038171003817100381710038171205771003817100381710038171003817119862(0119879) + 119906119862(0119879) (18)

for (120577 119906) isin 119862([0 119879] 119867119904(R) times 119867119904(R))



120573120601119909119909+ 120601

119910119910= 0


(19)


120578119905+ 120572120601

119909120578119909minus1

120573120601119910= 0

120578 + 120601119905+120572

2(1206012119909+1

1205731206012119910) minus 120573120590120597

119909(

120578119909

radic1 + 1205731205722 (120578119909)2

)

= 0

(20)


2


minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5




120601119910+120573

1205741198671015840 (

119909

120574)120601

119909= 0 (21)


119867(119909

120574) =

1 + 119899(119909

120574) when 0 lt 119909 lt 119871

1 when 119909 le 0 or 119909 ge 119871(22)







120601119909=1

|119869|[119910120577120601120585minus 119910

120585120601120577]

120601119910=1

|119869|[minus119909

120577120601120585+ 119909

120585120601120577]

1206012119909+ 1206012

119910=1

|119869|(1206012120585+ 1206012

120577

)

120597119909=1

|119869|(119910120577120597120585minus 119910

120585120597120577)

(23)

where

|119869| = 119909120585119910120577minus 119910

120585119909120577= 1199102

120577

+ 1199102120585 (24)


119872(120585) = 119910120577(120585 0)

= 1 +120587


minusinfin

119899 (119909 (1205850 minusradic120573) 120574)

cosh2 (1205872radic120573) (1205850minus 120585)1198891205850

(25)


120573120601120585120585+ 120601

120577120577= 0 at 0 lt 120577 lt 1 + 120572119873 (120585 119905) (26)


|119869|119873119905+ 120572120601

120585119873120585minus1

120573120601120577= 0

120578 + 120601119905+120572

2 |119869|(1206012120585+1

1205731206012120577) minus 120573120590

1

|119869|119872120597

120585(1

119872120578120585)

= 119874 (1205722 120572120573 1205732)

(27)


120601120577= 0 at 120577 = 0 (28)




120601 (120585 120577 119905) =119896

sum119895=0

(minus1)119895 1205731198951205772119895

(2119895)1205972119895

120585Φ (120585 119905) + 119874 (120573

119896+1) (29)


119872(120585) 120578119905+ [(1 +

120572

119872(120585)120578)Φ

120585]120585

minus120573

6Φ120585120585120585120585

= 119874 (1205722 120572120573 1205732)

(30)

120578 + Φ119905minus120573

2Φ120585120585119905+

120572

21198722 (120585)Φ2120585minus 120573120590

1

119872120597120585(1

119872120578120585)

= 119874 (120572120573 1205732)

(31)


|119869| (120585 119905) = 119872 (120585)2

+ 119874 (1205722) (32)



V119905= 119872120578

119905= minus1205972

120585Φ + 119874 (120572 120573) (33)


minus120573

61205974120585Φ =

120573

21205974120585Φ minus2120573

31205974120585Φ

= minus120573

21205972120585V119905minus2120573

31205974120585Φ + 119874(120572120573 1205732)

(34)


(119868 minus120573

21205972120585) V119905=2120573

31205974120585Φ minus 120597

120585((1 +

120572V1198722)Φ

120585)

(119868 minus120573

21205972120585)Φ

119905

= 120573120590(1

1198723V120585120585minus 31198721015840

1198724V120585+ 3(1198721015840)

2

1198725V minus11987210158401015840

1198724V)

minus1

119872V minus

120572

21198722Φ2120585

(35)


0(119909) with 119899



119872(120585) = 1 + radic1205731198990(120585) + 119874 (120573) (36)

Thus

1205731205901

1198723V120585120585= 120573120590(1 minus 3radic120573119899

0(120585) + 119874 (120573)) V

120585120585 (37)


(119868 minus120573

21205972120585) V119905=2120573

31205974120585Φ minus 120597

120585((1 +

120572V1198722)Φ

120585) (38)

(119868 minus120573

21205972120585)Φ

119905

= 120573120590(V120585120585minus 31198721015840

1198724V120585+ 3(1198721015840)

2

1198725V minus11987210158401015840

1198724V) minus

1

119872V

minus120572

21198722Φ2120585

(39)




(V

Φ)119905

= A(V

Φ) +G(

V

Φ) (40)

where


A(V

Φ) = (

0 (119868 minus120573

21205972120585)minus1

(119868 minus2120573

31205972120585) (minus1205972

120585)

(119868 minus120573

21205972120585)minus1

1205731205901205972120585

0

)(V

Φ)

G(V

Φ) = (


21205972120585)minus1

(120572

1198722VΦ120585)120585

(119868 minus120573

21205972120585)minus1

(minus31205731205901198721015840

1198724V120585+ 3120573120590

(1198721015840)2

1198725V minus 120573120590

11987210158401015840

1198724V minus1

119872V minus

120572

21198722Φ2120585)

)

(41)


V (120585 0) = V0(120585)

Φ (120585 0) = Φ0(120585)

(42)


119905= 119860 (119910) + G (119880)

(119910 0) = 0(119910)

(43)

where

119880 = (V

Φ)

1198800= (

V0

Φ0

)

119860 (119910) = (0

1199102 (1 + (21205733) 1199102)

1 + (1205732) 1199102

minus1205731205901199102

1 + (1205732) 11991020

)

(44)


119905= 119860 (119910)

(119910 0) = 0(119910)

(45)


(119910 119905) = 119890119860(119910)1199050(119910) (46)

where

119890119860(119910)119905 = (11988611(119910 119905) 119886

12(119910 119905)

11988621(119910 119905) 119886

22(119910 119905)

)

= (cos (120582 (119910) 119905) Γ (119910) sin (120582 (119910) 119905)

Ψ (119910) sin (120582 (119910) 119905) cos (120582 (119910) 119905))

120582 (119910) =21199102radic120573120590 (3 + 21205731199102)

radic3 (2 + 1205731199102)

Γ (119910) =radic3 + 21205731199102

radic3120573120590

Ψ (119910) = minusradic3120573120590

radic3 + 21205731199102

(47)


119880 (119909 119905) = 119878 (119905) 1198800(119909) (48)

where

119878 (119905) 119891 = Fminus1 (119890119860(sdot)119905119891 (sdot)) (49)

with

119891 = (1198911

1198912

) (50)


is a 1198620-



Proof Let 119891 = (1198911 1198912)119879 Then

1003817100381710038171003817119878 (119905) 11989110038171003817100381710038172


=

10038171003817100381710038171003817100381710038171003817100381710038171003817

(Fminus1 (119886

111198911+ 119886

121198912)

Fminus1 (119886211198911+ 119886

221198912))

10038171003817100381710038171003817100381710038171003817100381710038171003817

2


=10038171003817100381710038171003817F

minus1 (119886111198911) +Fminus1 (119886

121198912)100381710038171003817100381710038172

119867119904minus2

+10038171003817100381710038171003817F

minus1 (119886211198911) +Fminus1 (119886

221198912)100381710038171003817100381710038172

119867119904minus1

le 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381611988611

10038161003816100381610038162 100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381611988612

10038161003816100381610038162 100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381611988621

10038161003816100381610038162 100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381611988622

10038161003816100381610038162 100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

le 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381610038161198911

100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus2

(3 + 21205731199102)100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 1

3 + 21205731199102100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381610038161198912

100381610038161003816100381610038162

119889119910

le 119862 (1003817100381710038171003817119891110038171003817100381710038172

119867119904minus2 +1003817100381710038171003817119891210038171003817100381710038172

119867119904minus1)

(51)


1003817100381710038171003817119878 (119905) 119891 minus 11989110038171003817100381710038172


le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(Fminus1 (119886

111198911+ 119886

121198912minus 119891

1)

Fminus1 (119886211198911+ 119886

221198912minus 119891

2))

10038171003817100381710038171003817100381710038171003817100381710038171003817

2


le 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381611988611 minus 1

10038161003816100381610038162 100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381611988612

10038161003816100381610038162 100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381611988621

10038161003816100381610038162 100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381611988622 minus 1

10038161003816100381610038162 100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

(52)


21(119910 119905) rarr 0 as 119905 rarr 0+ and

11988611(119910 119905) rarr 1 119886





On the other hand

1003817100381710038171003817A11989110038171003817100381710038172


R

(1 + 1199102)119904minus2

sdot100381610038161003816100381610038161003816100381610038161003816F((119868 minus

120573

21205972120585)minus1

(119868 minus2120573

31205972120585) 12059721205851198912)100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ intR

(1 + 1199102)119904minus1

sdot100381610038161003816100381610038161003816100381610038161003816F((119868 minus

120573

21205972120585)minus1

12057312059012059721205851198911)100381610038161003816100381610038161003816100381610038161003816

2

119889119910

le intR

(1 + 1199102)119904minus2(1 + (21205733) 1199102)

2

1199104

(1 + (1205732) 1199102)2

100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

+ intR

(1 + 1199102)119904minus1 120573212059021199104

(1 + (1205732) 1199102)2

100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

le 119862 (1003817100381710038171003817119891110038171003817100381710038172

119904minus1+1003817100381710038171003817119891210038171003817100381710038172

119904) = 119862

100381710038171003817100381711989110038171003817100381710038172

119867119904minus1times119867119904

(54)


10038171003817100381710038171003817100381710038171003817

119878 (ℎ) 119891 minus 119891

ℎminusA119891

10038171003817100381710038171003817100381710038171003817

2


=100381710038171003817100381710038171003817100381710038171003817Fminus1 (119890119860ℎ minus 119868

ℎ119891

minus 119860119891)100381710038171003817100381710038171003817100381710038171003817

2


=

10038171003817100381710038171003817100381710038171003817100381710038171003817

Fminus1 (11988611(sdot ℎ) 119891

1+ 119886

12(sdot ℎ) 119891

2minus 119891

1)

ℎminus (119868

minus120573

21205972120585)minus1

(119868 minus2120573

31205972120585) (minus1205972

1205851198912)

10038171003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+

10038171003817100381710038171003817100381710038171003817100381710038171003817

Fminus1 (11988621(sdot ℎ) 119891

1+ 119886

22(sdot ℎ) 119891

2minus 119891

2)

ℎminus (119868

minus120573

21205972120585)minus1

12057312059012059721205851198911

10038171003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus1

= intR

(1 + 1199102)119904minus2


sdot

10038161003816100381610038161003816100381610038161003816100381610038161003816

(cos (120582ℎ) minus 1) 1198911+ Γ sin (120582ℎ) 119891

2

ℎ

minus1199102 (1 + (21205733) 1199102) 119891

2

1 + (1205732) 1199102

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910 + intR

(1 + 1199102)119904minus1

sdot

1003816100381610038161003816100381610038161003816100381610038161003816

Ψ sin (120582ℎ) 1198911+ (cos (120582ℎ) minus 1) 119891

2

ℎ

minusminus1205731205901199102119891

1

1 + (1205732) 1199102

1003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

(55)

But by virtue of

limℎrarr0


= 0

limℎrarr0


= Γ120582 =1199102 (1 + (21205733) 1199102)

1 + (1205732) 1199102

limℎrarr0


= Ψ120582 = minus1205731205901199102

1 + (1205732) 1199102

(56)


limℎrarr0

10038171003817100381710038171003817100381710038171003817

119878 (ℎ) 119891 minus 119891

ℎminusA119891

10038171003817100381710038171003817100381710038171003817

2


= 0 (57)



G (119880)119867119904minus1times119867119904 le 119862 119880

119867119904minus1times119867119904 (58)

Furthermore

1003817100381710038171003817G (1198801) minusG (1198802)10038171003817100381710038172

119867119904minus1times119867119904

le 119862 [(1 +1003817100381710038171003817Φ110038171003817100381710038172

119867119904)1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1

+ (1003817100381710038171003817V210038171003817100381710038172

119867119904minus1 +1003817100381710038171003817Φ1 + Φ2

10038171003817100381710038172

119867119904)1003817100381710038171003817Φ1 minus Φ2

10038171003817100381710038172

119867119904]

(59)


V119894Φ119894) 119894 = 1 2


G (119880)2

119867119904minus1times119867119904

le 119862intR

(1 + 1199102)119904minus1

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

2 100381610038161003816100381610038161003816100381610038161003816

(

V1198722Φ120585)120585

100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

210038161003816100381610038161003816100381610038161003816100381610038161003816

1015840

1198724V120585

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

2

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(1198721015840)

2

1198725V

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

210038161003816100381610038161003816100381610038161003816100381610038161003816

10158401015840

1198724V10038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

2 10038161003816100381610038161003816100381610038161003816

V119872

10038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

2100381610038161003816100381610038161003816100381610038161003816100381610038161003816

Φ2120585

1198722

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

(60)


1003817100381710038171003817100381710038171003817

V1198722Φ120585

1003817100381710038171003817100381710038171003817119867119904minus2le 1198621003817100381710038171003817100381710038171003817

V1198722

1003817100381710038171003817100381710038171003817119867119904minus210038171003817100381710038171003817Φ12058510038171003817100381710038171003817119867119904minus1

le 1198621003817100381710038171003817100381710038171003817

1

1198722

1003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1 Φ

119867119904

100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724V120585

100381710038171003817100381710038171003817100381710038171003817119867119904minus2le 119862100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724

100381710038171003817100381710038171003817100381710038171003817119867119904minus1

10038171003817100381710038171003817V12058510038171003817100381710038171003817119867119904minus2

le 119862100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724

100381710038171003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725V100381710038171003817100381710038171003817100381710038171003817100381710038171003817119867119904minus2

le 119862

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725

100381710038171003817100381710038171003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus2

le 119862

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725

100381710038171003817100381710038171003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1

100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724V100381710038171003817100381710038171003817100381710038171003817119867119904minus2

le 119862100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724

100381710038171003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1

1003817100381710038171003817100381710038171003817

V119872

1003817100381710038171003817100381710038171003817119867119904minus2le 1198621003817100381710038171003817100381710038171003817

1

119872

1003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus2

le 1198621003817100381710038171003817100381710038171003817

1

119872

1003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1


1003817100381710038171003817100381710038171003817100381710038171003817

Φ2120585

1198722

1003817100381710038171003817100381710038171003817100381710038171003817119867119904minus2le 11986210038171003817100381710038171003817100381710038171003817

Φ120585

1198722Φ120585

10038171003817100381710038171003817100381710038171003817119867119904minus2

le 11986210038171003817100381710038171003817Φ12058510038171003817100381710038171003817119867119904minus1

10038171003817100381710038171003817100381710038171003817

Φ120585

1198722

10038171003817100381710038171003817100381710038171003817119867119904minus2

le 119862 Φ119904

1003817100381710038171003817100381710038171003817

1

1198722

1003817100381710038171003817100381710038171003817119867119904minus110038171003817100381710038171003817Φ12058510038171003817100381710038171003817119867119904minus2

le 119862 Φ119904

1003817100381710038171003817100381710038171003817

1

1198722

1003817100381710038171003817100381710038171003817119867119904minus1Φ

119867119904

(61)


G (119880)2

119867119904minus1times119867119904 le 119862(

1003817100381710038171003817100381710038171003817

V1198722Φ120585

1003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724V120585

100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725V

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724V100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+1003817100381710038171003817100381710038171003817

V119872

1003817100381710038171003817100381710038171003817

2

119867119904minus2

+

1003817100381710038171003817100381710038171003817100381710038171003817

Φ2120585

1198722

1003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

) le 119862(V2119867119904minus1 + Φ

2

119867119904)

(62)



that

1003817100381710038171003817100381710038171003817100381710038171003817G(

V1

Φ1

) minusG(V2

Φ2

)

1003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus1times119867119904

le 119862(1003817100381710038171003817100381710038171003817

V1

1198722Φ1120585

minusV2

1198722Φ2120585

1003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724(V1minus V

2)120585

100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725(V1minus V

2)

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724(V1minus V

2)100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+1003817100381710038171003817100381710038171003817

1

119872(V1minus V

2)1003817100381710038171003817100381710038171003817

2

119867119904minus2

+1003817100381710038171003817100381710038171003817

1

1198722(Φ21120585minus Φ2

2120585)1003817100381710038171003817100381710038171003817

2

119867119904minus2

)

le 119862(10038171003817100381710038171003817V1Φ1120585 minus V2Φ2120585

100381710038171003817100381710038172

119867119904minus2+1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1 +10038171003817100381710038171003817Φ2

1120585

minus Φ22120585

100381710038171003817100381710038172

119867119904minus2) le 119862 [(

10038171003817100381710038171003817(V1 minus V2)Φ112058510038171003817100381710038171003817119867119904minus2

+10038171003817100381710038171003817V2 (Φ1120585 minus Φ2120585)

10038171003817100381710038171003817119867119904minus2)2

+1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1 +10038171003817100381710038171003817Φ1120585

+ Φ2120585

100381710038171003817100381710038172

119867119904minus1

10038171003817100381710038171003817Φ1120585 minus Φ2120585100381710038171003817100381710038172

119867119904minus1] le 119862 [

1003817100381710038171003817V1 minus V210038171003817100381710038172

119867119904minus1

sdot10038171003817100381710038171003817Φ1120585100381710038171003817100381710038172

119867119904minus2+1003817100381710038171003817V210038171003817100381710038172

119867119904minus1

10038171003817100381710038171003817Φ1120585 minus Φ2120585100381710038171003817100381710038172

119904minus2

+1003817100381710038171003817V1

minus V2

10038171003817100381710038172

119867119904minus1 +1003817100381710038171003817Φ1 + Φ2

10038171003817100381710038172

119867119904

1003817100381710038171003817Φ1 minus Φ210038171003817100381710038172

119867119904] le 119862 [(1

+1003817100381710038171003817Φ110038171003817100381710038172

119867119904)1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1 + (

1003817100381710038171003817V210038171003817100381710038172

119867119904minus1

+1003817100381710038171003817Φ1 + Φ2

10038171003817100381710038172

119867119904)1003817100381710038171003817Φ1 minus Φ2

10038171003817100381710038172

119867119904]

(63)




119880 (119905) = 119878 (119905) 120601 + int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840 (64)


(Ψ119880) (119905) = 119878 (119905) 120601 + int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840 (65)


X119872(119879) = 119880 isin 119862 ([0 119879] 119867

119904minus1 times 119867119904)

sup119905isin[0119879]


(66)




le1003817100381710038171003817(119878 (119905) minus 119878 (120591)) 120601

1003817100381710038171003817119867119904minus1times119867119904

+10038171003817100381710038171003817100381710038171003817int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840

minus 119878 (120591 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840

10038171003817100381710038171003817100381710038171003817119867119904minus1times119867119904

(67)



1003817100381710038171003817119867119904minus1times119867119904

+1003817100381710038171003817119878 (119905) 120601

1003817100381710038171003817119867119904minus1times119867119904

le 119872 +10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904

(68)


Given that (119878(119905))119905ge0




10038171003817100381710038171003817100381710038171003817int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840 minus int

120591

0

119878 (120591 minus 1199051015840)G (119880)

sdot (1199051015840) 119889119905101584010038171003817100381710038171003817100381710038171003817119867119904minus1times119867119904

le int119905

0



1015840 + int120591

119905

10038171003817100381710038171003817119878 (120591 minus 1199051015840)G (119880)

sdot (1199051015840)10038171003817100381710038171003817119867119904minus1times119867119904 119889119905

1015840 = 1198681+ 1198682

(69)


119894119895(119910 119905minus1199051015840) as 120591 rarr 119905 119894 119895 =


1rarr 0 as 120591 rarr 119905


1198682= int

120591

119905

10038171003817100381710038171003817119878 (120591 minus 1199051015840)G (119880) (119905

1015840)10038171003817100381710038171003817119867119904minus1times119867119904 119889119905

1015840

le 119862int120591

119905

10038171003817100381710038171003817G (119880) (1199051015840)10038171003817100381710038171003817119867119904minus1times119867119904 119889119905

1015840

le 119862int120591

119905

119880119867119904minus1times119867119904 1198891199051015840


(70)



Ψ(X119872(119879)) sub X

119872(119879) Let 119880 = (V Φ)119879 isin X

119872(119879) For any

119905 isin [0 119879]


le int119905

0

10038171003817100381710038171003817119878 (119905 minus 1199051015840)G (119880) (119905

1015840)10038171003817100381710038171003817119867119904minus1times119867119904

le 119862int119905

0

10038171003817100381710038171003817G (119880) (1199051015840)10038171003817100381710038171003817119867119904minus1times119867119904

le 119862119879 sup119905isin[0119879]

119880 (119905)119867119904minus1times119867119904 le 119862119879 (119872 +

10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904)

le 119872

(71)



119872() Let 119880

1= (V

1 Φ1)119879

1198802= (V

2 Φ2)119879 isin X



119905

0

10038171003817100381710038171003817119878 (119905 minus 1199051015840)

sdot (G (1198801) (1199051015840) minusG (119880

2) (1199051015840))

10038171003817100381710038171003817119867119904minus1times119867119904 1198891199051015840

le 119862int119905

0


1015840 le int119905

0

(1

+1003817100381710038171003817Φ11003817100381710038171003817119867119904)1003817100381710038171003817V1 minus V2

1003817100381710038171003817119867119904minus1 + (1003817100381710038171003817V21003817100381710038171003817119867119904minus1

+1003817100381710038171003817Φ1 + Φ2

1003817100381710038171003817119867119904)1003817100381710038171003817Φ1 minus Φ2

1003817100381710038171003817119867119904 1198891199051015840 le 119862119879 [(1 +119872

+10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904)

1003817100381710038171003817V1 minus V21003817100381710038171003817119867119904minus1 + (3119872 + 3

10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904)

sdot1003817100381710038171003817Φ1 minus Φ2

1003817100381710038171003817119867119904] le 119862119879 [3119872 + 310038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904]



(72)


119862 [3119872 + 310038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904] lt 119872 (73)



119880 isin 119862 ([0 ] 119867119904minus1 times 119867119904) (74)


5 Numerical Schemes


(119868 minus120573

21205972120585) V119905=2120573

31205973120585119906 minus 120597

120585((1 +

120572V1198722) 119906)

(119868 minus120573

21205972120585)119906

119905

= 1205731205901205973120585V + 120573120590(minus

31198721015840

1198724V120585+ 3(1198721015840)

2

1198725V minus11987210158401015840

1198724V)

minus 120597120585(1

119872V) minus

120572

2120597120585(1

11987221199062)

(75)



120585 In




V (120585 119905) = sum119895

V119895(119905) 119890

119894119908119895120585

119906 (120585 119905) = sum119895

119895(119905) 119890

119894119908119895120585(76)

with

119908119895=2120587119895

119871 119895 = minus

119873

2+ 1 0

119873

2 (77)



V119895(119905) =

1

119871int119871

0

V (120585 119905) 119890minus119894119908119895120585119889120585 (78)




⟨119891 119892⟩ = int119871

0

119891 (120585) 119892 (120585) 119889120585 (79)

it follows that

V1015840119895(119905) = 120574

1119895+ 119873

1

1015840119895(119905) = 120574

2V119895+ 119873

2

(80)

where

1205741=minus (21205733) 1198941199083

119895

1 + (1205732)1199082119895

1205742=minus1205731205901198941199083

119895

1 + (1205732)1199082119895

1198731= minus119894119908119895119875119895[(1 + 120572V1198722) 119906]1 + (1205732)1199082

119895

1198732=119894119908119895119875119895[120573120590 (minus (311987210158401198724) V

120585+ (3 (1198721015840)

2


1 + (1205732)1199082119895

(81)


119875119895[119892] =

1

119871int119871

0

119892 (120585) 119890minus119894119908119895120585119889120585 (82)



V119899+1119895minus V119899

119895

Δ119905=1205741(119899+1119895+ 119899

119895)

2+3

21198731198991minus1

2119873119899minus11

119899+1119895minus 119899

119895

Δ119905=1205742(V119899+1119895+ V119899

119895)

2+3

21198731198992minus1

2119873119899minus12

(83)






119895 119899+1119895




120585are


V120585(120585 119905) = sum

119895

119894119908119895119875119895[V (sdot 119905)] 119890119894119908119895120585 (84)





V (120585 119905) = V (120585 minus 119888119905)

119906 (120585 119905) = (120585 minus 119888119905) (85)



(V10158401015840

11990610158401015840)

=1

(1205734) 1198882 minus 21205731205903(

119888

2

2

3

120590119888

2

)(119888V minus ((1 + 120572V) 119906)

119888119906 minus V minus120572

21199062)

(86)


V (119909) asymp V0+1198732

sum119899=1

V119899cos(119899120587

119897119909)

119906 (119909) asymp 1199060+1198732

sum119899=1

119906119899cos(119899120587

119897119909)

(87)

where

V0=1

119897int119897

0

V (119909) 119889119909 =1

2119897int2119897

0

V (119909) 119889119909

V119899=2

119897int119897

0

V (119909) cos(119899120587119909119897) 119889119909

=2

2119897int2119897

0

V (119909) cos(119899120587119909119897) 119889119909

(88)


119909119895=2119897 (119895 minus 1)

119873 119895 = 1

119873

2+ 1 (89)


119865 (V0 V1 V

1198732 1199060 1199061 119906

1198732) = 0 (90)




119869119894119895119865 (119909) asymp

119865119894(119909 + ℎ119890

119895) minus 119865

119894(119909 minus ℎ119890

119895)

2ℎ

119895 = 1 119873 + 2

(91)




1199060(119909) = cos(120587119909

119897)

V0(119909) = cos(120587119909

119897)

(92)


V = minus(119868 minus120573

21205972120585)Φ

119905+ 120573120590V

120585120585minus120572

2Φ2120585

= minusΦ119905+ 119874 (120572 120573 120590)

(93)


minus Φ119905119905+ Φ

120585120585+ 120573 (1 minus 120590)Φ

120585120585119905119905minus2120573

3Φ120585120585120585120585minus 2120572Φ

120585Φ120585119905

minus 120572Φ119905Φ120585120585= 0

(94)





3)Φ1015840101584010158401015840

+3120572119888

2(Φ10158402)

1015840

= 0

(96)



3)Φ101584010158401015840 +

3120572119888

2(Φ1015840)

2

= 0

(97)


(1 minus 1198882) (Φ1015840)2

+ 120573((1 minus 120590) 1198882 minus2

3) (Φ10158401015840)

2

+ 120572119888 (Φ1015840)3

= 0

(98)


V = 119888119906 minus120573119888

211990610158401015840 + 12057312059011988811990610158401015840 minus

120572

21199062 (99)



119860 =1198882 minus 1

120572119888

119861 = radic1198882 minus 1

4120573 ((1 minus 120590) 1198882 minus 23)

(101)



2)

sdot [41198601198612sech2 (119861119909) tanh2 (119861119909)

minus 21198601198612sech4 (119861119909)]

(102)






119906 (120585 0) = V (120585 0) = 119890minus20(119909minus28)2

(103)




v u

2 4 6 8 100x

0

02

04

06

08

1

12

14

2 4 6 8 100x

0

05

1

15


v u

minus1

minus05

0

05

2 4 6 8 100x

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

02

2 4 6 8 100x






1198622120590=1199082

1198962

=1 + (21205733) 1198962

(1 + (1205732) 1198962)2+1205731205901198962 (1 + (21205733) 1198962)

(1 + (1205732) 1198962)2

gt 11986220

(104)


0119890119894(119896120585minus119908119905)

V = V0119890119894(119896120585minus119908119905) where 119906

0 V0are constants


v u

minus02

0

02

04

06

08

1

12

14

16

2 4 6 8 100

x

0

02

04

06

08

1

12

14

2 4 6 8 100

x


v

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

2 4 6 8 100

x

u

minus25

minus2

minus15

minus1

minus05

0

2 4 6 8 100

x





120590is larger







at t = 10

0

05

1

15u at t = 10

0

02

04

06

08

1

12

14

2 4 6 8 100

120585

2 4 6 8 100

120585


v

minus01

0

01

03

02

04

05

06

07

45 50 55 6040

x

u

45 50 55 6040

x

minus01

0

01

03

02

04

05

06

07





119906 (120585 0) = V (120585 0) = 119890minus20(119909minus198)2 (105)


v

0

005

01

015

02

025

03

035

04

045

45 50 55 6040

x

u

0

005

01

015

02

025

03

035

04

45 50 55 6040

x


v

minus05

0

05

1

15

2

25

45 50 55 6040

x

u

minus05

0

05

1

15

2

45 50 55 6040

x




7 Conclusions



at t = 10 u at t = 10

minus01

0

01

02

03

04

05

06

07

minus01

0

01

02

03

04

05

06

07

60 65 7055

120585

60 65 7055

120585


u at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

20 30 40 5010

120585

20 30 40 5010

120585






minus04

minus03

minus02

minus01

0

01

02

03

04

05u at t = 20

35 40 45 50 55 6030120585

v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 55 6030120585


0 5 10 15 20 25 30 35 40 45 50

C2120590

C20

81205903

radic120573k

0

02

04

06

08

1

12

14

16





05060708091

1112131415

30 32 34 36 38 40 4228

x





u at t = 20

35 40 45 50 5530120585

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 5530120585


v at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

290 300 310 320280

120585

u at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

280 300 310 320290

120585




Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











2 Preliminaries




10038171003817100381710038171198911003817100381710038171003817119871119901 = (int

R

1003816100381610038161003816119891 (119909)1003816100381610038161003816119901

)1119901

(7)



⟨119891 119892⟩ = intR

119891 (119909) 119892 (119909) 119889119909 (8)



F (119891) (119910) = 119891 (119910) = intR

119891 (119909) 119890minus119894119909119910119889119909 119910 isin R (9)


Fminus1 (119891) (119910) = (119910) =

1

2120587intR

119891 (119909) 119890119894119909119910119889119909 119910 isin R (10)


119891 lowast 119892 (119909) = intR

119891 (119909 minus 119910) 119892 (119910) 119889119910 (11)


119878 (R) = 119891 isin 119862infin

(R) 10038171003817100381710038171003817119909


isin Z+ (12)


10038171003817100381710038171198911003817100381710038171003817119867119904 = (int

R

(1 + 1199102)119904 10038161003816100381610038161003816119891 (119910)

100381610038161003816100381610038162

119889119910)12

(13)



⟨119891 119892⟩119904= int

R

(1 + 1199102)119904

119891 (119910) 119892 (119910) 119889119910 (14)


1003817100381710038171003817(120577 119906)1003817100381710038171003817119867119904times119867119904 =

10038171003817100381710038171205771003817100381710038171003817119867119904 + 119906119867119904 (15)


1003817100381710038171003817(120577 119906)1003817100381710038171003817119867119904times119867119904 = (

100381710038171003817100381712057710038171003817100381710038172

119867119904 + 119906

2

119867119904)12

(16)


10038171003817100381710038171198911003817100381710038171003817119862(0119879) = sup

119905isin[0119879]

1003817100381710038171003817119891 (119905 sdot)1003817100381710038171003817119867119904 (17)


1003817100381710038171003817(120577 119906)1003817100381710038171003817119862(0119879)2 =

10038171003817100381710038171205771003817100381710038171003817119862(0119879) + 119906119862(0119879) (18)

for (120577 119906) isin 119862([0 119879] 119867119904(R) times 119867119904(R))



120573120601119909119909+ 120601

119910119910= 0


(19)


120578119905+ 120572120601

119909120578119909minus1

120573120601119910= 0

120578 + 120601119905+120572

2(1206012119909+1

1205731206012119910) minus 120573120590120597

119909(

120578119909

radic1 + 1205731205722 (120578119909)2

)

= 0

(20)


2


minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5




120601119910+120573

1205741198671015840 (

119909

120574)120601

119909= 0 (21)


119867(119909

120574) =

1 + 119899(119909

120574) when 0 lt 119909 lt 119871

1 when 119909 le 0 or 119909 ge 119871(22)







120601119909=1

|119869|[119910120577120601120585minus 119910

120585120601120577]

120601119910=1

|119869|[minus119909

120577120601120585+ 119909

120585120601120577]

1206012119909+ 1206012

119910=1

|119869|(1206012120585+ 1206012

120577

)

120597119909=1

|119869|(119910120577120597120585minus 119910

120585120597120577)

(23)

where

|119869| = 119909120585119910120577minus 119910

120585119909120577= 1199102

120577

+ 1199102120585 (24)


119872(120585) = 119910120577(120585 0)

= 1 +120587


minusinfin

119899 (119909 (1205850 minusradic120573) 120574)

cosh2 (1205872radic120573) (1205850minus 120585)1198891205850

(25)


120573120601120585120585+ 120601

120577120577= 0 at 0 lt 120577 lt 1 + 120572119873 (120585 119905) (26)


|119869|119873119905+ 120572120601

120585119873120585minus1

120573120601120577= 0

120578 + 120601119905+120572

2 |119869|(1206012120585+1

1205731206012120577) minus 120573120590

1

|119869|119872120597

120585(1

119872120578120585)

= 119874 (1205722 120572120573 1205732)

(27)


120601120577= 0 at 120577 = 0 (28)




120601 (120585 120577 119905) =119896

sum119895=0

(minus1)119895 1205731198951205772119895

(2119895)1205972119895

120585Φ (120585 119905) + 119874 (120573

119896+1) (29)


119872(120585) 120578119905+ [(1 +

120572

119872(120585)120578)Φ

120585]120585

minus120573

6Φ120585120585120585120585

= 119874 (1205722 120572120573 1205732)

(30)

120578 + Φ119905minus120573

2Φ120585120585119905+

120572

21198722 (120585)Φ2120585minus 120573120590

1

119872120597120585(1

119872120578120585)

= 119874 (120572120573 1205732)

(31)


|119869| (120585 119905) = 119872 (120585)2

+ 119874 (1205722) (32)



V119905= 119872120578

119905= minus1205972

120585Φ + 119874 (120572 120573) (33)


minus120573

61205974120585Φ =

120573

21205974120585Φ minus2120573

31205974120585Φ

= minus120573

21205972120585V119905minus2120573

31205974120585Φ + 119874(120572120573 1205732)

(34)


(119868 minus120573

21205972120585) V119905=2120573

31205974120585Φ minus 120597

120585((1 +

120572V1198722)Φ

120585)

(119868 minus120573

21205972120585)Φ

119905

= 120573120590(1

1198723V120585120585minus 31198721015840

1198724V120585+ 3(1198721015840)

2

1198725V minus11987210158401015840

1198724V)

minus1

119872V minus

120572

21198722Φ2120585

(35)


0(119909) with 119899



119872(120585) = 1 + radic1205731198990(120585) + 119874 (120573) (36)

Thus

1205731205901

1198723V120585120585= 120573120590(1 minus 3radic120573119899

0(120585) + 119874 (120573)) V

120585120585 (37)


(119868 minus120573

21205972120585) V119905=2120573

31205974120585Φ minus 120597

120585((1 +

120572V1198722)Φ

120585) (38)

(119868 minus120573

21205972120585)Φ

119905

= 120573120590(V120585120585minus 31198721015840

1198724V120585+ 3(1198721015840)

2

1198725V minus11987210158401015840

1198724V) minus

1

119872V

minus120572

21198722Φ2120585

(39)




(V

Φ)119905

= A(V

Φ) +G(

V

Φ) (40)

where


A(V

Φ) = (

0 (119868 minus120573

21205972120585)minus1

(119868 minus2120573

31205972120585) (minus1205972

120585)

(119868 minus120573

21205972120585)minus1

1205731205901205972120585

0

)(V

Φ)

G(V

Φ) = (


21205972120585)minus1

(120572

1198722VΦ120585)120585

(119868 minus120573

21205972120585)minus1

(minus31205731205901198721015840

1198724V120585+ 3120573120590

(1198721015840)2

1198725V minus 120573120590

11987210158401015840

1198724V minus1

119872V minus

120572

21198722Φ2120585)

)

(41)


V (120585 0) = V0(120585)

Φ (120585 0) = Φ0(120585)

(42)


119905= 119860 (119910) + G (119880)

(119910 0) = 0(119910)

(43)

where

119880 = (V

Φ)

1198800= (

V0

Φ0

)

119860 (119910) = (0

1199102 (1 + (21205733) 1199102)

1 + (1205732) 1199102

minus1205731205901199102

1 + (1205732) 11991020

)

(44)


119905= 119860 (119910)

(119910 0) = 0(119910)

(45)


(119910 119905) = 119890119860(119910)1199050(119910) (46)

where

119890119860(119910)119905 = (11988611(119910 119905) 119886

12(119910 119905)

11988621(119910 119905) 119886

22(119910 119905)

)

= (cos (120582 (119910) 119905) Γ (119910) sin (120582 (119910) 119905)

Ψ (119910) sin (120582 (119910) 119905) cos (120582 (119910) 119905))

120582 (119910) =21199102radic120573120590 (3 + 21205731199102)

radic3 (2 + 1205731199102)

Γ (119910) =radic3 + 21205731199102

radic3120573120590

Ψ (119910) = minusradic3120573120590

radic3 + 21205731199102

(47)


119880 (119909 119905) = 119878 (119905) 1198800(119909) (48)

where

119878 (119905) 119891 = Fminus1 (119890119860(sdot)119905119891 (sdot)) (49)

with

119891 = (1198911

1198912

) (50)


is a 1198620-



Proof Let 119891 = (1198911 1198912)119879 Then

1003817100381710038171003817119878 (119905) 11989110038171003817100381710038172


=

10038171003817100381710038171003817100381710038171003817100381710038171003817

(Fminus1 (119886

111198911+ 119886

121198912)

Fminus1 (119886211198911+ 119886

221198912))

10038171003817100381710038171003817100381710038171003817100381710038171003817

2


=10038171003817100381710038171003817F

minus1 (119886111198911) +Fminus1 (119886

121198912)100381710038171003817100381710038172

119867119904minus2

+10038171003817100381710038171003817F

minus1 (119886211198911) +Fminus1 (119886

221198912)100381710038171003817100381710038172

119867119904minus1

le 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381611988611

10038161003816100381610038162 100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381611988612

10038161003816100381610038162 100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381611988621

10038161003816100381610038162 100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381611988622

10038161003816100381610038162 100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

le 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381610038161198911

100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus2

(3 + 21205731199102)100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 1

3 + 21205731199102100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381610038161198912

100381610038161003816100381610038162

119889119910

le 119862 (1003817100381710038171003817119891110038171003817100381710038172

119867119904minus2 +1003817100381710038171003817119891210038171003817100381710038172

119867119904minus1)

(51)


1003817100381710038171003817119878 (119905) 119891 minus 11989110038171003817100381710038172


le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(Fminus1 (119886

111198911+ 119886

121198912minus 119891

1)

Fminus1 (119886211198911+ 119886

221198912minus 119891

2))

10038171003817100381710038171003817100381710038171003817100381710038171003817

2


le 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381611988611 minus 1

10038161003816100381610038162 100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381611988612

10038161003816100381610038162 100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381611988621

10038161003816100381610038162 100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381611988622 minus 1

10038161003816100381610038162 100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

(52)


21(119910 119905) rarr 0 as 119905 rarr 0+ and

11988611(119910 119905) rarr 1 119886





On the other hand

1003817100381710038171003817A11989110038171003817100381710038172


R

(1 + 1199102)119904minus2

sdot100381610038161003816100381610038161003816100381610038161003816F((119868 minus

120573

21205972120585)minus1

(119868 minus2120573

31205972120585) 12059721205851198912)100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ intR

(1 + 1199102)119904minus1

sdot100381610038161003816100381610038161003816100381610038161003816F((119868 minus

120573

21205972120585)minus1

12057312059012059721205851198911)100381610038161003816100381610038161003816100381610038161003816

2

119889119910

le intR

(1 + 1199102)119904minus2(1 + (21205733) 1199102)

2

1199104

(1 + (1205732) 1199102)2

100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

+ intR

(1 + 1199102)119904minus1 120573212059021199104

(1 + (1205732) 1199102)2

100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

le 119862 (1003817100381710038171003817119891110038171003817100381710038172

119904minus1+1003817100381710038171003817119891210038171003817100381710038172

119904) = 119862

100381710038171003817100381711989110038171003817100381710038172

119867119904minus1times119867119904

(54)


10038171003817100381710038171003817100381710038171003817

119878 (ℎ) 119891 minus 119891

ℎminusA119891

10038171003817100381710038171003817100381710038171003817

2


=100381710038171003817100381710038171003817100381710038171003817Fminus1 (119890119860ℎ minus 119868

ℎ119891

minus 119860119891)100381710038171003817100381710038171003817100381710038171003817

2


=

10038171003817100381710038171003817100381710038171003817100381710038171003817

Fminus1 (11988611(sdot ℎ) 119891

1+ 119886

12(sdot ℎ) 119891

2minus 119891

1)

ℎminus (119868

minus120573

21205972120585)minus1

(119868 minus2120573

31205972120585) (minus1205972

1205851198912)

10038171003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+

10038171003817100381710038171003817100381710038171003817100381710038171003817

Fminus1 (11988621(sdot ℎ) 119891

1+ 119886

22(sdot ℎ) 119891

2minus 119891

2)

ℎminus (119868

minus120573

21205972120585)minus1

12057312059012059721205851198911

10038171003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus1

= intR

(1 + 1199102)119904minus2


sdot

10038161003816100381610038161003816100381610038161003816100381610038161003816

(cos (120582ℎ) minus 1) 1198911+ Γ sin (120582ℎ) 119891

2

ℎ

minus1199102 (1 + (21205733) 1199102) 119891

2

1 + (1205732) 1199102

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910 + intR

(1 + 1199102)119904minus1

sdot

1003816100381610038161003816100381610038161003816100381610038161003816

Ψ sin (120582ℎ) 1198911+ (cos (120582ℎ) minus 1) 119891

2

ℎ

minusminus1205731205901199102119891

1

1 + (1205732) 1199102

1003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

(55)

But by virtue of

limℎrarr0


= 0

limℎrarr0


= Γ120582 =1199102 (1 + (21205733) 1199102)

1 + (1205732) 1199102

limℎrarr0


= Ψ120582 = minus1205731205901199102

1 + (1205732) 1199102

(56)


limℎrarr0

10038171003817100381710038171003817100381710038171003817

119878 (ℎ) 119891 minus 119891

ℎminusA119891

10038171003817100381710038171003817100381710038171003817

2


= 0 (57)



G (119880)119867119904minus1times119867119904 le 119862 119880

119867119904minus1times119867119904 (58)

Furthermore

1003817100381710038171003817G (1198801) minusG (1198802)10038171003817100381710038172

119867119904minus1times119867119904

le 119862 [(1 +1003817100381710038171003817Φ110038171003817100381710038172

119867119904)1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1

+ (1003817100381710038171003817V210038171003817100381710038172

119867119904minus1 +1003817100381710038171003817Φ1 + Φ2

10038171003817100381710038172

119867119904)1003817100381710038171003817Φ1 minus Φ2

10038171003817100381710038172

119867119904]

(59)


V119894Φ119894) 119894 = 1 2


G (119880)2

119867119904minus1times119867119904

le 119862intR

(1 + 1199102)119904minus1

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

2 100381610038161003816100381610038161003816100381610038161003816

(

V1198722Φ120585)120585

100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

210038161003816100381610038161003816100381610038161003816100381610038161003816

1015840

1198724V120585

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

2

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(1198721015840)

2

1198725V

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

210038161003816100381610038161003816100381610038161003816100381610038161003816

10158401015840

1198724V10038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

2 10038161003816100381610038161003816100381610038161003816

V119872

10038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

2100381610038161003816100381610038161003816100381610038161003816100381610038161003816

Φ2120585

1198722

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

(60)


1003817100381710038171003817100381710038171003817

V1198722Φ120585

1003817100381710038171003817100381710038171003817119867119904minus2le 1198621003817100381710038171003817100381710038171003817

V1198722

1003817100381710038171003817100381710038171003817119867119904minus210038171003817100381710038171003817Φ12058510038171003817100381710038171003817119867119904minus1

le 1198621003817100381710038171003817100381710038171003817

1

1198722

1003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1 Φ

119867119904

100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724V120585

100381710038171003817100381710038171003817100381710038171003817119867119904minus2le 119862100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724

100381710038171003817100381710038171003817100381710038171003817119867119904minus1

10038171003817100381710038171003817V12058510038171003817100381710038171003817119867119904minus2

le 119862100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724

100381710038171003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725V100381710038171003817100381710038171003817100381710038171003817100381710038171003817119867119904minus2

le 119862

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725

100381710038171003817100381710038171003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus2

le 119862

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725

100381710038171003817100381710038171003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1

100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724V100381710038171003817100381710038171003817100381710038171003817119867119904minus2

le 119862100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724

100381710038171003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1

1003817100381710038171003817100381710038171003817

V119872

1003817100381710038171003817100381710038171003817119867119904minus2le 1198621003817100381710038171003817100381710038171003817

1

119872

1003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus2

le 1198621003817100381710038171003817100381710038171003817

1

119872

1003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1


1003817100381710038171003817100381710038171003817100381710038171003817

Φ2120585

1198722

1003817100381710038171003817100381710038171003817100381710038171003817119867119904minus2le 11986210038171003817100381710038171003817100381710038171003817

Φ120585

1198722Φ120585

10038171003817100381710038171003817100381710038171003817119867119904minus2

le 11986210038171003817100381710038171003817Φ12058510038171003817100381710038171003817119867119904minus1

10038171003817100381710038171003817100381710038171003817

Φ120585

1198722

10038171003817100381710038171003817100381710038171003817119867119904minus2

le 119862 Φ119904

1003817100381710038171003817100381710038171003817

1

1198722

1003817100381710038171003817100381710038171003817119867119904minus110038171003817100381710038171003817Φ12058510038171003817100381710038171003817119867119904minus2

le 119862 Φ119904

1003817100381710038171003817100381710038171003817

1

1198722

1003817100381710038171003817100381710038171003817119867119904minus1Φ

119867119904

(61)


G (119880)2

119867119904minus1times119867119904 le 119862(

1003817100381710038171003817100381710038171003817

V1198722Φ120585

1003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724V120585

100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725V

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724V100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+1003817100381710038171003817100381710038171003817

V119872

1003817100381710038171003817100381710038171003817

2

119867119904minus2

+

1003817100381710038171003817100381710038171003817100381710038171003817

Φ2120585

1198722

1003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

) le 119862(V2119867119904minus1 + Φ

2

119867119904)

(62)



that

1003817100381710038171003817100381710038171003817100381710038171003817G(

V1

Φ1

) minusG(V2

Φ2

)

1003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus1times119867119904

le 119862(1003817100381710038171003817100381710038171003817

V1

1198722Φ1120585

minusV2

1198722Φ2120585

1003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724(V1minus V

2)120585

100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725(V1minus V

2)

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724(V1minus V

2)100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+1003817100381710038171003817100381710038171003817

1

119872(V1minus V

2)1003817100381710038171003817100381710038171003817

2

119867119904minus2

+1003817100381710038171003817100381710038171003817

1

1198722(Φ21120585minus Φ2

2120585)1003817100381710038171003817100381710038171003817

2

119867119904minus2

)

le 119862(10038171003817100381710038171003817V1Φ1120585 minus V2Φ2120585

100381710038171003817100381710038172

119867119904minus2+1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1 +10038171003817100381710038171003817Φ2

1120585

minus Φ22120585

100381710038171003817100381710038172

119867119904minus2) le 119862 [(

10038171003817100381710038171003817(V1 minus V2)Φ112058510038171003817100381710038171003817119867119904minus2

+10038171003817100381710038171003817V2 (Φ1120585 minus Φ2120585)

10038171003817100381710038171003817119867119904minus2)2

+1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1 +10038171003817100381710038171003817Φ1120585

+ Φ2120585

100381710038171003817100381710038172

119867119904minus1

10038171003817100381710038171003817Φ1120585 minus Φ2120585100381710038171003817100381710038172

119867119904minus1] le 119862 [

1003817100381710038171003817V1 minus V210038171003817100381710038172

119867119904minus1

sdot10038171003817100381710038171003817Φ1120585100381710038171003817100381710038172

119867119904minus2+1003817100381710038171003817V210038171003817100381710038172

119867119904minus1

10038171003817100381710038171003817Φ1120585 minus Φ2120585100381710038171003817100381710038172

119904minus2

+1003817100381710038171003817V1

minus V2

10038171003817100381710038172

119867119904minus1 +1003817100381710038171003817Φ1 + Φ2

10038171003817100381710038172

119867119904

1003817100381710038171003817Φ1 minus Φ210038171003817100381710038172

119867119904] le 119862 [(1

+1003817100381710038171003817Φ110038171003817100381710038172

119867119904)1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1 + (

1003817100381710038171003817V210038171003817100381710038172

119867119904minus1

+1003817100381710038171003817Φ1 + Φ2

10038171003817100381710038172

119867119904)1003817100381710038171003817Φ1 minus Φ2

10038171003817100381710038172

119867119904]

(63)




119880 (119905) = 119878 (119905) 120601 + int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840 (64)


(Ψ119880) (119905) = 119878 (119905) 120601 + int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840 (65)


X119872(119879) = 119880 isin 119862 ([0 119879] 119867

119904minus1 times 119867119904)

sup119905isin[0119879]


(66)




le1003817100381710038171003817(119878 (119905) minus 119878 (120591)) 120601

1003817100381710038171003817119867119904minus1times119867119904

+10038171003817100381710038171003817100381710038171003817int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840

minus 119878 (120591 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840

10038171003817100381710038171003817100381710038171003817119867119904minus1times119867119904

(67)



1003817100381710038171003817119867119904minus1times119867119904

+1003817100381710038171003817119878 (119905) 120601

1003817100381710038171003817119867119904minus1times119867119904

le 119872 +10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904

(68)


Given that (119878(119905))119905ge0




10038171003817100381710038171003817100381710038171003817int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840 minus int

120591

0

119878 (120591 minus 1199051015840)G (119880)

sdot (1199051015840) 119889119905101584010038171003817100381710038171003817100381710038171003817119867119904minus1times119867119904

le int119905

0



1015840 + int120591

119905

10038171003817100381710038171003817119878 (120591 minus 1199051015840)G (119880)

sdot (1199051015840)10038171003817100381710038171003817119867119904minus1times119867119904 119889119905

1015840 = 1198681+ 1198682

(69)


119894119895(119910 119905minus1199051015840) as 120591 rarr 119905 119894 119895 =


1rarr 0 as 120591 rarr 119905


1198682= int

120591

119905

10038171003817100381710038171003817119878 (120591 minus 1199051015840)G (119880) (119905

1015840)10038171003817100381710038171003817119867119904minus1times119867119904 119889119905

1015840

le 119862int120591

119905

10038171003817100381710038171003817G (119880) (1199051015840)10038171003817100381710038171003817119867119904minus1times119867119904 119889119905

1015840

le 119862int120591

119905

119880119867119904minus1times119867119904 1198891199051015840


(70)



Ψ(X119872(119879)) sub X

119872(119879) Let 119880 = (V Φ)119879 isin X

119872(119879) For any

119905 isin [0 119879]


le int119905

0

10038171003817100381710038171003817119878 (119905 minus 1199051015840)G (119880) (119905

1015840)10038171003817100381710038171003817119867119904minus1times119867119904

le 119862int119905

0

10038171003817100381710038171003817G (119880) (1199051015840)10038171003817100381710038171003817119867119904minus1times119867119904

le 119862119879 sup119905isin[0119879]

119880 (119905)119867119904minus1times119867119904 le 119862119879 (119872 +

10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904)

le 119872

(71)



119872() Let 119880

1= (V

1 Φ1)119879

1198802= (V

2 Φ2)119879 isin X



119905

0

10038171003817100381710038171003817119878 (119905 minus 1199051015840)

sdot (G (1198801) (1199051015840) minusG (119880

2) (1199051015840))

10038171003817100381710038171003817119867119904minus1times119867119904 1198891199051015840

le 119862int119905

0


1015840 le int119905

0

(1

+1003817100381710038171003817Φ11003817100381710038171003817119867119904)1003817100381710038171003817V1 minus V2

1003817100381710038171003817119867119904minus1 + (1003817100381710038171003817V21003817100381710038171003817119867119904minus1

+1003817100381710038171003817Φ1 + Φ2

1003817100381710038171003817119867119904)1003817100381710038171003817Φ1 minus Φ2

1003817100381710038171003817119867119904 1198891199051015840 le 119862119879 [(1 +119872

+10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904)

1003817100381710038171003817V1 minus V21003817100381710038171003817119867119904minus1 + (3119872 + 3

10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904)

sdot1003817100381710038171003817Φ1 minus Φ2

1003817100381710038171003817119867119904] le 119862119879 [3119872 + 310038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904]



(72)


119862 [3119872 + 310038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904] lt 119872 (73)



119880 isin 119862 ([0 ] 119867119904minus1 times 119867119904) (74)


5 Numerical Schemes


(119868 minus120573

21205972120585) V119905=2120573

31205973120585119906 minus 120597

120585((1 +

120572V1198722) 119906)

(119868 minus120573

21205972120585)119906

119905

= 1205731205901205973120585V + 120573120590(minus

31198721015840

1198724V120585+ 3(1198721015840)

2

1198725V minus11987210158401015840

1198724V)

minus 120597120585(1

119872V) minus

120572

2120597120585(1

11987221199062)

(75)



120585 In




V (120585 119905) = sum119895

V119895(119905) 119890

119894119908119895120585

119906 (120585 119905) = sum119895

119895(119905) 119890

119894119908119895120585(76)

with

119908119895=2120587119895

119871 119895 = minus

119873

2+ 1 0

119873

2 (77)



V119895(119905) =

1

119871int119871

0

V (120585 119905) 119890minus119894119908119895120585119889120585 (78)




⟨119891 119892⟩ = int119871

0

119891 (120585) 119892 (120585) 119889120585 (79)

it follows that

V1015840119895(119905) = 120574

1119895+ 119873

1

1015840119895(119905) = 120574

2V119895+ 119873

2

(80)

where

1205741=minus (21205733) 1198941199083

119895

1 + (1205732)1199082119895

1205742=minus1205731205901198941199083

119895

1 + (1205732)1199082119895

1198731= minus119894119908119895119875119895[(1 + 120572V1198722) 119906]1 + (1205732)1199082

119895

1198732=119894119908119895119875119895[120573120590 (minus (311987210158401198724) V

120585+ (3 (1198721015840)

2


1 + (1205732)1199082119895

(81)


119875119895[119892] =

1

119871int119871

0

119892 (120585) 119890minus119894119908119895120585119889120585 (82)



V119899+1119895minus V119899

119895

Δ119905=1205741(119899+1119895+ 119899

119895)

2+3

21198731198991minus1

2119873119899minus11

119899+1119895minus 119899

119895

Δ119905=1205742(V119899+1119895+ V119899

119895)

2+3

21198731198992minus1

2119873119899minus12

(83)






119895 119899+1119895




120585are


V120585(120585 119905) = sum

119895

119894119908119895119875119895[V (sdot 119905)] 119890119894119908119895120585 (84)





V (120585 119905) = V (120585 minus 119888119905)

119906 (120585 119905) = (120585 minus 119888119905) (85)



(V10158401015840

11990610158401015840)

=1

(1205734) 1198882 minus 21205731205903(

119888

2

2

3

120590119888

2

)(119888V minus ((1 + 120572V) 119906)

119888119906 minus V minus120572

21199062)

(86)


V (119909) asymp V0+1198732

sum119899=1

V119899cos(119899120587

119897119909)

119906 (119909) asymp 1199060+1198732

sum119899=1

119906119899cos(119899120587

119897119909)

(87)

where

V0=1

119897int119897

0

V (119909) 119889119909 =1

2119897int2119897

0

V (119909) 119889119909

V119899=2

119897int119897

0

V (119909) cos(119899120587119909119897) 119889119909

=2

2119897int2119897

0

V (119909) cos(119899120587119909119897) 119889119909

(88)


119909119895=2119897 (119895 minus 1)

119873 119895 = 1

119873

2+ 1 (89)


119865 (V0 V1 V

1198732 1199060 1199061 119906

1198732) = 0 (90)




119869119894119895119865 (119909) asymp

119865119894(119909 + ℎ119890

119895) minus 119865

119894(119909 minus ℎ119890

119895)

2ℎ

119895 = 1 119873 + 2

(91)




1199060(119909) = cos(120587119909

119897)

V0(119909) = cos(120587119909

119897)

(92)


V = minus(119868 minus120573

21205972120585)Φ

119905+ 120573120590V

120585120585minus120572

2Φ2120585

= minusΦ119905+ 119874 (120572 120573 120590)

(93)


minus Φ119905119905+ Φ

120585120585+ 120573 (1 minus 120590)Φ

120585120585119905119905minus2120573

3Φ120585120585120585120585minus 2120572Φ

120585Φ120585119905

minus 120572Φ119905Φ120585120585= 0

(94)





3)Φ1015840101584010158401015840

+3120572119888

2(Φ10158402)

1015840

= 0

(96)



3)Φ101584010158401015840 +

3120572119888

2(Φ1015840)

2

= 0

(97)


(1 minus 1198882) (Φ1015840)2

+ 120573((1 minus 120590) 1198882 minus2

3) (Φ10158401015840)

2

+ 120572119888 (Φ1015840)3

= 0

(98)


V = 119888119906 minus120573119888

211990610158401015840 + 12057312059011988811990610158401015840 minus

120572

21199062 (99)



119860 =1198882 minus 1

120572119888

119861 = radic1198882 minus 1

4120573 ((1 minus 120590) 1198882 minus 23)

(101)



2)

sdot [41198601198612sech2 (119861119909) tanh2 (119861119909)

minus 21198601198612sech4 (119861119909)]

(102)






119906 (120585 0) = V (120585 0) = 119890minus20(119909minus28)2

(103)




v u

2 4 6 8 100x

0

02

04

06

08

1

12

14

2 4 6 8 100x

0

05

1

15


v u

minus1

minus05

0

05

2 4 6 8 100x

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

02

2 4 6 8 100x






1198622120590=1199082

1198962

=1 + (21205733) 1198962

(1 + (1205732) 1198962)2+1205731205901198962 (1 + (21205733) 1198962)

(1 + (1205732) 1198962)2

gt 11986220

(104)


0119890119894(119896120585minus119908119905)

V = V0119890119894(119896120585minus119908119905) where 119906

0 V0are constants


v u

minus02

0

02

04

06

08

1

12

14

16

2 4 6 8 100

x

0

02

04

06

08

1

12

14

2 4 6 8 100

x


v

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

2 4 6 8 100

x

u

minus25

minus2

minus15

minus1

minus05

0

2 4 6 8 100

x





120590is larger







at t = 10

0

05

1

15u at t = 10

0

02

04

06

08

1

12

14

2 4 6 8 100

120585

2 4 6 8 100

120585


v

minus01

0

01

03

02

04

05

06

07

45 50 55 6040

x

u

45 50 55 6040

x

minus01

0

01

03

02

04

05

06

07





119906 (120585 0) = V (120585 0) = 119890minus20(119909minus198)2 (105)


v

0

005

01

015

02

025

03

035

04

045

45 50 55 6040

x

u

0

005

01

015

02

025

03

035

04

45 50 55 6040

x


v

minus05

0

05

1

15

2

25

45 50 55 6040

x

u

minus05

0

05

1

15

2

45 50 55 6040

x




7 Conclusions



at t = 10 u at t = 10

minus01

0

01

02

03

04

05

06

07

minus01

0

01

02

03

04

05

06

07

60 65 7055

120585

60 65 7055

120585


u at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

20 30 40 5010

120585

20 30 40 5010

120585






minus04

minus03

minus02

minus01

0

01

02

03

04

05u at t = 20

35 40 45 50 55 6030120585

v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 55 6030120585


0 5 10 15 20 25 30 35 40 45 50

C2120590

C20

81205903

radic120573k

0

02

04

06

08

1

12

14

16





05060708091

1112131415

30 32 34 36 38 40 4228

x





u at t = 20

35 40 45 50 5530120585

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 5530120585


v at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

290 300 310 320280

120585

u at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

280 300 310 320290

120585




Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2


minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5




120601119910+120573

1205741198671015840 (

119909

120574)120601

119909= 0 (21)


119867(119909

120574) =

1 + 119899(119909

120574) when 0 lt 119909 lt 119871

1 when 119909 le 0 or 119909 ge 119871(22)







120601119909=1

|119869|[119910120577120601120585minus 119910

120585120601120577]

120601119910=1

|119869|[minus119909

120577120601120585+ 119909

120585120601120577]

1206012119909+ 1206012

119910=1

|119869|(1206012120585+ 1206012

120577

)

120597119909=1

|119869|(119910120577120597120585minus 119910

120585120597120577)

(23)

where

|119869| = 119909120585119910120577minus 119910

120585119909120577= 1199102

120577

+ 1199102120585 (24)


119872(120585) = 119910120577(120585 0)

= 1 +120587


minusinfin

119899 (119909 (1205850 minusradic120573) 120574)

cosh2 (1205872radic120573) (1205850minus 120585)1198891205850

(25)


120573120601120585120585+ 120601

120577120577= 0 at 0 lt 120577 lt 1 + 120572119873 (120585 119905) (26)


|119869|119873119905+ 120572120601

120585119873120585minus1

120573120601120577= 0

120578 + 120601119905+120572

2 |119869|(1206012120585+1

1205731206012120577) minus 120573120590

1

|119869|119872120597

120585(1

119872120578120585)

= 119874 (1205722 120572120573 1205732)

(27)


120601120577= 0 at 120577 = 0 (28)




120601 (120585 120577 119905) =119896

sum119895=0

(minus1)119895 1205731198951205772119895

(2119895)1205972119895

120585Φ (120585 119905) + 119874 (120573

119896+1) (29)


119872(120585) 120578119905+ [(1 +

120572

119872(120585)120578)Φ

120585]120585

minus120573

6Φ120585120585120585120585

= 119874 (1205722 120572120573 1205732)

(30)

120578 + Φ119905minus120573

2Φ120585120585119905+

120572

21198722 (120585)Φ2120585minus 120573120590

1

119872120597120585(1

119872120578120585)

= 119874 (120572120573 1205732)

(31)


|119869| (120585 119905) = 119872 (120585)2

+ 119874 (1205722) (32)



V119905= 119872120578

119905= minus1205972

120585Φ + 119874 (120572 120573) (33)


minus120573

61205974120585Φ =

120573

21205974120585Φ minus2120573

31205974120585Φ

= minus120573

21205972120585V119905minus2120573

31205974120585Φ + 119874(120572120573 1205732)

(34)


(119868 minus120573

21205972120585) V119905=2120573

31205974120585Φ minus 120597

120585((1 +

120572V1198722)Φ

120585)

(119868 minus120573

21205972120585)Φ

119905

= 120573120590(1

1198723V120585120585minus 31198721015840

1198724V120585+ 3(1198721015840)

2

1198725V minus11987210158401015840

1198724V)

minus1

119872V minus

120572

21198722Φ2120585

(35)


0(119909) with 119899



119872(120585) = 1 + radic1205731198990(120585) + 119874 (120573) (36)

Thus

1205731205901

1198723V120585120585= 120573120590(1 minus 3radic120573119899

0(120585) + 119874 (120573)) V

120585120585 (37)


(119868 minus120573

21205972120585) V119905=2120573

31205974120585Φ minus 120597

120585((1 +

120572V1198722)Φ

120585) (38)

(119868 minus120573

21205972120585)Φ

119905

= 120573120590(V120585120585minus 31198721015840

1198724V120585+ 3(1198721015840)

2

1198725V minus11987210158401015840

1198724V) minus

1

119872V

minus120572

21198722Φ2120585

(39)




(V

Φ)119905

= A(V

Φ) +G(

V

Φ) (40)

where


A(V

Φ) = (

0 (119868 minus120573

21205972120585)minus1

(119868 minus2120573

31205972120585) (minus1205972

120585)

(119868 minus120573

21205972120585)minus1

1205731205901205972120585

0

)(V

Φ)

G(V

Φ) = (


21205972120585)minus1

(120572

1198722VΦ120585)120585

(119868 minus120573

21205972120585)minus1

(minus31205731205901198721015840

1198724V120585+ 3120573120590

(1198721015840)2

1198725V minus 120573120590

11987210158401015840

1198724V minus1

119872V minus

120572

21198722Φ2120585)

)

(41)


V (120585 0) = V0(120585)

Φ (120585 0) = Φ0(120585)

(42)


119905= 119860 (119910) + G (119880)

(119910 0) = 0(119910)

(43)

where

119880 = (V

Φ)

1198800= (

V0

Φ0

)

119860 (119910) = (0

1199102 (1 + (21205733) 1199102)

1 + (1205732) 1199102

minus1205731205901199102

1 + (1205732) 11991020

)

(44)


119905= 119860 (119910)

(119910 0) = 0(119910)

(45)


(119910 119905) = 119890119860(119910)1199050(119910) (46)

where

119890119860(119910)119905 = (11988611(119910 119905) 119886

12(119910 119905)

11988621(119910 119905) 119886

22(119910 119905)

)

= (cos (120582 (119910) 119905) Γ (119910) sin (120582 (119910) 119905)

Ψ (119910) sin (120582 (119910) 119905) cos (120582 (119910) 119905))

120582 (119910) =21199102radic120573120590 (3 + 21205731199102)

radic3 (2 + 1205731199102)

Γ (119910) =radic3 + 21205731199102

radic3120573120590

Ψ (119910) = minusradic3120573120590

radic3 + 21205731199102

(47)


119880 (119909 119905) = 119878 (119905) 1198800(119909) (48)

where

119878 (119905) 119891 = Fminus1 (119890119860(sdot)119905119891 (sdot)) (49)

with

119891 = (1198911

1198912

) (50)


is a 1198620-



Proof Let 119891 = (1198911 1198912)119879 Then

1003817100381710038171003817119878 (119905) 11989110038171003817100381710038172


=

10038171003817100381710038171003817100381710038171003817100381710038171003817

(Fminus1 (119886

111198911+ 119886

121198912)

Fminus1 (119886211198911+ 119886

221198912))

10038171003817100381710038171003817100381710038171003817100381710038171003817

2


=10038171003817100381710038171003817F

minus1 (119886111198911) +Fminus1 (119886

121198912)100381710038171003817100381710038172

119867119904minus2

+10038171003817100381710038171003817F

minus1 (119886211198911) +Fminus1 (119886

221198912)100381710038171003817100381710038172

119867119904minus1

le 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381611988611

10038161003816100381610038162 100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381611988612

10038161003816100381610038162 100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381611988621

10038161003816100381610038162 100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381611988622

10038161003816100381610038162 100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

le 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381610038161198911

100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus2

(3 + 21205731199102)100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 1

3 + 21205731199102100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381610038161198912

100381610038161003816100381610038162

119889119910

le 119862 (1003817100381710038171003817119891110038171003817100381710038172

119867119904minus2 +1003817100381710038171003817119891210038171003817100381710038172

119867119904minus1)

(51)


1003817100381710038171003817119878 (119905) 119891 minus 11989110038171003817100381710038172


le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(Fminus1 (119886

111198911+ 119886

121198912minus 119891

1)

Fminus1 (119886211198911+ 119886

221198912minus 119891

2))

10038171003817100381710038171003817100381710038171003817100381710038171003817

2


le 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381611988611 minus 1

10038161003816100381610038162 100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381611988612

10038161003816100381610038162 100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381611988621

10038161003816100381610038162 100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381611988622 minus 1

10038161003816100381610038162 100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

(52)


21(119910 119905) rarr 0 as 119905 rarr 0+ and

11988611(119910 119905) rarr 1 119886





On the other hand

1003817100381710038171003817A11989110038171003817100381710038172


R

(1 + 1199102)119904minus2

sdot100381610038161003816100381610038161003816100381610038161003816F((119868 minus

120573

21205972120585)minus1

(119868 minus2120573

31205972120585) 12059721205851198912)100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ intR

(1 + 1199102)119904minus1

sdot100381610038161003816100381610038161003816100381610038161003816F((119868 minus

120573

21205972120585)minus1

12057312059012059721205851198911)100381610038161003816100381610038161003816100381610038161003816

2

119889119910

le intR

(1 + 1199102)119904minus2(1 + (21205733) 1199102)

2

1199104

(1 + (1205732) 1199102)2

100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

+ intR

(1 + 1199102)119904minus1 120573212059021199104

(1 + (1205732) 1199102)2

100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

le 119862 (1003817100381710038171003817119891110038171003817100381710038172

119904minus1+1003817100381710038171003817119891210038171003817100381710038172

119904) = 119862

100381710038171003817100381711989110038171003817100381710038172

119867119904minus1times119867119904

(54)


10038171003817100381710038171003817100381710038171003817

119878 (ℎ) 119891 minus 119891

ℎminusA119891

10038171003817100381710038171003817100381710038171003817

2


=100381710038171003817100381710038171003817100381710038171003817Fminus1 (119890119860ℎ minus 119868

ℎ119891

minus 119860119891)100381710038171003817100381710038171003817100381710038171003817

2


=

10038171003817100381710038171003817100381710038171003817100381710038171003817

Fminus1 (11988611(sdot ℎ) 119891

1+ 119886

12(sdot ℎ) 119891

2minus 119891

1)

ℎminus (119868

minus120573

21205972120585)minus1

(119868 minus2120573

31205972120585) (minus1205972

1205851198912)

10038171003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+

10038171003817100381710038171003817100381710038171003817100381710038171003817

Fminus1 (11988621(sdot ℎ) 119891

1+ 119886

22(sdot ℎ) 119891

2minus 119891

2)

ℎminus (119868

minus120573

21205972120585)minus1

12057312059012059721205851198911

10038171003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus1

= intR

(1 + 1199102)119904minus2


sdot

10038161003816100381610038161003816100381610038161003816100381610038161003816

(cos (120582ℎ) minus 1) 1198911+ Γ sin (120582ℎ) 119891

2

ℎ

minus1199102 (1 + (21205733) 1199102) 119891

2

1 + (1205732) 1199102

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910 + intR

(1 + 1199102)119904minus1

sdot

1003816100381610038161003816100381610038161003816100381610038161003816

Ψ sin (120582ℎ) 1198911+ (cos (120582ℎ) minus 1) 119891

2

ℎ

minusminus1205731205901199102119891

1

1 + (1205732) 1199102

1003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

(55)

But by virtue of

limℎrarr0


= 0

limℎrarr0


= Γ120582 =1199102 (1 + (21205733) 1199102)

1 + (1205732) 1199102

limℎrarr0


= Ψ120582 = minus1205731205901199102

1 + (1205732) 1199102

(56)


limℎrarr0

10038171003817100381710038171003817100381710038171003817

119878 (ℎ) 119891 minus 119891

ℎminusA119891

10038171003817100381710038171003817100381710038171003817

2


= 0 (57)



G (119880)119867119904minus1times119867119904 le 119862 119880

119867119904minus1times119867119904 (58)

Furthermore

1003817100381710038171003817G (1198801) minusG (1198802)10038171003817100381710038172

119867119904minus1times119867119904

le 119862 [(1 +1003817100381710038171003817Φ110038171003817100381710038172

119867119904)1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1

+ (1003817100381710038171003817V210038171003817100381710038172

119867119904minus1 +1003817100381710038171003817Φ1 + Φ2

10038171003817100381710038172

119867119904)1003817100381710038171003817Φ1 minus Φ2

10038171003817100381710038172

119867119904]

(59)


V119894Φ119894) 119894 = 1 2


G (119880)2

119867119904minus1times119867119904

le 119862intR

(1 + 1199102)119904minus1

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

2 100381610038161003816100381610038161003816100381610038161003816

(

V1198722Φ120585)120585

100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

210038161003816100381610038161003816100381610038161003816100381610038161003816

1015840

1198724V120585

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

2

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(1198721015840)

2

1198725V

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

210038161003816100381610038161003816100381610038161003816100381610038161003816

10158401015840

1198724V10038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

2 10038161003816100381610038161003816100381610038161003816

V119872

10038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

2100381610038161003816100381610038161003816100381610038161003816100381610038161003816

Φ2120585

1198722

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

(60)


1003817100381710038171003817100381710038171003817

V1198722Φ120585

1003817100381710038171003817100381710038171003817119867119904minus2le 1198621003817100381710038171003817100381710038171003817

V1198722

1003817100381710038171003817100381710038171003817119867119904minus210038171003817100381710038171003817Φ12058510038171003817100381710038171003817119867119904minus1

le 1198621003817100381710038171003817100381710038171003817

1

1198722

1003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1 Φ

119867119904

100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724V120585

100381710038171003817100381710038171003817100381710038171003817119867119904minus2le 119862100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724

100381710038171003817100381710038171003817100381710038171003817119867119904minus1

10038171003817100381710038171003817V12058510038171003817100381710038171003817119867119904minus2

le 119862100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724

100381710038171003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725V100381710038171003817100381710038171003817100381710038171003817100381710038171003817119867119904minus2

le 119862

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725

100381710038171003817100381710038171003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus2

le 119862

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725

100381710038171003817100381710038171003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1

100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724V100381710038171003817100381710038171003817100381710038171003817119867119904minus2

le 119862100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724

100381710038171003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1

1003817100381710038171003817100381710038171003817

V119872

1003817100381710038171003817100381710038171003817119867119904minus2le 1198621003817100381710038171003817100381710038171003817

1

119872

1003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus2

le 1198621003817100381710038171003817100381710038171003817

1

119872

1003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1


1003817100381710038171003817100381710038171003817100381710038171003817

Φ2120585

1198722

1003817100381710038171003817100381710038171003817100381710038171003817119867119904minus2le 11986210038171003817100381710038171003817100381710038171003817

Φ120585

1198722Φ120585

10038171003817100381710038171003817100381710038171003817119867119904minus2

le 11986210038171003817100381710038171003817Φ12058510038171003817100381710038171003817119867119904minus1

10038171003817100381710038171003817100381710038171003817

Φ120585

1198722

10038171003817100381710038171003817100381710038171003817119867119904minus2

le 119862 Φ119904

1003817100381710038171003817100381710038171003817

1

1198722

1003817100381710038171003817100381710038171003817119867119904minus110038171003817100381710038171003817Φ12058510038171003817100381710038171003817119867119904minus2

le 119862 Φ119904

1003817100381710038171003817100381710038171003817

1

1198722

1003817100381710038171003817100381710038171003817119867119904minus1Φ

119867119904

(61)


G (119880)2

119867119904minus1times119867119904 le 119862(

1003817100381710038171003817100381710038171003817

V1198722Φ120585

1003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724V120585

100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725V

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724V100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+1003817100381710038171003817100381710038171003817

V119872

1003817100381710038171003817100381710038171003817

2

119867119904minus2

+

1003817100381710038171003817100381710038171003817100381710038171003817

Φ2120585

1198722

1003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

) le 119862(V2119867119904minus1 + Φ

2

119867119904)

(62)



that

1003817100381710038171003817100381710038171003817100381710038171003817G(

V1

Φ1

) minusG(V2

Φ2

)

1003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus1times119867119904

le 119862(1003817100381710038171003817100381710038171003817

V1

1198722Φ1120585

minusV2

1198722Φ2120585

1003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724(V1minus V

2)120585

100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725(V1minus V

2)

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724(V1minus V

2)100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+1003817100381710038171003817100381710038171003817

1

119872(V1minus V

2)1003817100381710038171003817100381710038171003817

2

119867119904minus2

+1003817100381710038171003817100381710038171003817

1

1198722(Φ21120585minus Φ2

2120585)1003817100381710038171003817100381710038171003817

2

119867119904minus2

)

le 119862(10038171003817100381710038171003817V1Φ1120585 minus V2Φ2120585

100381710038171003817100381710038172

119867119904minus2+1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1 +10038171003817100381710038171003817Φ2

1120585

minus Φ22120585

100381710038171003817100381710038172

119867119904minus2) le 119862 [(

10038171003817100381710038171003817(V1 minus V2)Φ112058510038171003817100381710038171003817119867119904minus2

+10038171003817100381710038171003817V2 (Φ1120585 minus Φ2120585)

10038171003817100381710038171003817119867119904minus2)2

+1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1 +10038171003817100381710038171003817Φ1120585

+ Φ2120585

100381710038171003817100381710038172

119867119904minus1

10038171003817100381710038171003817Φ1120585 minus Φ2120585100381710038171003817100381710038172

119867119904minus1] le 119862 [

1003817100381710038171003817V1 minus V210038171003817100381710038172

119867119904minus1

sdot10038171003817100381710038171003817Φ1120585100381710038171003817100381710038172

119867119904minus2+1003817100381710038171003817V210038171003817100381710038172

119867119904minus1

10038171003817100381710038171003817Φ1120585 minus Φ2120585100381710038171003817100381710038172

119904minus2

+1003817100381710038171003817V1

minus V2

10038171003817100381710038172

119867119904minus1 +1003817100381710038171003817Φ1 + Φ2

10038171003817100381710038172

119867119904

1003817100381710038171003817Φ1 minus Φ210038171003817100381710038172

119867119904] le 119862 [(1

+1003817100381710038171003817Φ110038171003817100381710038172

119867119904)1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1 + (

1003817100381710038171003817V210038171003817100381710038172

119867119904minus1

+1003817100381710038171003817Φ1 + Φ2

10038171003817100381710038172

119867119904)1003817100381710038171003817Φ1 minus Φ2

10038171003817100381710038172

119867119904]

(63)




119880 (119905) = 119878 (119905) 120601 + int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840 (64)


(Ψ119880) (119905) = 119878 (119905) 120601 + int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840 (65)


X119872(119879) = 119880 isin 119862 ([0 119879] 119867

119904minus1 times 119867119904)

sup119905isin[0119879]


(66)




le1003817100381710038171003817(119878 (119905) minus 119878 (120591)) 120601

1003817100381710038171003817119867119904minus1times119867119904

+10038171003817100381710038171003817100381710038171003817int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840

minus 119878 (120591 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840

10038171003817100381710038171003817100381710038171003817119867119904minus1times119867119904

(67)



1003817100381710038171003817119867119904minus1times119867119904

+1003817100381710038171003817119878 (119905) 120601

1003817100381710038171003817119867119904minus1times119867119904

le 119872 +10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904

(68)


Given that (119878(119905))119905ge0




10038171003817100381710038171003817100381710038171003817int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840 minus int

120591

0

119878 (120591 minus 1199051015840)G (119880)

sdot (1199051015840) 119889119905101584010038171003817100381710038171003817100381710038171003817119867119904minus1times119867119904

le int119905

0



1015840 + int120591

119905

10038171003817100381710038171003817119878 (120591 minus 1199051015840)G (119880)

sdot (1199051015840)10038171003817100381710038171003817119867119904minus1times119867119904 119889119905

1015840 = 1198681+ 1198682

(69)


119894119895(119910 119905minus1199051015840) as 120591 rarr 119905 119894 119895 =


1rarr 0 as 120591 rarr 119905


1198682= int

120591

119905

10038171003817100381710038171003817119878 (120591 minus 1199051015840)G (119880) (119905

1015840)10038171003817100381710038171003817119867119904minus1times119867119904 119889119905

1015840

le 119862int120591

119905

10038171003817100381710038171003817G (119880) (1199051015840)10038171003817100381710038171003817119867119904minus1times119867119904 119889119905

1015840

le 119862int120591

119905

119880119867119904minus1times119867119904 1198891199051015840


(70)



Ψ(X119872(119879)) sub X

119872(119879) Let 119880 = (V Φ)119879 isin X

119872(119879) For any

119905 isin [0 119879]


le int119905

0

10038171003817100381710038171003817119878 (119905 minus 1199051015840)G (119880) (119905

1015840)10038171003817100381710038171003817119867119904minus1times119867119904

le 119862int119905

0

10038171003817100381710038171003817G (119880) (1199051015840)10038171003817100381710038171003817119867119904minus1times119867119904

le 119862119879 sup119905isin[0119879]

119880 (119905)119867119904minus1times119867119904 le 119862119879 (119872 +

10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904)

le 119872

(71)



119872() Let 119880

1= (V

1 Φ1)119879

1198802= (V

2 Φ2)119879 isin X



119905

0

10038171003817100381710038171003817119878 (119905 minus 1199051015840)

sdot (G (1198801) (1199051015840) minusG (119880

2) (1199051015840))

10038171003817100381710038171003817119867119904minus1times119867119904 1198891199051015840

le 119862int119905

0


1015840 le int119905

0

(1

+1003817100381710038171003817Φ11003817100381710038171003817119867119904)1003817100381710038171003817V1 minus V2

1003817100381710038171003817119867119904minus1 + (1003817100381710038171003817V21003817100381710038171003817119867119904minus1

+1003817100381710038171003817Φ1 + Φ2

1003817100381710038171003817119867119904)1003817100381710038171003817Φ1 minus Φ2

1003817100381710038171003817119867119904 1198891199051015840 le 119862119879 [(1 +119872

+10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904)

1003817100381710038171003817V1 minus V21003817100381710038171003817119867119904minus1 + (3119872 + 3

10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904)

sdot1003817100381710038171003817Φ1 minus Φ2

1003817100381710038171003817119867119904] le 119862119879 [3119872 + 310038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904]



(72)


119862 [3119872 + 310038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904] lt 119872 (73)



119880 isin 119862 ([0 ] 119867119904minus1 times 119867119904) (74)


5 Numerical Schemes


(119868 minus120573

21205972120585) V119905=2120573

31205973120585119906 minus 120597

120585((1 +

120572V1198722) 119906)

(119868 minus120573

21205972120585)119906

119905

= 1205731205901205973120585V + 120573120590(minus

31198721015840

1198724V120585+ 3(1198721015840)

2

1198725V minus11987210158401015840

1198724V)

minus 120597120585(1

119872V) minus

120572

2120597120585(1

11987221199062)

(75)



120585 In




V (120585 119905) = sum119895

V119895(119905) 119890

119894119908119895120585

119906 (120585 119905) = sum119895

119895(119905) 119890

119894119908119895120585(76)

with

119908119895=2120587119895

119871 119895 = minus

119873

2+ 1 0

119873

2 (77)



V119895(119905) =

1

119871int119871

0

V (120585 119905) 119890minus119894119908119895120585119889120585 (78)




⟨119891 119892⟩ = int119871

0

119891 (120585) 119892 (120585) 119889120585 (79)

it follows that

V1015840119895(119905) = 120574

1119895+ 119873

1

1015840119895(119905) = 120574

2V119895+ 119873

2

(80)

where

1205741=minus (21205733) 1198941199083

119895

1 + (1205732)1199082119895

1205742=minus1205731205901198941199083

119895

1 + (1205732)1199082119895

1198731= minus119894119908119895119875119895[(1 + 120572V1198722) 119906]1 + (1205732)1199082

119895

1198732=119894119908119895119875119895[120573120590 (minus (311987210158401198724) V

120585+ (3 (1198721015840)

2


1 + (1205732)1199082119895

(81)


119875119895[119892] =

1

119871int119871

0

119892 (120585) 119890minus119894119908119895120585119889120585 (82)



V119899+1119895minus V119899

119895

Δ119905=1205741(119899+1119895+ 119899

119895)

2+3

21198731198991minus1

2119873119899minus11

119899+1119895minus 119899

119895

Δ119905=1205742(V119899+1119895+ V119899

119895)

2+3

21198731198992minus1

2119873119899minus12

(83)






119895 119899+1119895




120585are


V120585(120585 119905) = sum

119895

119894119908119895119875119895[V (sdot 119905)] 119890119894119908119895120585 (84)





V (120585 119905) = V (120585 minus 119888119905)

119906 (120585 119905) = (120585 minus 119888119905) (85)



(V10158401015840

11990610158401015840)

=1

(1205734) 1198882 minus 21205731205903(

119888

2

2

3

120590119888

2

)(119888V minus ((1 + 120572V) 119906)

119888119906 minus V minus120572

21199062)

(86)


V (119909) asymp V0+1198732

sum119899=1

V119899cos(119899120587

119897119909)

119906 (119909) asymp 1199060+1198732

sum119899=1

119906119899cos(119899120587

119897119909)

(87)

where

V0=1

119897int119897

0

V (119909) 119889119909 =1

2119897int2119897

0

V (119909) 119889119909

V119899=2

119897int119897

0

V (119909) cos(119899120587119909119897) 119889119909

=2

2119897int2119897

0

V (119909) cos(119899120587119909119897) 119889119909

(88)


119909119895=2119897 (119895 minus 1)

119873 119895 = 1

119873

2+ 1 (89)


119865 (V0 V1 V

1198732 1199060 1199061 119906

1198732) = 0 (90)




119869119894119895119865 (119909) asymp

119865119894(119909 + ℎ119890

119895) minus 119865

119894(119909 minus ℎ119890

119895)

2ℎ

119895 = 1 119873 + 2

(91)




1199060(119909) = cos(120587119909

119897)

V0(119909) = cos(120587119909

119897)

(92)


V = minus(119868 minus120573

21205972120585)Φ

119905+ 120573120590V

120585120585minus120572

2Φ2120585

= minusΦ119905+ 119874 (120572 120573 120590)

(93)


minus Φ119905119905+ Φ

120585120585+ 120573 (1 minus 120590)Φ

120585120585119905119905minus2120573

3Φ120585120585120585120585minus 2120572Φ

120585Φ120585119905

minus 120572Φ119905Φ120585120585= 0

(94)





3)Φ1015840101584010158401015840

+3120572119888

2(Φ10158402)

1015840

= 0

(96)



3)Φ101584010158401015840 +

3120572119888

2(Φ1015840)

2

= 0

(97)


(1 minus 1198882) (Φ1015840)2

+ 120573((1 minus 120590) 1198882 minus2

3) (Φ10158401015840)

2

+ 120572119888 (Φ1015840)3

= 0

(98)


V = 119888119906 minus120573119888

211990610158401015840 + 12057312059011988811990610158401015840 minus

120572

21199062 (99)



119860 =1198882 minus 1

120572119888

119861 = radic1198882 minus 1

4120573 ((1 minus 120590) 1198882 minus 23)

(101)



2)

sdot [41198601198612sech2 (119861119909) tanh2 (119861119909)

minus 21198601198612sech4 (119861119909)]

(102)






119906 (120585 0) = V (120585 0) = 119890minus20(119909minus28)2

(103)




v u

2 4 6 8 100x

0

02

04

06

08

1

12

14

2 4 6 8 100x

0

05

1

15


v u

minus1

minus05

0

05

2 4 6 8 100x

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

02

2 4 6 8 100x






1198622120590=1199082

1198962

=1 + (21205733) 1198962

(1 + (1205732) 1198962)2+1205731205901198962 (1 + (21205733) 1198962)

(1 + (1205732) 1198962)2

gt 11986220

(104)


0119890119894(119896120585minus119908119905)

V = V0119890119894(119896120585minus119908119905) where 119906

0 V0are constants


v u

minus02

0

02

04

06

08

1

12

14

16

2 4 6 8 100

x

0

02

04

06

08

1

12

14

2 4 6 8 100

x


v

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

2 4 6 8 100

x

u

minus25

minus2

minus15

minus1

minus05

0

2 4 6 8 100

x





120590is larger







at t = 10

0

05

1

15u at t = 10

0

02

04

06

08

1

12

14

2 4 6 8 100

120585

2 4 6 8 100

120585


v

minus01

0

01

03

02

04

05

06

07

45 50 55 6040

x

u

45 50 55 6040

x

minus01

0

01

03

02

04

05

06

07





119906 (120585 0) = V (120585 0) = 119890minus20(119909minus198)2 (105)


v

0

005

01

015

02

025

03

035

04

045

45 50 55 6040

x

u

0

005

01

015

02

025

03

035

04

45 50 55 6040

x


v

minus05

0

05

1

15

2

25

45 50 55 6040

x

u

minus05

0

05

1

15

2

45 50 55 6040

x




7 Conclusions



at t = 10 u at t = 10

minus01

0

01

02

03

04

05

06

07

minus01

0

01

02

03

04

05

06

07

60 65 7055

120585

60 65 7055

120585


u at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

20 30 40 5010

120585

20 30 40 5010

120585






minus04

minus03

minus02

minus01

0

01

02

03

04

05u at t = 20

35 40 45 50 55 6030120585

v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 55 6030120585


0 5 10 15 20 25 30 35 40 45 50

C2120590

C20

81205903

radic120573k

0

02

04

06

08

1

12

14

16





05060708091

1112131415

30 32 34 36 38 40 4228

x





u at t = 20

35 40 45 50 5530120585

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 5530120585


v at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

290 300 310 320280

120585

u at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

280 300 310 320290

120585




Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120601 (120585 120577 119905) =119896

sum119895=0

(minus1)119895 1205731198951205772119895

(2119895)1205972119895

120585Φ (120585 119905) + 119874 (120573

119896+1) (29)


119872(120585) 120578119905+ [(1 +

120572

119872(120585)120578)Φ

120585]120585

minus120573

6Φ120585120585120585120585

= 119874 (1205722 120572120573 1205732)

(30)

120578 + Φ119905minus120573

2Φ120585120585119905+

120572

21198722 (120585)Φ2120585minus 120573120590

1

119872120597120585(1

119872120578120585)

= 119874 (120572120573 1205732)

(31)


|119869| (120585 119905) = 119872 (120585)2

+ 119874 (1205722) (32)



V119905= 119872120578

119905= minus1205972

120585Φ + 119874 (120572 120573) (33)


minus120573

61205974120585Φ =

120573

21205974120585Φ minus2120573

31205974120585Φ

= minus120573

21205972120585V119905minus2120573

31205974120585Φ + 119874(120572120573 1205732)

(34)


(119868 minus120573

21205972120585) V119905=2120573

31205974120585Φ minus 120597

120585((1 +

120572V1198722)Φ

120585)

(119868 minus120573

21205972120585)Φ

119905

= 120573120590(1

1198723V120585120585minus 31198721015840

1198724V120585+ 3(1198721015840)

2

1198725V minus11987210158401015840

1198724V)

minus1

119872V minus

120572

21198722Φ2120585

(35)


0(119909) with 119899



119872(120585) = 1 + radic1205731198990(120585) + 119874 (120573) (36)

Thus

1205731205901

1198723V120585120585= 120573120590(1 minus 3radic120573119899

0(120585) + 119874 (120573)) V

120585120585 (37)


(119868 minus120573

21205972120585) V119905=2120573

31205974120585Φ minus 120597

120585((1 +

120572V1198722)Φ

120585) (38)

(119868 minus120573

21205972120585)Φ

119905

= 120573120590(V120585120585minus 31198721015840

1198724V120585+ 3(1198721015840)

2

1198725V minus11987210158401015840

1198724V) minus

1

119872V

minus120572

21198722Φ2120585

(39)




(V

Φ)119905

= A(V

Φ) +G(

V

Φ) (40)

where


A(V

Φ) = (

0 (119868 minus120573

21205972120585)minus1

(119868 minus2120573

31205972120585) (minus1205972

120585)

(119868 minus120573

21205972120585)minus1

1205731205901205972120585

0

)(V

Φ)

G(V

Φ) = (


21205972120585)minus1

(120572

1198722VΦ120585)120585

(119868 minus120573

21205972120585)minus1

(minus31205731205901198721015840

1198724V120585+ 3120573120590

(1198721015840)2

1198725V minus 120573120590

11987210158401015840

1198724V minus1

119872V minus

120572

21198722Φ2120585)

)

(41)


V (120585 0) = V0(120585)

Φ (120585 0) = Φ0(120585)

(42)


119905= 119860 (119910) + G (119880)

(119910 0) = 0(119910)

(43)

where

119880 = (V

Φ)

1198800= (

V0

Φ0

)

119860 (119910) = (0

1199102 (1 + (21205733) 1199102)

1 + (1205732) 1199102

minus1205731205901199102

1 + (1205732) 11991020

)

(44)


119905= 119860 (119910)

(119910 0) = 0(119910)

(45)


(119910 119905) = 119890119860(119910)1199050(119910) (46)

where

119890119860(119910)119905 = (11988611(119910 119905) 119886

12(119910 119905)

11988621(119910 119905) 119886

22(119910 119905)

)

= (cos (120582 (119910) 119905) Γ (119910) sin (120582 (119910) 119905)

Ψ (119910) sin (120582 (119910) 119905) cos (120582 (119910) 119905))

120582 (119910) =21199102radic120573120590 (3 + 21205731199102)

radic3 (2 + 1205731199102)

Γ (119910) =radic3 + 21205731199102

radic3120573120590

Ψ (119910) = minusradic3120573120590

radic3 + 21205731199102

(47)


119880 (119909 119905) = 119878 (119905) 1198800(119909) (48)

where

119878 (119905) 119891 = Fminus1 (119890119860(sdot)119905119891 (sdot)) (49)

with

119891 = (1198911

1198912

) (50)


is a 1198620-



Proof Let 119891 = (1198911 1198912)119879 Then

1003817100381710038171003817119878 (119905) 11989110038171003817100381710038172


=

10038171003817100381710038171003817100381710038171003817100381710038171003817

(Fminus1 (119886

111198911+ 119886

121198912)

Fminus1 (119886211198911+ 119886

221198912))

10038171003817100381710038171003817100381710038171003817100381710038171003817

2


=10038171003817100381710038171003817F

minus1 (119886111198911) +Fminus1 (119886

121198912)100381710038171003817100381710038172

119867119904minus2

+10038171003817100381710038171003817F

minus1 (119886211198911) +Fminus1 (119886

221198912)100381710038171003817100381710038172

119867119904minus1

le 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381611988611

10038161003816100381610038162 100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381611988612

10038161003816100381610038162 100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381611988621

10038161003816100381610038162 100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381611988622

10038161003816100381610038162 100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

le 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381610038161198911

100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus2

(3 + 21205731199102)100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 1

3 + 21205731199102100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381610038161198912

100381610038161003816100381610038162

119889119910

le 119862 (1003817100381710038171003817119891110038171003817100381710038172

119867119904minus2 +1003817100381710038171003817119891210038171003817100381710038172

119867119904minus1)

(51)


1003817100381710038171003817119878 (119905) 119891 minus 11989110038171003817100381710038172


le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(Fminus1 (119886

111198911+ 119886

121198912minus 119891

1)

Fminus1 (119886211198911+ 119886

221198912minus 119891

2))

10038171003817100381710038171003817100381710038171003817100381710038171003817

2


le 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381611988611 minus 1

10038161003816100381610038162 100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381611988612

10038161003816100381610038162 100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381611988621

10038161003816100381610038162 100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381611988622 minus 1

10038161003816100381610038162 100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

(52)


21(119910 119905) rarr 0 as 119905 rarr 0+ and

11988611(119910 119905) rarr 1 119886





On the other hand

1003817100381710038171003817A11989110038171003817100381710038172


R

(1 + 1199102)119904minus2

sdot100381610038161003816100381610038161003816100381610038161003816F((119868 minus

120573

21205972120585)minus1

(119868 minus2120573

31205972120585) 12059721205851198912)100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ intR

(1 + 1199102)119904minus1

sdot100381610038161003816100381610038161003816100381610038161003816F((119868 minus

120573

21205972120585)minus1

12057312059012059721205851198911)100381610038161003816100381610038161003816100381610038161003816

2

119889119910

le intR

(1 + 1199102)119904minus2(1 + (21205733) 1199102)

2

1199104

(1 + (1205732) 1199102)2

100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

+ intR

(1 + 1199102)119904minus1 120573212059021199104

(1 + (1205732) 1199102)2

100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

le 119862 (1003817100381710038171003817119891110038171003817100381710038172

119904minus1+1003817100381710038171003817119891210038171003817100381710038172

119904) = 119862

100381710038171003817100381711989110038171003817100381710038172

119867119904minus1times119867119904

(54)


10038171003817100381710038171003817100381710038171003817

119878 (ℎ) 119891 minus 119891

ℎminusA119891

10038171003817100381710038171003817100381710038171003817

2


=100381710038171003817100381710038171003817100381710038171003817Fminus1 (119890119860ℎ minus 119868

ℎ119891

minus 119860119891)100381710038171003817100381710038171003817100381710038171003817

2


=

10038171003817100381710038171003817100381710038171003817100381710038171003817

Fminus1 (11988611(sdot ℎ) 119891

1+ 119886

12(sdot ℎ) 119891

2minus 119891

1)

ℎminus (119868

minus120573

21205972120585)minus1

(119868 minus2120573

31205972120585) (minus1205972

1205851198912)

10038171003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+

10038171003817100381710038171003817100381710038171003817100381710038171003817

Fminus1 (11988621(sdot ℎ) 119891

1+ 119886

22(sdot ℎ) 119891

2minus 119891

2)

ℎminus (119868

minus120573

21205972120585)minus1

12057312059012059721205851198911

10038171003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus1

= intR

(1 + 1199102)119904minus2


sdot

10038161003816100381610038161003816100381610038161003816100381610038161003816

(cos (120582ℎ) minus 1) 1198911+ Γ sin (120582ℎ) 119891

2

ℎ

minus1199102 (1 + (21205733) 1199102) 119891

2

1 + (1205732) 1199102

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910 + intR

(1 + 1199102)119904minus1

sdot

1003816100381610038161003816100381610038161003816100381610038161003816

Ψ sin (120582ℎ) 1198911+ (cos (120582ℎ) minus 1) 119891

2

ℎ

minusminus1205731205901199102119891

1

1 + (1205732) 1199102

1003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

(55)

But by virtue of

limℎrarr0


= 0

limℎrarr0


= Γ120582 =1199102 (1 + (21205733) 1199102)

1 + (1205732) 1199102

limℎrarr0


= Ψ120582 = minus1205731205901199102

1 + (1205732) 1199102

(56)


limℎrarr0

10038171003817100381710038171003817100381710038171003817

119878 (ℎ) 119891 minus 119891

ℎminusA119891

10038171003817100381710038171003817100381710038171003817

2


= 0 (57)



G (119880)119867119904minus1times119867119904 le 119862 119880

119867119904minus1times119867119904 (58)

Furthermore

1003817100381710038171003817G (1198801) minusG (1198802)10038171003817100381710038172

119867119904minus1times119867119904

le 119862 [(1 +1003817100381710038171003817Φ110038171003817100381710038172

119867119904)1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1

+ (1003817100381710038171003817V210038171003817100381710038172

119867119904minus1 +1003817100381710038171003817Φ1 + Φ2

10038171003817100381710038172

119867119904)1003817100381710038171003817Φ1 minus Φ2

10038171003817100381710038172

119867119904]

(59)


V119894Φ119894) 119894 = 1 2


G (119880)2

119867119904minus1times119867119904

le 119862intR

(1 + 1199102)119904minus1

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

2 100381610038161003816100381610038161003816100381610038161003816

(

V1198722Φ120585)120585

100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

210038161003816100381610038161003816100381610038161003816100381610038161003816

1015840

1198724V120585

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

2

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(1198721015840)

2

1198725V

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

210038161003816100381610038161003816100381610038161003816100381610038161003816

10158401015840

1198724V10038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

2 10038161003816100381610038161003816100381610038161003816

V119872

10038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

2100381610038161003816100381610038161003816100381610038161003816100381610038161003816

Φ2120585

1198722

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

(60)


1003817100381710038171003817100381710038171003817

V1198722Φ120585

1003817100381710038171003817100381710038171003817119867119904minus2le 1198621003817100381710038171003817100381710038171003817

V1198722

1003817100381710038171003817100381710038171003817119867119904minus210038171003817100381710038171003817Φ12058510038171003817100381710038171003817119867119904minus1

le 1198621003817100381710038171003817100381710038171003817

1

1198722

1003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1 Φ

119867119904

100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724V120585

100381710038171003817100381710038171003817100381710038171003817119867119904minus2le 119862100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724

100381710038171003817100381710038171003817100381710038171003817119867119904minus1

10038171003817100381710038171003817V12058510038171003817100381710038171003817119867119904minus2

le 119862100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724

100381710038171003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725V100381710038171003817100381710038171003817100381710038171003817100381710038171003817119867119904minus2

le 119862

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725

100381710038171003817100381710038171003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus2

le 119862

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725

100381710038171003817100381710038171003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1

100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724V100381710038171003817100381710038171003817100381710038171003817119867119904minus2

le 119862100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724

100381710038171003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1

1003817100381710038171003817100381710038171003817

V119872

1003817100381710038171003817100381710038171003817119867119904minus2le 1198621003817100381710038171003817100381710038171003817

1

119872

1003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus2

le 1198621003817100381710038171003817100381710038171003817

1

119872

1003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1


1003817100381710038171003817100381710038171003817100381710038171003817

Φ2120585

1198722

1003817100381710038171003817100381710038171003817100381710038171003817119867119904minus2le 11986210038171003817100381710038171003817100381710038171003817

Φ120585

1198722Φ120585

10038171003817100381710038171003817100381710038171003817119867119904minus2

le 11986210038171003817100381710038171003817Φ12058510038171003817100381710038171003817119867119904minus1

10038171003817100381710038171003817100381710038171003817

Φ120585

1198722

10038171003817100381710038171003817100381710038171003817119867119904minus2

le 119862 Φ119904

1003817100381710038171003817100381710038171003817

1

1198722

1003817100381710038171003817100381710038171003817119867119904minus110038171003817100381710038171003817Φ12058510038171003817100381710038171003817119867119904minus2

le 119862 Φ119904

1003817100381710038171003817100381710038171003817

1

1198722

1003817100381710038171003817100381710038171003817119867119904minus1Φ

119867119904

(61)


G (119880)2

119867119904minus1times119867119904 le 119862(

1003817100381710038171003817100381710038171003817

V1198722Φ120585

1003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724V120585

100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725V

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724V100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+1003817100381710038171003817100381710038171003817

V119872

1003817100381710038171003817100381710038171003817

2

119867119904minus2

+

1003817100381710038171003817100381710038171003817100381710038171003817

Φ2120585

1198722

1003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

) le 119862(V2119867119904minus1 + Φ

2

119867119904)

(62)



that

1003817100381710038171003817100381710038171003817100381710038171003817G(

V1

Φ1

) minusG(V2

Φ2

)

1003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus1times119867119904

le 119862(1003817100381710038171003817100381710038171003817

V1

1198722Φ1120585

minusV2

1198722Φ2120585

1003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724(V1minus V

2)120585

100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725(V1minus V

2)

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724(V1minus V

2)100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+1003817100381710038171003817100381710038171003817

1

119872(V1minus V

2)1003817100381710038171003817100381710038171003817

2

119867119904minus2

+1003817100381710038171003817100381710038171003817

1

1198722(Φ21120585minus Φ2

2120585)1003817100381710038171003817100381710038171003817

2

119867119904minus2

)

le 119862(10038171003817100381710038171003817V1Φ1120585 minus V2Φ2120585

100381710038171003817100381710038172

119867119904minus2+1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1 +10038171003817100381710038171003817Φ2

1120585

minus Φ22120585

100381710038171003817100381710038172

119867119904minus2) le 119862 [(

10038171003817100381710038171003817(V1 minus V2)Φ112058510038171003817100381710038171003817119867119904minus2

+10038171003817100381710038171003817V2 (Φ1120585 minus Φ2120585)

10038171003817100381710038171003817119867119904minus2)2

+1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1 +10038171003817100381710038171003817Φ1120585

+ Φ2120585

100381710038171003817100381710038172

119867119904minus1

10038171003817100381710038171003817Φ1120585 minus Φ2120585100381710038171003817100381710038172

119867119904minus1] le 119862 [

1003817100381710038171003817V1 minus V210038171003817100381710038172

119867119904minus1

sdot10038171003817100381710038171003817Φ1120585100381710038171003817100381710038172

119867119904minus2+1003817100381710038171003817V210038171003817100381710038172

119867119904minus1

10038171003817100381710038171003817Φ1120585 minus Φ2120585100381710038171003817100381710038172

119904minus2

+1003817100381710038171003817V1

minus V2

10038171003817100381710038172

119867119904minus1 +1003817100381710038171003817Φ1 + Φ2

10038171003817100381710038172

119867119904

1003817100381710038171003817Φ1 minus Φ210038171003817100381710038172

119867119904] le 119862 [(1

+1003817100381710038171003817Φ110038171003817100381710038172

119867119904)1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1 + (

1003817100381710038171003817V210038171003817100381710038172

119867119904minus1

+1003817100381710038171003817Φ1 + Φ2

10038171003817100381710038172

119867119904)1003817100381710038171003817Φ1 minus Φ2

10038171003817100381710038172

119867119904]

(63)




119880 (119905) = 119878 (119905) 120601 + int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840 (64)


(Ψ119880) (119905) = 119878 (119905) 120601 + int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840 (65)


X119872(119879) = 119880 isin 119862 ([0 119879] 119867

119904minus1 times 119867119904)

sup119905isin[0119879]


(66)




le1003817100381710038171003817(119878 (119905) minus 119878 (120591)) 120601

1003817100381710038171003817119867119904minus1times119867119904

+10038171003817100381710038171003817100381710038171003817int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840

minus 119878 (120591 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840

10038171003817100381710038171003817100381710038171003817119867119904minus1times119867119904

(67)



1003817100381710038171003817119867119904minus1times119867119904

+1003817100381710038171003817119878 (119905) 120601

1003817100381710038171003817119867119904minus1times119867119904

le 119872 +10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904

(68)


Given that (119878(119905))119905ge0




10038171003817100381710038171003817100381710038171003817int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840 minus int

120591

0

119878 (120591 minus 1199051015840)G (119880)

sdot (1199051015840) 119889119905101584010038171003817100381710038171003817100381710038171003817119867119904minus1times119867119904

le int119905

0



1015840 + int120591

119905

10038171003817100381710038171003817119878 (120591 minus 1199051015840)G (119880)

sdot (1199051015840)10038171003817100381710038171003817119867119904minus1times119867119904 119889119905

1015840 = 1198681+ 1198682

(69)


119894119895(119910 119905minus1199051015840) as 120591 rarr 119905 119894 119895 =


1rarr 0 as 120591 rarr 119905


1198682= int

120591

119905

10038171003817100381710038171003817119878 (120591 minus 1199051015840)G (119880) (119905

1015840)10038171003817100381710038171003817119867119904minus1times119867119904 119889119905

1015840

le 119862int120591

119905

10038171003817100381710038171003817G (119880) (1199051015840)10038171003817100381710038171003817119867119904minus1times119867119904 119889119905

1015840

le 119862int120591

119905

119880119867119904minus1times119867119904 1198891199051015840


(70)



Ψ(X119872(119879)) sub X

119872(119879) Let 119880 = (V Φ)119879 isin X

119872(119879) For any

119905 isin [0 119879]


le int119905

0

10038171003817100381710038171003817119878 (119905 minus 1199051015840)G (119880) (119905

1015840)10038171003817100381710038171003817119867119904minus1times119867119904

le 119862int119905

0

10038171003817100381710038171003817G (119880) (1199051015840)10038171003817100381710038171003817119867119904minus1times119867119904

le 119862119879 sup119905isin[0119879]

119880 (119905)119867119904minus1times119867119904 le 119862119879 (119872 +

10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904)

le 119872

(71)



119872() Let 119880

1= (V

1 Φ1)119879

1198802= (V

2 Φ2)119879 isin X



119905

0

10038171003817100381710038171003817119878 (119905 minus 1199051015840)

sdot (G (1198801) (1199051015840) minusG (119880

2) (1199051015840))

10038171003817100381710038171003817119867119904minus1times119867119904 1198891199051015840

le 119862int119905

0


1015840 le int119905

0

(1

+1003817100381710038171003817Φ11003817100381710038171003817119867119904)1003817100381710038171003817V1 minus V2

1003817100381710038171003817119867119904minus1 + (1003817100381710038171003817V21003817100381710038171003817119867119904minus1

+1003817100381710038171003817Φ1 + Φ2

1003817100381710038171003817119867119904)1003817100381710038171003817Φ1 minus Φ2

1003817100381710038171003817119867119904 1198891199051015840 le 119862119879 [(1 +119872

+10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904)

1003817100381710038171003817V1 minus V21003817100381710038171003817119867119904minus1 + (3119872 + 3

10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904)

sdot1003817100381710038171003817Φ1 minus Φ2

1003817100381710038171003817119867119904] le 119862119879 [3119872 + 310038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904]



(72)


119862 [3119872 + 310038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904] lt 119872 (73)



119880 isin 119862 ([0 ] 119867119904minus1 times 119867119904) (74)


5 Numerical Schemes


(119868 minus120573

21205972120585) V119905=2120573

31205973120585119906 minus 120597

120585((1 +

120572V1198722) 119906)

(119868 minus120573

21205972120585)119906

119905

= 1205731205901205973120585V + 120573120590(minus

31198721015840

1198724V120585+ 3(1198721015840)

2

1198725V minus11987210158401015840

1198724V)

minus 120597120585(1

119872V) minus

120572

2120597120585(1

11987221199062)

(75)



120585 In




V (120585 119905) = sum119895

V119895(119905) 119890

119894119908119895120585

119906 (120585 119905) = sum119895

119895(119905) 119890

119894119908119895120585(76)

with

119908119895=2120587119895

119871 119895 = minus

119873

2+ 1 0

119873

2 (77)



V119895(119905) =

1

119871int119871

0

V (120585 119905) 119890minus119894119908119895120585119889120585 (78)




⟨119891 119892⟩ = int119871

0

119891 (120585) 119892 (120585) 119889120585 (79)

it follows that

V1015840119895(119905) = 120574

1119895+ 119873

1

1015840119895(119905) = 120574

2V119895+ 119873

2

(80)

where

1205741=minus (21205733) 1198941199083

119895

1 + (1205732)1199082119895

1205742=minus1205731205901198941199083

119895

1 + (1205732)1199082119895

1198731= minus119894119908119895119875119895[(1 + 120572V1198722) 119906]1 + (1205732)1199082

119895

1198732=119894119908119895119875119895[120573120590 (minus (311987210158401198724) V

120585+ (3 (1198721015840)

2


1 + (1205732)1199082119895

(81)


119875119895[119892] =

1

119871int119871

0

119892 (120585) 119890minus119894119908119895120585119889120585 (82)



V119899+1119895minus V119899

119895

Δ119905=1205741(119899+1119895+ 119899

119895)

2+3

21198731198991minus1

2119873119899minus11

119899+1119895minus 119899

119895

Δ119905=1205742(V119899+1119895+ V119899

119895)

2+3

21198731198992minus1

2119873119899minus12

(83)






119895 119899+1119895




120585are


V120585(120585 119905) = sum

119895

119894119908119895119875119895[V (sdot 119905)] 119890119894119908119895120585 (84)





V (120585 119905) = V (120585 minus 119888119905)

119906 (120585 119905) = (120585 minus 119888119905) (85)



(V10158401015840

11990610158401015840)

=1

(1205734) 1198882 minus 21205731205903(

119888

2

2

3

120590119888

2

)(119888V minus ((1 + 120572V) 119906)

119888119906 minus V minus120572

21199062)

(86)


V (119909) asymp V0+1198732

sum119899=1

V119899cos(119899120587

119897119909)

119906 (119909) asymp 1199060+1198732

sum119899=1

119906119899cos(119899120587

119897119909)

(87)

where

V0=1

119897int119897

0

V (119909) 119889119909 =1

2119897int2119897

0

V (119909) 119889119909

V119899=2

119897int119897

0

V (119909) cos(119899120587119909119897) 119889119909

=2

2119897int2119897

0

V (119909) cos(119899120587119909119897) 119889119909

(88)


119909119895=2119897 (119895 minus 1)

119873 119895 = 1

119873

2+ 1 (89)


119865 (V0 V1 V

1198732 1199060 1199061 119906

1198732) = 0 (90)




119869119894119895119865 (119909) asymp

119865119894(119909 + ℎ119890

119895) minus 119865

119894(119909 minus ℎ119890

119895)

2ℎ

119895 = 1 119873 + 2

(91)




1199060(119909) = cos(120587119909

119897)

V0(119909) = cos(120587119909

119897)

(92)


V = minus(119868 minus120573

21205972120585)Φ

119905+ 120573120590V

120585120585minus120572

2Φ2120585

= minusΦ119905+ 119874 (120572 120573 120590)

(93)


minus Φ119905119905+ Φ

120585120585+ 120573 (1 minus 120590)Φ

120585120585119905119905minus2120573

3Φ120585120585120585120585minus 2120572Φ

120585Φ120585119905

minus 120572Φ119905Φ120585120585= 0

(94)





3)Φ1015840101584010158401015840

+3120572119888

2(Φ10158402)

1015840

= 0

(96)



3)Φ101584010158401015840 +

3120572119888

2(Φ1015840)

2

= 0

(97)


(1 minus 1198882) (Φ1015840)2

+ 120573((1 minus 120590) 1198882 minus2

3) (Φ10158401015840)

2

+ 120572119888 (Φ1015840)3

= 0

(98)


V = 119888119906 minus120573119888

211990610158401015840 + 12057312059011988811990610158401015840 minus

120572

21199062 (99)



119860 =1198882 minus 1

120572119888

119861 = radic1198882 minus 1

4120573 ((1 minus 120590) 1198882 minus 23)

(101)



2)

sdot [41198601198612sech2 (119861119909) tanh2 (119861119909)

minus 21198601198612sech4 (119861119909)]

(102)






119906 (120585 0) = V (120585 0) = 119890minus20(119909minus28)2

(103)




v u

2 4 6 8 100x

0

02

04

06

08

1

12

14

2 4 6 8 100x

0

05

1

15


v u

minus1

minus05

0

05

2 4 6 8 100x

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

02

2 4 6 8 100x






1198622120590=1199082

1198962

=1 + (21205733) 1198962

(1 + (1205732) 1198962)2+1205731205901198962 (1 + (21205733) 1198962)

(1 + (1205732) 1198962)2

gt 11986220

(104)


0119890119894(119896120585minus119908119905)

V = V0119890119894(119896120585minus119908119905) where 119906

0 V0are constants


v u

minus02

0

02

04

06

08

1

12

14

16

2 4 6 8 100

x

0

02

04

06

08

1

12

14

2 4 6 8 100

x


v

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

2 4 6 8 100

x

u

minus25

minus2

minus15

minus1

minus05

0

2 4 6 8 100

x





120590is larger







at t = 10

0

05

1

15u at t = 10

0

02

04

06

08

1

12

14

2 4 6 8 100

120585

2 4 6 8 100

120585


v

minus01

0

01

03

02

04

05

06

07

45 50 55 6040

x

u

45 50 55 6040

x

minus01

0

01

03

02

04

05

06

07





119906 (120585 0) = V (120585 0) = 119890minus20(119909minus198)2 (105)


v

0

005

01

015

02

025

03

035

04

045

45 50 55 6040

x

u

0

005

01

015

02

025

03

035

04

45 50 55 6040

x


v

minus05

0

05

1

15

2

25

45 50 55 6040

x

u

minus05

0

05

1

15

2

45 50 55 6040

x




7 Conclusions



at t = 10 u at t = 10

minus01

0

01

02

03

04

05

06

07

minus01

0

01

02

03

04

05

06

07

60 65 7055

120585

60 65 7055

120585


u at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

20 30 40 5010

120585

20 30 40 5010

120585






minus04

minus03

minus02

minus01

0

01

02

03

04

05u at t = 20

35 40 45 50 55 6030120585

v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 55 6030120585


0 5 10 15 20 25 30 35 40 45 50

C2120590

C20

81205903

radic120573k

0

02

04

06

08

1

12

14

16





05060708091

1112131415

30 32 34 36 38 40 4228

x





u at t = 20

35 40 45 50 5530120585

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 5530120585


v at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

290 300 310 320280

120585

u at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

280 300 310 320290

120585




Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










A(V

Φ) = (

0 (119868 minus120573

21205972120585)minus1

(119868 minus2120573

31205972120585) (minus1205972

120585)

(119868 minus120573

21205972120585)minus1

1205731205901205972120585

0

)(V

Φ)

G(V

Φ) = (


21205972120585)minus1

(120572

1198722VΦ120585)120585

(119868 minus120573

21205972120585)minus1

(minus31205731205901198721015840

1198724V120585+ 3120573120590

(1198721015840)2

1198725V minus 120573120590

11987210158401015840

1198724V minus1

119872V minus

120572

21198722Φ2120585)

)

(41)


V (120585 0) = V0(120585)

Φ (120585 0) = Φ0(120585)

(42)


119905= 119860 (119910) + G (119880)

(119910 0) = 0(119910)

(43)

where

119880 = (V

Φ)

1198800= (

V0

Φ0

)

119860 (119910) = (0

1199102 (1 + (21205733) 1199102)

1 + (1205732) 1199102

minus1205731205901199102

1 + (1205732) 11991020

)

(44)


119905= 119860 (119910)

(119910 0) = 0(119910)

(45)


(119910 119905) = 119890119860(119910)1199050(119910) (46)

where

119890119860(119910)119905 = (11988611(119910 119905) 119886

12(119910 119905)

11988621(119910 119905) 119886

22(119910 119905)

)

= (cos (120582 (119910) 119905) Γ (119910) sin (120582 (119910) 119905)

Ψ (119910) sin (120582 (119910) 119905) cos (120582 (119910) 119905))

120582 (119910) =21199102radic120573120590 (3 + 21205731199102)

radic3 (2 + 1205731199102)

Γ (119910) =radic3 + 21205731199102

radic3120573120590

Ψ (119910) = minusradic3120573120590

radic3 + 21205731199102

(47)


119880 (119909 119905) = 119878 (119905) 1198800(119909) (48)

where

119878 (119905) 119891 = Fminus1 (119890119860(sdot)119905119891 (sdot)) (49)

with

119891 = (1198911

1198912

) (50)


is a 1198620-



Proof Let 119891 = (1198911 1198912)119879 Then

1003817100381710038171003817119878 (119905) 11989110038171003817100381710038172


=

10038171003817100381710038171003817100381710038171003817100381710038171003817

(Fminus1 (119886

111198911+ 119886

121198912)

Fminus1 (119886211198911+ 119886

221198912))

10038171003817100381710038171003817100381710038171003817100381710038171003817

2


=10038171003817100381710038171003817F

minus1 (119886111198911) +Fminus1 (119886

121198912)100381710038171003817100381710038172

119867119904minus2

+10038171003817100381710038171003817F

minus1 (119886211198911) +Fminus1 (119886

221198912)100381710038171003817100381710038172

119867119904minus1

le 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381611988611

10038161003816100381610038162 100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381611988612

10038161003816100381610038162 100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381611988621

10038161003816100381610038162 100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381611988622

10038161003816100381610038162 100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

le 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381610038161198911

100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus2

(3 + 21205731199102)100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 1

3 + 21205731199102100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381610038161198912

100381610038161003816100381610038162

119889119910

le 119862 (1003817100381710038171003817119891110038171003817100381710038172

119867119904minus2 +1003817100381710038171003817119891210038171003817100381710038172

119867119904minus1)

(51)


1003817100381710038171003817119878 (119905) 119891 minus 11989110038171003817100381710038172


le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(Fminus1 (119886

111198911+ 119886

121198912minus 119891

1)

Fminus1 (119886211198911+ 119886

221198912minus 119891

2))

10038171003817100381710038171003817100381710038171003817100381710038171003817

2


le 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381611988611 minus 1

10038161003816100381610038162 100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381611988612

10038161003816100381610038162 100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381611988621

10038161003816100381610038162 100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381611988622 minus 1

10038161003816100381610038162 100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

(52)


21(119910 119905) rarr 0 as 119905 rarr 0+ and

11988611(119910 119905) rarr 1 119886





On the other hand

1003817100381710038171003817A11989110038171003817100381710038172


R

(1 + 1199102)119904minus2

sdot100381610038161003816100381610038161003816100381610038161003816F((119868 minus

120573

21205972120585)minus1

(119868 minus2120573

31205972120585) 12059721205851198912)100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ intR

(1 + 1199102)119904minus1

sdot100381610038161003816100381610038161003816100381610038161003816F((119868 minus

120573

21205972120585)minus1

12057312059012059721205851198911)100381610038161003816100381610038161003816100381610038161003816

2

119889119910

le intR

(1 + 1199102)119904minus2(1 + (21205733) 1199102)

2

1199104

(1 + (1205732) 1199102)2

100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

+ intR

(1 + 1199102)119904minus1 120573212059021199104

(1 + (1205732) 1199102)2

100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

le 119862 (1003817100381710038171003817119891110038171003817100381710038172

119904minus1+1003817100381710038171003817119891210038171003817100381710038172

119904) = 119862

100381710038171003817100381711989110038171003817100381710038172

119867119904minus1times119867119904

(54)


10038171003817100381710038171003817100381710038171003817

119878 (ℎ) 119891 minus 119891

ℎminusA119891

10038171003817100381710038171003817100381710038171003817

2


=100381710038171003817100381710038171003817100381710038171003817Fminus1 (119890119860ℎ minus 119868

ℎ119891

minus 119860119891)100381710038171003817100381710038171003817100381710038171003817

2


=

10038171003817100381710038171003817100381710038171003817100381710038171003817

Fminus1 (11988611(sdot ℎ) 119891

1+ 119886

12(sdot ℎ) 119891

2minus 119891

1)

ℎminus (119868

minus120573

21205972120585)minus1

(119868 minus2120573

31205972120585) (minus1205972

1205851198912)

10038171003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+

10038171003817100381710038171003817100381710038171003817100381710038171003817

Fminus1 (11988621(sdot ℎ) 119891

1+ 119886

22(sdot ℎ) 119891

2minus 119891

2)

ℎminus (119868

minus120573

21205972120585)minus1

12057312059012059721205851198911

10038171003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus1

= intR

(1 + 1199102)119904minus2


sdot

10038161003816100381610038161003816100381610038161003816100381610038161003816

(cos (120582ℎ) minus 1) 1198911+ Γ sin (120582ℎ) 119891

2

ℎ

minus1199102 (1 + (21205733) 1199102) 119891

2

1 + (1205732) 1199102

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910 + intR

(1 + 1199102)119904minus1

sdot

1003816100381610038161003816100381610038161003816100381610038161003816

Ψ sin (120582ℎ) 1198911+ (cos (120582ℎ) minus 1) 119891

2

ℎ

minusminus1205731205901199102119891

1

1 + (1205732) 1199102

1003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

(55)

But by virtue of

limℎrarr0


= 0

limℎrarr0


= Γ120582 =1199102 (1 + (21205733) 1199102)

1 + (1205732) 1199102

limℎrarr0


= Ψ120582 = minus1205731205901199102

1 + (1205732) 1199102

(56)


limℎrarr0

10038171003817100381710038171003817100381710038171003817

119878 (ℎ) 119891 minus 119891

ℎminusA119891

10038171003817100381710038171003817100381710038171003817

2


= 0 (57)



G (119880)119867119904minus1times119867119904 le 119862 119880

119867119904minus1times119867119904 (58)

Furthermore

1003817100381710038171003817G (1198801) minusG (1198802)10038171003817100381710038172

119867119904minus1times119867119904

le 119862 [(1 +1003817100381710038171003817Φ110038171003817100381710038172

119867119904)1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1

+ (1003817100381710038171003817V210038171003817100381710038172

119867119904minus1 +1003817100381710038171003817Φ1 + Φ2

10038171003817100381710038172

119867119904)1003817100381710038171003817Φ1 minus Φ2

10038171003817100381710038172

119867119904]

(59)


V119894Φ119894) 119894 = 1 2


G (119880)2

119867119904minus1times119867119904

le 119862intR

(1 + 1199102)119904minus1

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

2 100381610038161003816100381610038161003816100381610038161003816

(

V1198722Φ120585)120585

100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

210038161003816100381610038161003816100381610038161003816100381610038161003816

1015840

1198724V120585

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

2

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(1198721015840)

2

1198725V

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

210038161003816100381610038161003816100381610038161003816100381610038161003816

10158401015840

1198724V10038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

2 10038161003816100381610038161003816100381610038161003816

V119872

10038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

2100381610038161003816100381610038161003816100381610038161003816100381610038161003816

Φ2120585

1198722

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

(60)


1003817100381710038171003817100381710038171003817

V1198722Φ120585

1003817100381710038171003817100381710038171003817119867119904minus2le 1198621003817100381710038171003817100381710038171003817

V1198722

1003817100381710038171003817100381710038171003817119867119904minus210038171003817100381710038171003817Φ12058510038171003817100381710038171003817119867119904minus1

le 1198621003817100381710038171003817100381710038171003817

1

1198722

1003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1 Φ

119867119904

100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724V120585

100381710038171003817100381710038171003817100381710038171003817119867119904minus2le 119862100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724

100381710038171003817100381710038171003817100381710038171003817119867119904minus1

10038171003817100381710038171003817V12058510038171003817100381710038171003817119867119904minus2

le 119862100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724

100381710038171003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725V100381710038171003817100381710038171003817100381710038171003817100381710038171003817119867119904minus2

le 119862

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725

100381710038171003817100381710038171003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus2

le 119862

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725

100381710038171003817100381710038171003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1

100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724V100381710038171003817100381710038171003817100381710038171003817119867119904minus2

le 119862100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724

100381710038171003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1

1003817100381710038171003817100381710038171003817

V119872

1003817100381710038171003817100381710038171003817119867119904minus2le 1198621003817100381710038171003817100381710038171003817

1

119872

1003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus2

le 1198621003817100381710038171003817100381710038171003817

1

119872

1003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1


1003817100381710038171003817100381710038171003817100381710038171003817

Φ2120585

1198722

1003817100381710038171003817100381710038171003817100381710038171003817119867119904minus2le 11986210038171003817100381710038171003817100381710038171003817

Φ120585

1198722Φ120585

10038171003817100381710038171003817100381710038171003817119867119904minus2

le 11986210038171003817100381710038171003817Φ12058510038171003817100381710038171003817119867119904minus1

10038171003817100381710038171003817100381710038171003817

Φ120585

1198722

10038171003817100381710038171003817100381710038171003817119867119904minus2

le 119862 Φ119904

1003817100381710038171003817100381710038171003817

1

1198722

1003817100381710038171003817100381710038171003817119867119904minus110038171003817100381710038171003817Φ12058510038171003817100381710038171003817119867119904minus2

le 119862 Φ119904

1003817100381710038171003817100381710038171003817

1

1198722

1003817100381710038171003817100381710038171003817119867119904minus1Φ

119867119904

(61)


G (119880)2

119867119904minus1times119867119904 le 119862(

1003817100381710038171003817100381710038171003817

V1198722Φ120585

1003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724V120585

100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725V

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724V100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+1003817100381710038171003817100381710038171003817

V119872

1003817100381710038171003817100381710038171003817

2

119867119904minus2

+

1003817100381710038171003817100381710038171003817100381710038171003817

Φ2120585

1198722

1003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

) le 119862(V2119867119904minus1 + Φ

2

119867119904)

(62)



that

1003817100381710038171003817100381710038171003817100381710038171003817G(

V1

Φ1

) minusG(V2

Φ2

)

1003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus1times119867119904

le 119862(1003817100381710038171003817100381710038171003817

V1

1198722Φ1120585

minusV2

1198722Φ2120585

1003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724(V1minus V

2)120585

100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725(V1minus V

2)

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724(V1minus V

2)100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+1003817100381710038171003817100381710038171003817

1

119872(V1minus V

2)1003817100381710038171003817100381710038171003817

2

119867119904minus2

+1003817100381710038171003817100381710038171003817

1

1198722(Φ21120585minus Φ2

2120585)1003817100381710038171003817100381710038171003817

2

119867119904minus2

)

le 119862(10038171003817100381710038171003817V1Φ1120585 minus V2Φ2120585

100381710038171003817100381710038172

119867119904minus2+1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1 +10038171003817100381710038171003817Φ2

1120585

minus Φ22120585

100381710038171003817100381710038172

119867119904minus2) le 119862 [(

10038171003817100381710038171003817(V1 minus V2)Φ112058510038171003817100381710038171003817119867119904minus2

+10038171003817100381710038171003817V2 (Φ1120585 minus Φ2120585)

10038171003817100381710038171003817119867119904minus2)2

+1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1 +10038171003817100381710038171003817Φ1120585

+ Φ2120585

100381710038171003817100381710038172

119867119904minus1

10038171003817100381710038171003817Φ1120585 minus Φ2120585100381710038171003817100381710038172

119867119904minus1] le 119862 [

1003817100381710038171003817V1 minus V210038171003817100381710038172

119867119904minus1

sdot10038171003817100381710038171003817Φ1120585100381710038171003817100381710038172

119867119904minus2+1003817100381710038171003817V210038171003817100381710038172

119867119904minus1

10038171003817100381710038171003817Φ1120585 minus Φ2120585100381710038171003817100381710038172

119904minus2

+1003817100381710038171003817V1

minus V2

10038171003817100381710038172

119867119904minus1 +1003817100381710038171003817Φ1 + Φ2

10038171003817100381710038172

119867119904

1003817100381710038171003817Φ1 minus Φ210038171003817100381710038172

119867119904] le 119862 [(1

+1003817100381710038171003817Φ110038171003817100381710038172

119867119904)1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1 + (

1003817100381710038171003817V210038171003817100381710038172

119867119904minus1

+1003817100381710038171003817Φ1 + Φ2

10038171003817100381710038172

119867119904)1003817100381710038171003817Φ1 minus Φ2

10038171003817100381710038172

119867119904]

(63)




119880 (119905) = 119878 (119905) 120601 + int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840 (64)


(Ψ119880) (119905) = 119878 (119905) 120601 + int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840 (65)


X119872(119879) = 119880 isin 119862 ([0 119879] 119867

119904minus1 times 119867119904)

sup119905isin[0119879]


(66)




le1003817100381710038171003817(119878 (119905) minus 119878 (120591)) 120601

1003817100381710038171003817119867119904minus1times119867119904

+10038171003817100381710038171003817100381710038171003817int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840

minus 119878 (120591 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840

10038171003817100381710038171003817100381710038171003817119867119904minus1times119867119904

(67)



1003817100381710038171003817119867119904minus1times119867119904

+1003817100381710038171003817119878 (119905) 120601

1003817100381710038171003817119867119904minus1times119867119904

le 119872 +10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904

(68)


Given that (119878(119905))119905ge0




10038171003817100381710038171003817100381710038171003817int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840 minus int

120591

0

119878 (120591 minus 1199051015840)G (119880)

sdot (1199051015840) 119889119905101584010038171003817100381710038171003817100381710038171003817119867119904minus1times119867119904

le int119905

0



1015840 + int120591

119905

10038171003817100381710038171003817119878 (120591 minus 1199051015840)G (119880)

sdot (1199051015840)10038171003817100381710038171003817119867119904minus1times119867119904 119889119905

1015840 = 1198681+ 1198682

(69)


119894119895(119910 119905minus1199051015840) as 120591 rarr 119905 119894 119895 =


1rarr 0 as 120591 rarr 119905


1198682= int

120591

119905

10038171003817100381710038171003817119878 (120591 minus 1199051015840)G (119880) (119905

1015840)10038171003817100381710038171003817119867119904minus1times119867119904 119889119905

1015840

le 119862int120591

119905

10038171003817100381710038171003817G (119880) (1199051015840)10038171003817100381710038171003817119867119904minus1times119867119904 119889119905

1015840

le 119862int120591

119905

119880119867119904minus1times119867119904 1198891199051015840


(70)



Ψ(X119872(119879)) sub X

119872(119879) Let 119880 = (V Φ)119879 isin X

119872(119879) For any

119905 isin [0 119879]


le int119905

0

10038171003817100381710038171003817119878 (119905 minus 1199051015840)G (119880) (119905

1015840)10038171003817100381710038171003817119867119904minus1times119867119904

le 119862int119905

0

10038171003817100381710038171003817G (119880) (1199051015840)10038171003817100381710038171003817119867119904minus1times119867119904

le 119862119879 sup119905isin[0119879]

119880 (119905)119867119904minus1times119867119904 le 119862119879 (119872 +

10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904)

le 119872

(71)



119872() Let 119880

1= (V

1 Φ1)119879

1198802= (V

2 Φ2)119879 isin X



119905

0

10038171003817100381710038171003817119878 (119905 minus 1199051015840)

sdot (G (1198801) (1199051015840) minusG (119880

2) (1199051015840))

10038171003817100381710038171003817119867119904minus1times119867119904 1198891199051015840

le 119862int119905

0


1015840 le int119905

0

(1

+1003817100381710038171003817Φ11003817100381710038171003817119867119904)1003817100381710038171003817V1 minus V2

1003817100381710038171003817119867119904minus1 + (1003817100381710038171003817V21003817100381710038171003817119867119904minus1

+1003817100381710038171003817Φ1 + Φ2

1003817100381710038171003817119867119904)1003817100381710038171003817Φ1 minus Φ2

1003817100381710038171003817119867119904 1198891199051015840 le 119862119879 [(1 +119872

+10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904)

1003817100381710038171003817V1 minus V21003817100381710038171003817119867119904minus1 + (3119872 + 3

10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904)

sdot1003817100381710038171003817Φ1 minus Φ2

1003817100381710038171003817119867119904] le 119862119879 [3119872 + 310038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904]



(72)


119862 [3119872 + 310038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904] lt 119872 (73)



119880 isin 119862 ([0 ] 119867119904minus1 times 119867119904) (74)


5 Numerical Schemes


(119868 minus120573

21205972120585) V119905=2120573

31205973120585119906 minus 120597

120585((1 +

120572V1198722) 119906)

(119868 minus120573

21205972120585)119906

119905

= 1205731205901205973120585V + 120573120590(minus

31198721015840

1198724V120585+ 3(1198721015840)

2

1198725V minus11987210158401015840

1198724V)

minus 120597120585(1

119872V) minus

120572

2120597120585(1

11987221199062)

(75)



120585 In




V (120585 119905) = sum119895

V119895(119905) 119890

119894119908119895120585

119906 (120585 119905) = sum119895

119895(119905) 119890

119894119908119895120585(76)

with

119908119895=2120587119895

119871 119895 = minus

119873

2+ 1 0

119873

2 (77)



V119895(119905) =

1

119871int119871

0

V (120585 119905) 119890minus119894119908119895120585119889120585 (78)




⟨119891 119892⟩ = int119871

0

119891 (120585) 119892 (120585) 119889120585 (79)

it follows that

V1015840119895(119905) = 120574

1119895+ 119873

1

1015840119895(119905) = 120574

2V119895+ 119873

2

(80)

where

1205741=minus (21205733) 1198941199083

119895

1 + (1205732)1199082119895

1205742=minus1205731205901198941199083

119895

1 + (1205732)1199082119895

1198731= minus119894119908119895119875119895[(1 + 120572V1198722) 119906]1 + (1205732)1199082

119895

1198732=119894119908119895119875119895[120573120590 (minus (311987210158401198724) V

120585+ (3 (1198721015840)

2


1 + (1205732)1199082119895

(81)


119875119895[119892] =

1

119871int119871

0

119892 (120585) 119890minus119894119908119895120585119889120585 (82)



V119899+1119895minus V119899

119895

Δ119905=1205741(119899+1119895+ 119899

119895)

2+3

21198731198991minus1

2119873119899minus11

119899+1119895minus 119899

119895

Δ119905=1205742(V119899+1119895+ V119899

119895)

2+3

21198731198992minus1

2119873119899minus12

(83)






119895 119899+1119895




120585are


V120585(120585 119905) = sum

119895

119894119908119895119875119895[V (sdot 119905)] 119890119894119908119895120585 (84)





V (120585 119905) = V (120585 minus 119888119905)

119906 (120585 119905) = (120585 minus 119888119905) (85)



(V10158401015840

11990610158401015840)

=1

(1205734) 1198882 minus 21205731205903(

119888

2

2

3

120590119888

2

)(119888V minus ((1 + 120572V) 119906)

119888119906 minus V minus120572

21199062)

(86)


V (119909) asymp V0+1198732

sum119899=1

V119899cos(119899120587

119897119909)

119906 (119909) asymp 1199060+1198732

sum119899=1

119906119899cos(119899120587

119897119909)

(87)

where

V0=1

119897int119897

0

V (119909) 119889119909 =1

2119897int2119897

0

V (119909) 119889119909

V119899=2

119897int119897

0

V (119909) cos(119899120587119909119897) 119889119909

=2

2119897int2119897

0

V (119909) cos(119899120587119909119897) 119889119909

(88)


119909119895=2119897 (119895 minus 1)

119873 119895 = 1

119873

2+ 1 (89)


119865 (V0 V1 V

1198732 1199060 1199061 119906

1198732) = 0 (90)




119869119894119895119865 (119909) asymp

119865119894(119909 + ℎ119890

119895) minus 119865

119894(119909 minus ℎ119890

119895)

2ℎ

119895 = 1 119873 + 2

(91)




1199060(119909) = cos(120587119909

119897)

V0(119909) = cos(120587119909

119897)

(92)


V = minus(119868 minus120573

21205972120585)Φ

119905+ 120573120590V

120585120585minus120572

2Φ2120585

= minusΦ119905+ 119874 (120572 120573 120590)

(93)


minus Φ119905119905+ Φ

120585120585+ 120573 (1 minus 120590)Φ

120585120585119905119905minus2120573

3Φ120585120585120585120585minus 2120572Φ

120585Φ120585119905

minus 120572Φ119905Φ120585120585= 0

(94)





3)Φ1015840101584010158401015840

+3120572119888

2(Φ10158402)

1015840

= 0

(96)



3)Φ101584010158401015840 +

3120572119888

2(Φ1015840)

2

= 0

(97)


(1 minus 1198882) (Φ1015840)2

+ 120573((1 minus 120590) 1198882 minus2

3) (Φ10158401015840)

2

+ 120572119888 (Φ1015840)3

= 0

(98)


V = 119888119906 minus120573119888

211990610158401015840 + 12057312059011988811990610158401015840 minus

120572

21199062 (99)



119860 =1198882 minus 1

120572119888

119861 = radic1198882 minus 1

4120573 ((1 minus 120590) 1198882 minus 23)

(101)



2)

sdot [41198601198612sech2 (119861119909) tanh2 (119861119909)

minus 21198601198612sech4 (119861119909)]

(102)






119906 (120585 0) = V (120585 0) = 119890minus20(119909minus28)2

(103)




v u

2 4 6 8 100x

0

02

04

06

08

1

12

14

2 4 6 8 100x

0

05

1

15


v u

minus1

minus05

0

05

2 4 6 8 100x

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

02

2 4 6 8 100x






1198622120590=1199082

1198962

=1 + (21205733) 1198962

(1 + (1205732) 1198962)2+1205731205901198962 (1 + (21205733) 1198962)

(1 + (1205732) 1198962)2

gt 11986220

(104)


0119890119894(119896120585minus119908119905)

V = V0119890119894(119896120585minus119908119905) where 119906

0 V0are constants


v u

minus02

0

02

04

06

08

1

12

14

16

2 4 6 8 100

x

0

02

04

06

08

1

12

14

2 4 6 8 100

x


v

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

2 4 6 8 100

x

u

minus25

minus2

minus15

minus1

minus05

0

2 4 6 8 100

x





120590is larger







at t = 10

0

05

1

15u at t = 10

0

02

04

06

08

1

12

14

2 4 6 8 100

120585

2 4 6 8 100

120585


v

minus01

0

01

03

02

04

05

06

07

45 50 55 6040

x

u

45 50 55 6040

x

minus01

0

01

03

02

04

05

06

07





119906 (120585 0) = V (120585 0) = 119890minus20(119909minus198)2 (105)


v

0

005

01

015

02

025

03

035

04

045

45 50 55 6040

x

u

0

005

01

015

02

025

03

035

04

45 50 55 6040

x


v

minus05

0

05

1

15

2

25

45 50 55 6040

x

u

minus05

0

05

1

15

2

45 50 55 6040

x




7 Conclusions



at t = 10 u at t = 10

minus01

0

01

02

03

04

05

06

07

minus01

0

01

02

03

04

05

06

07

60 65 7055

120585

60 65 7055

120585


u at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

20 30 40 5010

120585

20 30 40 5010

120585






minus04

minus03

minus02

minus01

0

01

02

03

04

05u at t = 20

35 40 45 50 55 6030120585

v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 55 6030120585


0 5 10 15 20 25 30 35 40 45 50

C2120590

C20

81205903

radic120573k

0

02

04

06

08

1

12

14

16





05060708091

1112131415

30 32 34 36 38 40 4228

x





u at t = 20

35 40 45 50 5530120585

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 5530120585


v at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

290 300 310 320280

120585

u at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

280 300 310 320290

120585




Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Proof Let 119891 = (1198911 1198912)119879 Then

1003817100381710038171003817119878 (119905) 11989110038171003817100381710038172


=

10038171003817100381710038171003817100381710038171003817100381710038171003817

(Fminus1 (119886

111198911+ 119886

121198912)

Fminus1 (119886211198911+ 119886

221198912))

10038171003817100381710038171003817100381710038171003817100381710038171003817

2


=10038171003817100381710038171003817F

minus1 (119886111198911) +Fminus1 (119886

121198912)100381710038171003817100381710038172

119867119904minus2

+10038171003817100381710038171003817F

minus1 (119886211198911) +Fminus1 (119886

221198912)100381710038171003817100381710038172

119867119904minus1

le 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381611988611

10038161003816100381610038162 100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381611988612

10038161003816100381610038162 100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381611988621

10038161003816100381610038162 100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381611988622

10038161003816100381610038162 100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

le 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381610038161198911

100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus2

(3 + 21205731199102)100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 1

3 + 21205731199102100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381610038161198912

100381610038161003816100381610038162

119889119910

le 119862 (1003817100381710038171003817119891110038171003817100381710038172

119867119904minus2 +1003817100381710038171003817119891210038171003817100381710038172

119867119904minus1)

(51)


1003817100381710038171003817119878 (119905) 119891 minus 11989110038171003817100381710038172


le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(Fminus1 (119886

111198911+ 119886

121198912minus 119891

1)

Fminus1 (119886211198911+ 119886

221198912minus 119891

2))

10038171003817100381710038171003817100381710038171003817100381710038171003817

2


le 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381611988611 minus 1

10038161003816100381610038162 100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus2 100381610038161003816100381611988612

10038161003816100381610038162 100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381611988621

10038161003816100381610038162 100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

+ 119862intR

(1 + 1199102)119904minus1 100381610038161003816100381611988622 minus 1

10038161003816100381610038162 100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

(52)


21(119910 119905) rarr 0 as 119905 rarr 0+ and

11988611(119910 119905) rarr 1 119886





On the other hand

1003817100381710038171003817A11989110038171003817100381710038172


R

(1 + 1199102)119904minus2

sdot100381610038161003816100381610038161003816100381610038161003816F((119868 minus

120573

21205972120585)minus1

(119868 minus2120573

31205972120585) 12059721205851198912)100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ intR

(1 + 1199102)119904minus1

sdot100381610038161003816100381610038161003816100381610038161003816F((119868 minus

120573

21205972120585)minus1

12057312059012059721205851198911)100381610038161003816100381610038161003816100381610038161003816

2

119889119910

le intR

(1 + 1199102)119904minus2(1 + (21205733) 1199102)

2

1199104

(1 + (1205732) 1199102)2

100381610038161003816100381610038161198912100381610038161003816100381610038162

119889119910

+ intR

(1 + 1199102)119904minus1 120573212059021199104

(1 + (1205732) 1199102)2

100381610038161003816100381610038161198911100381610038161003816100381610038162

119889119910

le 119862 (1003817100381710038171003817119891110038171003817100381710038172

119904minus1+1003817100381710038171003817119891210038171003817100381710038172

119904) = 119862

100381710038171003817100381711989110038171003817100381710038172

119867119904minus1times119867119904

(54)


10038171003817100381710038171003817100381710038171003817

119878 (ℎ) 119891 minus 119891

ℎminusA119891

10038171003817100381710038171003817100381710038171003817

2


=100381710038171003817100381710038171003817100381710038171003817Fminus1 (119890119860ℎ minus 119868

ℎ119891

minus 119860119891)100381710038171003817100381710038171003817100381710038171003817

2


=

10038171003817100381710038171003817100381710038171003817100381710038171003817

Fminus1 (11988611(sdot ℎ) 119891

1+ 119886

12(sdot ℎ) 119891

2minus 119891

1)

ℎminus (119868

minus120573

21205972120585)minus1

(119868 minus2120573

31205972120585) (minus1205972

1205851198912)

10038171003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+

10038171003817100381710038171003817100381710038171003817100381710038171003817

Fminus1 (11988621(sdot ℎ) 119891

1+ 119886

22(sdot ℎ) 119891

2minus 119891

2)

ℎminus (119868

minus120573

21205972120585)minus1

12057312059012059721205851198911

10038171003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus1

= intR

(1 + 1199102)119904minus2


sdot

10038161003816100381610038161003816100381610038161003816100381610038161003816

(cos (120582ℎ) minus 1) 1198911+ Γ sin (120582ℎ) 119891

2

ℎ

minus1199102 (1 + (21205733) 1199102) 119891

2

1 + (1205732) 1199102

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910 + intR

(1 + 1199102)119904minus1

sdot

1003816100381610038161003816100381610038161003816100381610038161003816

Ψ sin (120582ℎ) 1198911+ (cos (120582ℎ) minus 1) 119891

2

ℎ

minusminus1205731205901199102119891

1

1 + (1205732) 1199102

1003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

(55)

But by virtue of

limℎrarr0


= 0

limℎrarr0


= Γ120582 =1199102 (1 + (21205733) 1199102)

1 + (1205732) 1199102

limℎrarr0


= Ψ120582 = minus1205731205901199102

1 + (1205732) 1199102

(56)


limℎrarr0

10038171003817100381710038171003817100381710038171003817

119878 (ℎ) 119891 minus 119891

ℎminusA119891

10038171003817100381710038171003817100381710038171003817

2


= 0 (57)



G (119880)119867119904minus1times119867119904 le 119862 119880

119867119904minus1times119867119904 (58)

Furthermore

1003817100381710038171003817G (1198801) minusG (1198802)10038171003817100381710038172

119867119904minus1times119867119904

le 119862 [(1 +1003817100381710038171003817Φ110038171003817100381710038172

119867119904)1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1

+ (1003817100381710038171003817V210038171003817100381710038172

119867119904minus1 +1003817100381710038171003817Φ1 + Φ2

10038171003817100381710038172

119867119904)1003817100381710038171003817Φ1 minus Φ2

10038171003817100381710038172

119867119904]

(59)


V119894Φ119894) 119894 = 1 2


G (119880)2

119867119904minus1times119867119904

le 119862intR

(1 + 1199102)119904minus1

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

2 100381610038161003816100381610038161003816100381610038161003816

(

V1198722Φ120585)120585

100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

210038161003816100381610038161003816100381610038161003816100381610038161003816

1015840

1198724V120585

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

2

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(1198721015840)

2

1198725V

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

210038161003816100381610038161003816100381610038161003816100381610038161003816

10158401015840

1198724V10038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

2 10038161003816100381610038161003816100381610038161003816

V119872

10038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

2100381610038161003816100381610038161003816100381610038161003816100381610038161003816

Φ2120585

1198722

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

(60)


1003817100381710038171003817100381710038171003817

V1198722Φ120585

1003817100381710038171003817100381710038171003817119867119904minus2le 1198621003817100381710038171003817100381710038171003817

V1198722

1003817100381710038171003817100381710038171003817119867119904minus210038171003817100381710038171003817Φ12058510038171003817100381710038171003817119867119904minus1

le 1198621003817100381710038171003817100381710038171003817

1

1198722

1003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1 Φ

119867119904

100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724V120585

100381710038171003817100381710038171003817100381710038171003817119867119904minus2le 119862100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724

100381710038171003817100381710038171003817100381710038171003817119867119904minus1

10038171003817100381710038171003817V12058510038171003817100381710038171003817119867119904minus2

le 119862100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724

100381710038171003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725V100381710038171003817100381710038171003817100381710038171003817100381710038171003817119867119904minus2

le 119862

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725

100381710038171003817100381710038171003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus2

le 119862

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725

100381710038171003817100381710038171003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1

100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724V100381710038171003817100381710038171003817100381710038171003817119867119904minus2

le 119862100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724

100381710038171003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1

1003817100381710038171003817100381710038171003817

V119872

1003817100381710038171003817100381710038171003817119867119904minus2le 1198621003817100381710038171003817100381710038171003817

1

119872

1003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus2

le 1198621003817100381710038171003817100381710038171003817

1

119872

1003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1


1003817100381710038171003817100381710038171003817100381710038171003817

Φ2120585

1198722

1003817100381710038171003817100381710038171003817100381710038171003817119867119904minus2le 11986210038171003817100381710038171003817100381710038171003817

Φ120585

1198722Φ120585

10038171003817100381710038171003817100381710038171003817119867119904minus2

le 11986210038171003817100381710038171003817Φ12058510038171003817100381710038171003817119867119904minus1

10038171003817100381710038171003817100381710038171003817

Φ120585

1198722

10038171003817100381710038171003817100381710038171003817119867119904minus2

le 119862 Φ119904

1003817100381710038171003817100381710038171003817

1

1198722

1003817100381710038171003817100381710038171003817119867119904minus110038171003817100381710038171003817Φ12058510038171003817100381710038171003817119867119904minus2

le 119862 Φ119904

1003817100381710038171003817100381710038171003817

1

1198722

1003817100381710038171003817100381710038171003817119867119904minus1Φ

119867119904

(61)


G (119880)2

119867119904minus1times119867119904 le 119862(

1003817100381710038171003817100381710038171003817

V1198722Φ120585

1003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724V120585

100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725V

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724V100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+1003817100381710038171003817100381710038171003817

V119872

1003817100381710038171003817100381710038171003817

2

119867119904minus2

+

1003817100381710038171003817100381710038171003817100381710038171003817

Φ2120585

1198722

1003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

) le 119862(V2119867119904minus1 + Φ

2

119867119904)

(62)



that

1003817100381710038171003817100381710038171003817100381710038171003817G(

V1

Φ1

) minusG(V2

Φ2

)

1003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus1times119867119904

le 119862(1003817100381710038171003817100381710038171003817

V1

1198722Φ1120585

minusV2

1198722Φ2120585

1003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724(V1minus V

2)120585

100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725(V1minus V

2)

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724(V1minus V

2)100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+1003817100381710038171003817100381710038171003817

1

119872(V1minus V

2)1003817100381710038171003817100381710038171003817

2

119867119904minus2

+1003817100381710038171003817100381710038171003817

1

1198722(Φ21120585minus Φ2

2120585)1003817100381710038171003817100381710038171003817

2

119867119904minus2

)

le 119862(10038171003817100381710038171003817V1Φ1120585 minus V2Φ2120585

100381710038171003817100381710038172

119867119904minus2+1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1 +10038171003817100381710038171003817Φ2

1120585

minus Φ22120585

100381710038171003817100381710038172

119867119904minus2) le 119862 [(

10038171003817100381710038171003817(V1 minus V2)Φ112058510038171003817100381710038171003817119867119904minus2

+10038171003817100381710038171003817V2 (Φ1120585 minus Φ2120585)

10038171003817100381710038171003817119867119904minus2)2

+1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1 +10038171003817100381710038171003817Φ1120585

+ Φ2120585

100381710038171003817100381710038172

119867119904minus1

10038171003817100381710038171003817Φ1120585 minus Φ2120585100381710038171003817100381710038172

119867119904minus1] le 119862 [

1003817100381710038171003817V1 minus V210038171003817100381710038172

119867119904minus1

sdot10038171003817100381710038171003817Φ1120585100381710038171003817100381710038172

119867119904minus2+1003817100381710038171003817V210038171003817100381710038172

119867119904minus1

10038171003817100381710038171003817Φ1120585 minus Φ2120585100381710038171003817100381710038172

119904minus2

+1003817100381710038171003817V1

minus V2

10038171003817100381710038172

119867119904minus1 +1003817100381710038171003817Φ1 + Φ2

10038171003817100381710038172

119867119904

1003817100381710038171003817Φ1 minus Φ210038171003817100381710038172

119867119904] le 119862 [(1

+1003817100381710038171003817Φ110038171003817100381710038172

119867119904)1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1 + (

1003817100381710038171003817V210038171003817100381710038172

119867119904minus1

+1003817100381710038171003817Φ1 + Φ2

10038171003817100381710038172

119867119904)1003817100381710038171003817Φ1 minus Φ2

10038171003817100381710038172

119867119904]

(63)




119880 (119905) = 119878 (119905) 120601 + int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840 (64)


(Ψ119880) (119905) = 119878 (119905) 120601 + int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840 (65)


X119872(119879) = 119880 isin 119862 ([0 119879] 119867

119904minus1 times 119867119904)

sup119905isin[0119879]


(66)




le1003817100381710038171003817(119878 (119905) minus 119878 (120591)) 120601

1003817100381710038171003817119867119904minus1times119867119904

+10038171003817100381710038171003817100381710038171003817int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840

minus 119878 (120591 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840

10038171003817100381710038171003817100381710038171003817119867119904minus1times119867119904

(67)



1003817100381710038171003817119867119904minus1times119867119904

+1003817100381710038171003817119878 (119905) 120601

1003817100381710038171003817119867119904minus1times119867119904

le 119872 +10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904

(68)


Given that (119878(119905))119905ge0




10038171003817100381710038171003817100381710038171003817int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840 minus int

120591

0

119878 (120591 minus 1199051015840)G (119880)

sdot (1199051015840) 119889119905101584010038171003817100381710038171003817100381710038171003817119867119904minus1times119867119904

le int119905

0



1015840 + int120591

119905

10038171003817100381710038171003817119878 (120591 minus 1199051015840)G (119880)

sdot (1199051015840)10038171003817100381710038171003817119867119904minus1times119867119904 119889119905

1015840 = 1198681+ 1198682

(69)


119894119895(119910 119905minus1199051015840) as 120591 rarr 119905 119894 119895 =


1rarr 0 as 120591 rarr 119905


1198682= int

120591

119905

10038171003817100381710038171003817119878 (120591 minus 1199051015840)G (119880) (119905

1015840)10038171003817100381710038171003817119867119904minus1times119867119904 119889119905

1015840

le 119862int120591

119905

10038171003817100381710038171003817G (119880) (1199051015840)10038171003817100381710038171003817119867119904minus1times119867119904 119889119905

1015840

le 119862int120591

119905

119880119867119904minus1times119867119904 1198891199051015840


(70)



Ψ(X119872(119879)) sub X

119872(119879) Let 119880 = (V Φ)119879 isin X

119872(119879) For any

119905 isin [0 119879]


le int119905

0

10038171003817100381710038171003817119878 (119905 minus 1199051015840)G (119880) (119905

1015840)10038171003817100381710038171003817119867119904minus1times119867119904

le 119862int119905

0

10038171003817100381710038171003817G (119880) (1199051015840)10038171003817100381710038171003817119867119904minus1times119867119904

le 119862119879 sup119905isin[0119879]

119880 (119905)119867119904minus1times119867119904 le 119862119879 (119872 +

10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904)

le 119872

(71)



119872() Let 119880

1= (V

1 Φ1)119879

1198802= (V

2 Φ2)119879 isin X



119905

0

10038171003817100381710038171003817119878 (119905 minus 1199051015840)

sdot (G (1198801) (1199051015840) minusG (119880

2) (1199051015840))

10038171003817100381710038171003817119867119904minus1times119867119904 1198891199051015840

le 119862int119905

0


1015840 le int119905

0

(1

+1003817100381710038171003817Φ11003817100381710038171003817119867119904)1003817100381710038171003817V1 minus V2

1003817100381710038171003817119867119904minus1 + (1003817100381710038171003817V21003817100381710038171003817119867119904minus1

+1003817100381710038171003817Φ1 + Φ2

1003817100381710038171003817119867119904)1003817100381710038171003817Φ1 minus Φ2

1003817100381710038171003817119867119904 1198891199051015840 le 119862119879 [(1 +119872

+10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904)

1003817100381710038171003817V1 minus V21003817100381710038171003817119867119904minus1 + (3119872 + 3

10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904)

sdot1003817100381710038171003817Φ1 minus Φ2

1003817100381710038171003817119867119904] le 119862119879 [3119872 + 310038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904]



(72)


119862 [3119872 + 310038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904] lt 119872 (73)



119880 isin 119862 ([0 ] 119867119904minus1 times 119867119904) (74)


5 Numerical Schemes


(119868 minus120573

21205972120585) V119905=2120573

31205973120585119906 minus 120597

120585((1 +

120572V1198722) 119906)

(119868 minus120573

21205972120585)119906

119905

= 1205731205901205973120585V + 120573120590(minus

31198721015840

1198724V120585+ 3(1198721015840)

2

1198725V minus11987210158401015840

1198724V)

minus 120597120585(1

119872V) minus

120572

2120597120585(1

11987221199062)

(75)



120585 In




V (120585 119905) = sum119895

V119895(119905) 119890

119894119908119895120585

119906 (120585 119905) = sum119895

119895(119905) 119890

119894119908119895120585(76)

with

119908119895=2120587119895

119871 119895 = minus

119873

2+ 1 0

119873

2 (77)



V119895(119905) =

1

119871int119871

0

V (120585 119905) 119890minus119894119908119895120585119889120585 (78)




⟨119891 119892⟩ = int119871

0

119891 (120585) 119892 (120585) 119889120585 (79)

it follows that

V1015840119895(119905) = 120574

1119895+ 119873

1

1015840119895(119905) = 120574

2V119895+ 119873

2

(80)

where

1205741=minus (21205733) 1198941199083

119895

1 + (1205732)1199082119895

1205742=minus1205731205901198941199083

119895

1 + (1205732)1199082119895

1198731= minus119894119908119895119875119895[(1 + 120572V1198722) 119906]1 + (1205732)1199082

119895

1198732=119894119908119895119875119895[120573120590 (minus (311987210158401198724) V

120585+ (3 (1198721015840)

2


1 + (1205732)1199082119895

(81)


119875119895[119892] =

1

119871int119871

0

119892 (120585) 119890minus119894119908119895120585119889120585 (82)



V119899+1119895minus V119899

119895

Δ119905=1205741(119899+1119895+ 119899

119895)

2+3

21198731198991minus1

2119873119899minus11

119899+1119895minus 119899

119895

Δ119905=1205742(V119899+1119895+ V119899

119895)

2+3

21198731198992minus1

2119873119899minus12

(83)






119895 119899+1119895




120585are


V120585(120585 119905) = sum

119895

119894119908119895119875119895[V (sdot 119905)] 119890119894119908119895120585 (84)





V (120585 119905) = V (120585 minus 119888119905)

119906 (120585 119905) = (120585 minus 119888119905) (85)



(V10158401015840

11990610158401015840)

=1

(1205734) 1198882 minus 21205731205903(

119888

2

2

3

120590119888

2

)(119888V minus ((1 + 120572V) 119906)

119888119906 minus V minus120572

21199062)

(86)


V (119909) asymp V0+1198732

sum119899=1

V119899cos(119899120587

119897119909)

119906 (119909) asymp 1199060+1198732

sum119899=1

119906119899cos(119899120587

119897119909)

(87)

where

V0=1

119897int119897

0

V (119909) 119889119909 =1

2119897int2119897

0

V (119909) 119889119909

V119899=2

119897int119897

0

V (119909) cos(119899120587119909119897) 119889119909

=2

2119897int2119897

0

V (119909) cos(119899120587119909119897) 119889119909

(88)


119909119895=2119897 (119895 minus 1)

119873 119895 = 1

119873

2+ 1 (89)


119865 (V0 V1 V

1198732 1199060 1199061 119906

1198732) = 0 (90)




119869119894119895119865 (119909) asymp

119865119894(119909 + ℎ119890

119895) minus 119865

119894(119909 minus ℎ119890

119895)

2ℎ

119895 = 1 119873 + 2

(91)




1199060(119909) = cos(120587119909

119897)

V0(119909) = cos(120587119909

119897)

(92)


V = minus(119868 minus120573

21205972120585)Φ

119905+ 120573120590V

120585120585minus120572

2Φ2120585

= minusΦ119905+ 119874 (120572 120573 120590)

(93)


minus Φ119905119905+ Φ

120585120585+ 120573 (1 minus 120590)Φ

120585120585119905119905minus2120573

3Φ120585120585120585120585minus 2120572Φ

120585Φ120585119905

minus 120572Φ119905Φ120585120585= 0

(94)





3)Φ1015840101584010158401015840

+3120572119888

2(Φ10158402)

1015840

= 0

(96)



3)Φ101584010158401015840 +

3120572119888

2(Φ1015840)

2

= 0

(97)


(1 minus 1198882) (Φ1015840)2

+ 120573((1 minus 120590) 1198882 minus2

3) (Φ10158401015840)

2

+ 120572119888 (Φ1015840)3

= 0

(98)


V = 119888119906 minus120573119888

211990610158401015840 + 12057312059011988811990610158401015840 minus

120572

21199062 (99)



119860 =1198882 minus 1

120572119888

119861 = radic1198882 minus 1

4120573 ((1 minus 120590) 1198882 minus 23)

(101)



2)

sdot [41198601198612sech2 (119861119909) tanh2 (119861119909)

minus 21198601198612sech4 (119861119909)]

(102)






119906 (120585 0) = V (120585 0) = 119890minus20(119909minus28)2

(103)




v u

2 4 6 8 100x

0

02

04

06

08

1

12

14

2 4 6 8 100x

0

05

1

15


v u

minus1

minus05

0

05

2 4 6 8 100x

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

02

2 4 6 8 100x






1198622120590=1199082

1198962

=1 + (21205733) 1198962

(1 + (1205732) 1198962)2+1205731205901198962 (1 + (21205733) 1198962)

(1 + (1205732) 1198962)2

gt 11986220

(104)


0119890119894(119896120585minus119908119905)

V = V0119890119894(119896120585minus119908119905) where 119906

0 V0are constants


v u

minus02

0

02

04

06

08

1

12

14

16

2 4 6 8 100

x

0

02

04

06

08

1

12

14

2 4 6 8 100

x


v

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

2 4 6 8 100

x

u

minus25

minus2

minus15

minus1

minus05

0

2 4 6 8 100

x





120590is larger







at t = 10

0

05

1

15u at t = 10

0

02

04

06

08

1

12

14

2 4 6 8 100

120585

2 4 6 8 100

120585


v

minus01

0

01

03

02

04

05

06

07

45 50 55 6040

x

u

45 50 55 6040

x

minus01

0

01

03

02

04

05

06

07





119906 (120585 0) = V (120585 0) = 119890minus20(119909minus198)2 (105)


v

0

005

01

015

02

025

03

035

04

045

45 50 55 6040

x

u

0

005

01

015

02

025

03

035

04

45 50 55 6040

x


v

minus05

0

05

1

15

2

25

45 50 55 6040

x

u

minus05

0

05

1

15

2

45 50 55 6040

x




7 Conclusions



at t = 10 u at t = 10

minus01

0

01

02

03

04

05

06

07

minus01

0

01

02

03

04

05

06

07

60 65 7055

120585

60 65 7055

120585


u at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

20 30 40 5010

120585

20 30 40 5010

120585






minus04

minus03

minus02

minus01

0

01

02

03

04

05u at t = 20

35 40 45 50 55 6030120585

v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 55 6030120585


0 5 10 15 20 25 30 35 40 45 50

C2120590

C20

81205903

radic120573k

0

02

04

06

08

1

12

14

16





05060708091

1112131415

30 32 34 36 38 40 4228

x





u at t = 20

35 40 45 50 5530120585

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 5530120585


v at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

290 300 310 320280

120585

u at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

280 300 310 320290

120585




Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










sdot

10038161003816100381610038161003816100381610038161003816100381610038161003816

(cos (120582ℎ) minus 1) 1198911+ Γ sin (120582ℎ) 119891

2

ℎ

minus1199102 (1 + (21205733) 1199102) 119891

2

1 + (1205732) 1199102

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910 + intR

(1 + 1199102)119904minus1

sdot

1003816100381610038161003816100381610038161003816100381610038161003816

Ψ sin (120582ℎ) 1198911+ (cos (120582ℎ) minus 1) 119891

2

ℎ

minusminus1205731205901199102119891

1

1 + (1205732) 1199102

1003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

(55)

But by virtue of

limℎrarr0


= 0

limℎrarr0


= Γ120582 =1199102 (1 + (21205733) 1199102)

1 + (1205732) 1199102

limℎrarr0


= Ψ120582 = minus1205731205901199102

1 + (1205732) 1199102

(56)


limℎrarr0

10038171003817100381710038171003817100381710038171003817

119878 (ℎ) 119891 minus 119891

ℎminusA119891

10038171003817100381710038171003817100381710038171003817

2


= 0 (57)



G (119880)119867119904minus1times119867119904 le 119862 119880

119867119904minus1times119867119904 (58)

Furthermore

1003817100381710038171003817G (1198801) minusG (1198802)10038171003817100381710038172

119867119904minus1times119867119904

le 119862 [(1 +1003817100381710038171003817Φ110038171003817100381710038172

119867119904)1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1

+ (1003817100381710038171003817V210038171003817100381710038172

119867119904minus1 +1003817100381710038171003817Φ1 + Φ2

10038171003817100381710038172

119867119904)1003817100381710038171003817Φ1 minus Φ2

10038171003817100381710038172

119867119904]

(59)


V119894Φ119894) 119894 = 1 2


G (119880)2

119867119904minus1times119867119904

le 119862intR

(1 + 1199102)119904minus1

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

2 100381610038161003816100381610038161003816100381610038161003816

(

V1198722Φ120585)120585

100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

210038161003816100381610038161003816100381610038161003816100381610038161003816

1015840

1198724V120585

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

2

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(1198721015840)

2

1198725V

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

210038161003816100381610038161003816100381610038161003816100381610038161003816

10158401015840

1198724V10038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

2 10038161003816100381610038161003816100381610038161003816

V119872

10038161003816100381610038161003816100381610038161003816

2

119889119910

+ 119862intR

(1 + 1199102)119904

100381610038161003816100381610038161003816100381610038161003816

1

1 + (1205732) 1199102

100381610038161003816100381610038161003816100381610038161003816

2100381610038161003816100381610038161003816100381610038161003816100381610038161003816

Φ2120585

1198722

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

(60)


1003817100381710038171003817100381710038171003817

V1198722Φ120585

1003817100381710038171003817100381710038171003817119867119904minus2le 1198621003817100381710038171003817100381710038171003817

V1198722

1003817100381710038171003817100381710038171003817119867119904minus210038171003817100381710038171003817Φ12058510038171003817100381710038171003817119867119904minus1

le 1198621003817100381710038171003817100381710038171003817

1

1198722

1003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1 Φ

119867119904

100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724V120585

100381710038171003817100381710038171003817100381710038171003817119867119904minus2le 119862100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724

100381710038171003817100381710038171003817100381710038171003817119867119904minus1

10038171003817100381710038171003817V12058510038171003817100381710038171003817119867119904minus2

le 119862100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724

100381710038171003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725V100381710038171003817100381710038171003817100381710038171003817100381710038171003817119867119904minus2

le 119862

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725

100381710038171003817100381710038171003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus2

le 119862

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725

100381710038171003817100381710038171003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1

100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724V100381710038171003817100381710038171003817100381710038171003817119867119904minus2

le 119862100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724

100381710038171003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1

1003817100381710038171003817100381710038171003817

V119872

1003817100381710038171003817100381710038171003817119867119904minus2le 1198621003817100381710038171003817100381710038171003817

1

119872

1003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus2

le 1198621003817100381710038171003817100381710038171003817

1

119872

1003817100381710038171003817100381710038171003817119867119904minus1V

119867119904minus1


1003817100381710038171003817100381710038171003817100381710038171003817

Φ2120585

1198722

1003817100381710038171003817100381710038171003817100381710038171003817119867119904minus2le 11986210038171003817100381710038171003817100381710038171003817

Φ120585

1198722Φ120585

10038171003817100381710038171003817100381710038171003817119867119904minus2

le 11986210038171003817100381710038171003817Φ12058510038171003817100381710038171003817119867119904minus1

10038171003817100381710038171003817100381710038171003817

Φ120585

1198722

10038171003817100381710038171003817100381710038171003817119867119904minus2

le 119862 Φ119904

1003817100381710038171003817100381710038171003817

1

1198722

1003817100381710038171003817100381710038171003817119867119904minus110038171003817100381710038171003817Φ12058510038171003817100381710038171003817119867119904minus2

le 119862 Φ119904

1003817100381710038171003817100381710038171003817

1

1198722

1003817100381710038171003817100381710038171003817119867119904minus1Φ

119867119904

(61)


G (119880)2

119867119904minus1times119867119904 le 119862(

1003817100381710038171003817100381710038171003817

V1198722Φ120585

1003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724V120585

100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725V

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724V100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+1003817100381710038171003817100381710038171003817

V119872

1003817100381710038171003817100381710038171003817

2

119867119904minus2

+

1003817100381710038171003817100381710038171003817100381710038171003817

Φ2120585

1198722

1003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

) le 119862(V2119867119904minus1 + Φ

2

119867119904)

(62)



that

1003817100381710038171003817100381710038171003817100381710038171003817G(

V1

Φ1

) minusG(V2

Φ2

)

1003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus1times119867119904

le 119862(1003817100381710038171003817100381710038171003817

V1

1198722Φ1120585

minusV2

1198722Φ2120585

1003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724(V1minus V

2)120585

100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725(V1minus V

2)

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724(V1minus V

2)100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+1003817100381710038171003817100381710038171003817

1

119872(V1minus V

2)1003817100381710038171003817100381710038171003817

2

119867119904minus2

+1003817100381710038171003817100381710038171003817

1

1198722(Φ21120585minus Φ2

2120585)1003817100381710038171003817100381710038171003817

2

119867119904minus2

)

le 119862(10038171003817100381710038171003817V1Φ1120585 minus V2Φ2120585

100381710038171003817100381710038172

119867119904minus2+1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1 +10038171003817100381710038171003817Φ2

1120585

minus Φ22120585

100381710038171003817100381710038172

119867119904minus2) le 119862 [(

10038171003817100381710038171003817(V1 minus V2)Φ112058510038171003817100381710038171003817119867119904minus2

+10038171003817100381710038171003817V2 (Φ1120585 minus Φ2120585)

10038171003817100381710038171003817119867119904minus2)2

+1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1 +10038171003817100381710038171003817Φ1120585

+ Φ2120585

100381710038171003817100381710038172

119867119904minus1

10038171003817100381710038171003817Φ1120585 minus Φ2120585100381710038171003817100381710038172

119867119904minus1] le 119862 [

1003817100381710038171003817V1 minus V210038171003817100381710038172

119867119904minus1

sdot10038171003817100381710038171003817Φ1120585100381710038171003817100381710038172

119867119904minus2+1003817100381710038171003817V210038171003817100381710038172

119867119904minus1

10038171003817100381710038171003817Φ1120585 minus Φ2120585100381710038171003817100381710038172

119904minus2

+1003817100381710038171003817V1

minus V2

10038171003817100381710038172

119867119904minus1 +1003817100381710038171003817Φ1 + Φ2

10038171003817100381710038172

119867119904

1003817100381710038171003817Φ1 minus Φ210038171003817100381710038172

119867119904] le 119862 [(1

+1003817100381710038171003817Φ110038171003817100381710038172

119867119904)1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1 + (

1003817100381710038171003817V210038171003817100381710038172

119867119904minus1

+1003817100381710038171003817Φ1 + Φ2

10038171003817100381710038172

119867119904)1003817100381710038171003817Φ1 minus Φ2

10038171003817100381710038172

119867119904]

(63)




119880 (119905) = 119878 (119905) 120601 + int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840 (64)


(Ψ119880) (119905) = 119878 (119905) 120601 + int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840 (65)


X119872(119879) = 119880 isin 119862 ([0 119879] 119867

119904minus1 times 119867119904)

sup119905isin[0119879]


(66)




le1003817100381710038171003817(119878 (119905) minus 119878 (120591)) 120601

1003817100381710038171003817119867119904minus1times119867119904

+10038171003817100381710038171003817100381710038171003817int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840

minus 119878 (120591 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840

10038171003817100381710038171003817100381710038171003817119867119904minus1times119867119904

(67)



1003817100381710038171003817119867119904minus1times119867119904

+1003817100381710038171003817119878 (119905) 120601

1003817100381710038171003817119867119904minus1times119867119904

le 119872 +10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904

(68)


Given that (119878(119905))119905ge0




10038171003817100381710038171003817100381710038171003817int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840 minus int

120591

0

119878 (120591 minus 1199051015840)G (119880)

sdot (1199051015840) 119889119905101584010038171003817100381710038171003817100381710038171003817119867119904minus1times119867119904

le int119905

0



1015840 + int120591

119905

10038171003817100381710038171003817119878 (120591 minus 1199051015840)G (119880)

sdot (1199051015840)10038171003817100381710038171003817119867119904minus1times119867119904 119889119905

1015840 = 1198681+ 1198682

(69)


119894119895(119910 119905minus1199051015840) as 120591 rarr 119905 119894 119895 =


1rarr 0 as 120591 rarr 119905


1198682= int

120591

119905

10038171003817100381710038171003817119878 (120591 minus 1199051015840)G (119880) (119905

1015840)10038171003817100381710038171003817119867119904minus1times119867119904 119889119905

1015840

le 119862int120591

119905

10038171003817100381710038171003817G (119880) (1199051015840)10038171003817100381710038171003817119867119904minus1times119867119904 119889119905

1015840

le 119862int120591

119905

119880119867119904minus1times119867119904 1198891199051015840


(70)



Ψ(X119872(119879)) sub X

119872(119879) Let 119880 = (V Φ)119879 isin X

119872(119879) For any

119905 isin [0 119879]


le int119905

0

10038171003817100381710038171003817119878 (119905 minus 1199051015840)G (119880) (119905

1015840)10038171003817100381710038171003817119867119904minus1times119867119904

le 119862int119905

0

10038171003817100381710038171003817G (119880) (1199051015840)10038171003817100381710038171003817119867119904minus1times119867119904

le 119862119879 sup119905isin[0119879]

119880 (119905)119867119904minus1times119867119904 le 119862119879 (119872 +

10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904)

le 119872

(71)



119872() Let 119880

1= (V

1 Φ1)119879

1198802= (V

2 Φ2)119879 isin X



119905

0

10038171003817100381710038171003817119878 (119905 minus 1199051015840)

sdot (G (1198801) (1199051015840) minusG (119880

2) (1199051015840))

10038171003817100381710038171003817119867119904minus1times119867119904 1198891199051015840

le 119862int119905

0


1015840 le int119905

0

(1

+1003817100381710038171003817Φ11003817100381710038171003817119867119904)1003817100381710038171003817V1 minus V2

1003817100381710038171003817119867119904minus1 + (1003817100381710038171003817V21003817100381710038171003817119867119904minus1

+1003817100381710038171003817Φ1 + Φ2

1003817100381710038171003817119867119904)1003817100381710038171003817Φ1 minus Φ2

1003817100381710038171003817119867119904 1198891199051015840 le 119862119879 [(1 +119872

+10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904)

1003817100381710038171003817V1 minus V21003817100381710038171003817119867119904minus1 + (3119872 + 3

10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904)

sdot1003817100381710038171003817Φ1 minus Φ2

1003817100381710038171003817119867119904] le 119862119879 [3119872 + 310038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904]



(72)


119862 [3119872 + 310038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904] lt 119872 (73)



119880 isin 119862 ([0 ] 119867119904minus1 times 119867119904) (74)


5 Numerical Schemes


(119868 minus120573

21205972120585) V119905=2120573

31205973120585119906 minus 120597

120585((1 +

120572V1198722) 119906)

(119868 minus120573

21205972120585)119906

119905

= 1205731205901205973120585V + 120573120590(minus

31198721015840

1198724V120585+ 3(1198721015840)

2

1198725V minus11987210158401015840

1198724V)

minus 120597120585(1

119872V) minus

120572

2120597120585(1

11987221199062)

(75)



120585 In




V (120585 119905) = sum119895

V119895(119905) 119890

119894119908119895120585

119906 (120585 119905) = sum119895

119895(119905) 119890

119894119908119895120585(76)

with

119908119895=2120587119895

119871 119895 = minus

119873

2+ 1 0

119873

2 (77)



V119895(119905) =

1

119871int119871

0

V (120585 119905) 119890minus119894119908119895120585119889120585 (78)




⟨119891 119892⟩ = int119871

0

119891 (120585) 119892 (120585) 119889120585 (79)

it follows that

V1015840119895(119905) = 120574

1119895+ 119873

1

1015840119895(119905) = 120574

2V119895+ 119873

2

(80)

where

1205741=minus (21205733) 1198941199083

119895

1 + (1205732)1199082119895

1205742=minus1205731205901198941199083

119895

1 + (1205732)1199082119895

1198731= minus119894119908119895119875119895[(1 + 120572V1198722) 119906]1 + (1205732)1199082

119895

1198732=119894119908119895119875119895[120573120590 (minus (311987210158401198724) V

120585+ (3 (1198721015840)

2


1 + (1205732)1199082119895

(81)


119875119895[119892] =

1

119871int119871

0

119892 (120585) 119890minus119894119908119895120585119889120585 (82)



V119899+1119895minus V119899

119895

Δ119905=1205741(119899+1119895+ 119899

119895)

2+3

21198731198991minus1

2119873119899minus11

119899+1119895minus 119899

119895

Δ119905=1205742(V119899+1119895+ V119899

119895)

2+3

21198731198992minus1

2119873119899minus12

(83)






119895 119899+1119895




120585are


V120585(120585 119905) = sum

119895

119894119908119895119875119895[V (sdot 119905)] 119890119894119908119895120585 (84)





V (120585 119905) = V (120585 minus 119888119905)

119906 (120585 119905) = (120585 minus 119888119905) (85)



(V10158401015840

11990610158401015840)

=1

(1205734) 1198882 minus 21205731205903(

119888

2

2

3

120590119888

2

)(119888V minus ((1 + 120572V) 119906)

119888119906 minus V minus120572

21199062)

(86)


V (119909) asymp V0+1198732

sum119899=1

V119899cos(119899120587

119897119909)

119906 (119909) asymp 1199060+1198732

sum119899=1

119906119899cos(119899120587

119897119909)

(87)

where

V0=1

119897int119897

0

V (119909) 119889119909 =1

2119897int2119897

0

V (119909) 119889119909

V119899=2

119897int119897

0

V (119909) cos(119899120587119909119897) 119889119909

=2

2119897int2119897

0

V (119909) cos(119899120587119909119897) 119889119909

(88)


119909119895=2119897 (119895 minus 1)

119873 119895 = 1

119873

2+ 1 (89)


119865 (V0 V1 V

1198732 1199060 1199061 119906

1198732) = 0 (90)




119869119894119895119865 (119909) asymp

119865119894(119909 + ℎ119890

119895) minus 119865

119894(119909 minus ℎ119890

119895)

2ℎ

119895 = 1 119873 + 2

(91)




1199060(119909) = cos(120587119909

119897)

V0(119909) = cos(120587119909

119897)

(92)


V = minus(119868 minus120573

21205972120585)Φ

119905+ 120573120590V

120585120585minus120572

2Φ2120585

= minusΦ119905+ 119874 (120572 120573 120590)

(93)


minus Φ119905119905+ Φ

120585120585+ 120573 (1 minus 120590)Φ

120585120585119905119905minus2120573

3Φ120585120585120585120585minus 2120572Φ

120585Φ120585119905

minus 120572Φ119905Φ120585120585= 0

(94)





3)Φ1015840101584010158401015840

+3120572119888

2(Φ10158402)

1015840

= 0

(96)



3)Φ101584010158401015840 +

3120572119888

2(Φ1015840)

2

= 0

(97)


(1 minus 1198882) (Φ1015840)2

+ 120573((1 minus 120590) 1198882 minus2

3) (Φ10158401015840)

2

+ 120572119888 (Φ1015840)3

= 0

(98)


V = 119888119906 minus120573119888

211990610158401015840 + 12057312059011988811990610158401015840 minus

120572

21199062 (99)



119860 =1198882 minus 1

120572119888

119861 = radic1198882 minus 1

4120573 ((1 minus 120590) 1198882 minus 23)

(101)



2)

sdot [41198601198612sech2 (119861119909) tanh2 (119861119909)

minus 21198601198612sech4 (119861119909)]

(102)






119906 (120585 0) = V (120585 0) = 119890minus20(119909minus28)2

(103)




v u

2 4 6 8 100x

0

02

04

06

08

1

12

14

2 4 6 8 100x

0

05

1

15


v u

minus1

minus05

0

05

2 4 6 8 100x

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

02

2 4 6 8 100x






1198622120590=1199082

1198962

=1 + (21205733) 1198962

(1 + (1205732) 1198962)2+1205731205901198962 (1 + (21205733) 1198962)

(1 + (1205732) 1198962)2

gt 11986220

(104)


0119890119894(119896120585minus119908119905)

V = V0119890119894(119896120585minus119908119905) where 119906

0 V0are constants


v u

minus02

0

02

04

06

08

1

12

14

16

2 4 6 8 100

x

0

02

04

06

08

1

12

14

2 4 6 8 100

x


v

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

2 4 6 8 100

x

u

minus25

minus2

minus15

minus1

minus05

0

2 4 6 8 100

x





120590is larger







at t = 10

0

05

1

15u at t = 10

0

02

04

06

08

1

12

14

2 4 6 8 100

120585

2 4 6 8 100

120585


v

minus01

0

01

03

02

04

05

06

07

45 50 55 6040

x

u

45 50 55 6040

x

minus01

0

01

03

02

04

05

06

07





119906 (120585 0) = V (120585 0) = 119890minus20(119909minus198)2 (105)


v

0

005

01

015

02

025

03

035

04

045

45 50 55 6040

x

u

0

005

01

015

02

025

03

035

04

45 50 55 6040

x


v

minus05

0

05

1

15

2

25

45 50 55 6040

x

u

minus05

0

05

1

15

2

45 50 55 6040

x




7 Conclusions



at t = 10 u at t = 10

minus01

0

01

02

03

04

05

06

07

minus01

0

01

02

03

04

05

06

07

60 65 7055

120585

60 65 7055

120585


u at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

20 30 40 5010

120585

20 30 40 5010

120585






minus04

minus03

minus02

minus01

0

01

02

03

04

05u at t = 20

35 40 45 50 55 6030120585

v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 55 6030120585


0 5 10 15 20 25 30 35 40 45 50

C2120590

C20

81205903

radic120573k

0

02

04

06

08

1

12

14

16





05060708091

1112131415

30 32 34 36 38 40 4228

x





u at t = 20

35 40 45 50 5530120585

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 5530120585


v at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

290 300 310 320280

120585

u at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

280 300 310 320290

120585




Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1003817100381710038171003817100381710038171003817100381710038171003817

Φ2120585

1198722

1003817100381710038171003817100381710038171003817100381710038171003817119867119904minus2le 11986210038171003817100381710038171003817100381710038171003817

Φ120585

1198722Φ120585

10038171003817100381710038171003817100381710038171003817119867119904minus2

le 11986210038171003817100381710038171003817Φ12058510038171003817100381710038171003817119867119904minus1

10038171003817100381710038171003817100381710038171003817

Φ120585

1198722

10038171003817100381710038171003817100381710038171003817119867119904minus2

le 119862 Φ119904

1003817100381710038171003817100381710038171003817

1

1198722

1003817100381710038171003817100381710038171003817119867119904minus110038171003817100381710038171003817Φ12058510038171003817100381710038171003817119867119904minus2

le 119862 Φ119904

1003817100381710038171003817100381710038171003817

1

1198722

1003817100381710038171003817100381710038171003817119867119904minus1Φ

119867119904

(61)


G (119880)2

119867119904minus1times119867119904 le 119862(

1003817100381710038171003817100381710038171003817

V1198722Φ120585

1003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724V120585

100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725V

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724V100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+1003817100381710038171003817100381710038171003817

V119872

1003817100381710038171003817100381710038171003817

2

119867119904minus2

+

1003817100381710038171003817100381710038171003817100381710038171003817

Φ2120585

1198722

1003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

) le 119862(V2119867119904minus1 + Φ

2

119867119904)

(62)



that

1003817100381710038171003817100381710038171003817100381710038171003817G(

V1

Φ1

) minusG(V2

Φ2

)

1003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus1times119867119904

le 119862(1003817100381710038171003817100381710038171003817

V1

1198722Φ1120585

minusV2

1198722Φ2120585

1003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

1198721015840

1198724(V1minus V

2)120585

100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(1198721015840)2

1198725(V1minus V

2)

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+100381710038171003817100381710038171003817100381710038171003817

11987210158401015840

1198724(V1minus V

2)100381710038171003817100381710038171003817100381710038171003817

2

119867119904minus2

+1003817100381710038171003817100381710038171003817

1

119872(V1minus V

2)1003817100381710038171003817100381710038171003817

2

119867119904minus2

+1003817100381710038171003817100381710038171003817

1

1198722(Φ21120585minus Φ2

2120585)1003817100381710038171003817100381710038171003817

2

119867119904minus2

)

le 119862(10038171003817100381710038171003817V1Φ1120585 minus V2Φ2120585

100381710038171003817100381710038172

119867119904minus2+1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1 +10038171003817100381710038171003817Φ2

1120585

minus Φ22120585

100381710038171003817100381710038172

119867119904minus2) le 119862 [(

10038171003817100381710038171003817(V1 minus V2)Φ112058510038171003817100381710038171003817119867119904minus2

+10038171003817100381710038171003817V2 (Φ1120585 minus Φ2120585)

10038171003817100381710038171003817119867119904minus2)2

+1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1 +10038171003817100381710038171003817Φ1120585

+ Φ2120585

100381710038171003817100381710038172

119867119904minus1

10038171003817100381710038171003817Φ1120585 minus Φ2120585100381710038171003817100381710038172

119867119904minus1] le 119862 [

1003817100381710038171003817V1 minus V210038171003817100381710038172

119867119904minus1

sdot10038171003817100381710038171003817Φ1120585100381710038171003817100381710038172

119867119904minus2+1003817100381710038171003817V210038171003817100381710038172

119867119904minus1

10038171003817100381710038171003817Φ1120585 minus Φ2120585100381710038171003817100381710038172

119904minus2

+1003817100381710038171003817V1

minus V2

10038171003817100381710038172

119867119904minus1 +1003817100381710038171003817Φ1 + Φ2

10038171003817100381710038172

119867119904

1003817100381710038171003817Φ1 minus Φ210038171003817100381710038172

119867119904] le 119862 [(1

+1003817100381710038171003817Φ110038171003817100381710038172

119867119904)1003817100381710038171003817V1 minus V2

10038171003817100381710038172

119867119904minus1 + (

1003817100381710038171003817V210038171003817100381710038172

119867119904minus1

+1003817100381710038171003817Φ1 + Φ2

10038171003817100381710038172

119867119904)1003817100381710038171003817Φ1 minus Φ2

10038171003817100381710038172

119867119904]

(63)




119880 (119905) = 119878 (119905) 120601 + int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840 (64)


(Ψ119880) (119905) = 119878 (119905) 120601 + int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840 (65)


X119872(119879) = 119880 isin 119862 ([0 119879] 119867

119904minus1 times 119867119904)

sup119905isin[0119879]


(66)




le1003817100381710038171003817(119878 (119905) minus 119878 (120591)) 120601

1003817100381710038171003817119867119904minus1times119867119904

+10038171003817100381710038171003817100381710038171003817int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840

minus 119878 (120591 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840

10038171003817100381710038171003817100381710038171003817119867119904minus1times119867119904

(67)



1003817100381710038171003817119867119904minus1times119867119904

+1003817100381710038171003817119878 (119905) 120601

1003817100381710038171003817119867119904minus1times119867119904

le 119872 +10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904

(68)


Given that (119878(119905))119905ge0




10038171003817100381710038171003817100381710038171003817int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840 minus int

120591

0

119878 (120591 minus 1199051015840)G (119880)

sdot (1199051015840) 119889119905101584010038171003817100381710038171003817100381710038171003817119867119904minus1times119867119904

le int119905

0



1015840 + int120591

119905

10038171003817100381710038171003817119878 (120591 minus 1199051015840)G (119880)

sdot (1199051015840)10038171003817100381710038171003817119867119904minus1times119867119904 119889119905

1015840 = 1198681+ 1198682

(69)


119894119895(119910 119905minus1199051015840) as 120591 rarr 119905 119894 119895 =


1rarr 0 as 120591 rarr 119905


1198682= int

120591

119905

10038171003817100381710038171003817119878 (120591 minus 1199051015840)G (119880) (119905

1015840)10038171003817100381710038171003817119867119904minus1times119867119904 119889119905

1015840

le 119862int120591

119905

10038171003817100381710038171003817G (119880) (1199051015840)10038171003817100381710038171003817119867119904minus1times119867119904 119889119905

1015840

le 119862int120591

119905

119880119867119904minus1times119867119904 1198891199051015840


(70)



Ψ(X119872(119879)) sub X

119872(119879) Let 119880 = (V Φ)119879 isin X

119872(119879) For any

119905 isin [0 119879]


le int119905

0

10038171003817100381710038171003817119878 (119905 minus 1199051015840)G (119880) (119905

1015840)10038171003817100381710038171003817119867119904minus1times119867119904

le 119862int119905

0

10038171003817100381710038171003817G (119880) (1199051015840)10038171003817100381710038171003817119867119904minus1times119867119904

le 119862119879 sup119905isin[0119879]

119880 (119905)119867119904minus1times119867119904 le 119862119879 (119872 +

10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904)

le 119872

(71)



119872() Let 119880

1= (V

1 Φ1)119879

1198802= (V

2 Φ2)119879 isin X



119905

0

10038171003817100381710038171003817119878 (119905 minus 1199051015840)

sdot (G (1198801) (1199051015840) minusG (119880

2) (1199051015840))

10038171003817100381710038171003817119867119904minus1times119867119904 1198891199051015840

le 119862int119905

0


1015840 le int119905

0

(1

+1003817100381710038171003817Φ11003817100381710038171003817119867119904)1003817100381710038171003817V1 minus V2

1003817100381710038171003817119867119904minus1 + (1003817100381710038171003817V21003817100381710038171003817119867119904minus1

+1003817100381710038171003817Φ1 + Φ2

1003817100381710038171003817119867119904)1003817100381710038171003817Φ1 minus Φ2

1003817100381710038171003817119867119904 1198891199051015840 le 119862119879 [(1 +119872

+10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904)

1003817100381710038171003817V1 minus V21003817100381710038171003817119867119904minus1 + (3119872 + 3

10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904)

sdot1003817100381710038171003817Φ1 minus Φ2

1003817100381710038171003817119867119904] le 119862119879 [3119872 + 310038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904]



(72)


119862 [3119872 + 310038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904] lt 119872 (73)



119880 isin 119862 ([0 ] 119867119904minus1 times 119867119904) (74)


5 Numerical Schemes


(119868 minus120573

21205972120585) V119905=2120573

31205973120585119906 minus 120597

120585((1 +

120572V1198722) 119906)

(119868 minus120573

21205972120585)119906

119905

= 1205731205901205973120585V + 120573120590(minus

31198721015840

1198724V120585+ 3(1198721015840)

2

1198725V minus11987210158401015840

1198724V)

minus 120597120585(1

119872V) minus

120572

2120597120585(1

11987221199062)

(75)



120585 In




V (120585 119905) = sum119895

V119895(119905) 119890

119894119908119895120585

119906 (120585 119905) = sum119895

119895(119905) 119890

119894119908119895120585(76)

with

119908119895=2120587119895

119871 119895 = minus

119873

2+ 1 0

119873

2 (77)



V119895(119905) =

1

119871int119871

0

V (120585 119905) 119890minus119894119908119895120585119889120585 (78)




⟨119891 119892⟩ = int119871

0

119891 (120585) 119892 (120585) 119889120585 (79)

it follows that

V1015840119895(119905) = 120574

1119895+ 119873

1

1015840119895(119905) = 120574

2V119895+ 119873

2

(80)

where

1205741=minus (21205733) 1198941199083

119895

1 + (1205732)1199082119895

1205742=minus1205731205901198941199083

119895

1 + (1205732)1199082119895

1198731= minus119894119908119895119875119895[(1 + 120572V1198722) 119906]1 + (1205732)1199082

119895

1198732=119894119908119895119875119895[120573120590 (minus (311987210158401198724) V

120585+ (3 (1198721015840)

2


1 + (1205732)1199082119895

(81)


119875119895[119892] =

1

119871int119871

0

119892 (120585) 119890minus119894119908119895120585119889120585 (82)



V119899+1119895minus V119899

119895

Δ119905=1205741(119899+1119895+ 119899

119895)

2+3

21198731198991minus1

2119873119899minus11

119899+1119895minus 119899

119895

Δ119905=1205742(V119899+1119895+ V119899

119895)

2+3

21198731198992minus1

2119873119899minus12

(83)






119895 119899+1119895




120585are


V120585(120585 119905) = sum

119895

119894119908119895119875119895[V (sdot 119905)] 119890119894119908119895120585 (84)





V (120585 119905) = V (120585 minus 119888119905)

119906 (120585 119905) = (120585 minus 119888119905) (85)



(V10158401015840

11990610158401015840)

=1

(1205734) 1198882 minus 21205731205903(

119888

2

2

3

120590119888

2

)(119888V minus ((1 + 120572V) 119906)

119888119906 minus V minus120572

21199062)

(86)


V (119909) asymp V0+1198732

sum119899=1

V119899cos(119899120587

119897119909)

119906 (119909) asymp 1199060+1198732

sum119899=1

119906119899cos(119899120587

119897119909)

(87)

where

V0=1

119897int119897

0

V (119909) 119889119909 =1

2119897int2119897

0

V (119909) 119889119909

V119899=2

119897int119897

0

V (119909) cos(119899120587119909119897) 119889119909

=2

2119897int2119897

0

V (119909) cos(119899120587119909119897) 119889119909

(88)


119909119895=2119897 (119895 minus 1)

119873 119895 = 1

119873

2+ 1 (89)


119865 (V0 V1 V

1198732 1199060 1199061 119906

1198732) = 0 (90)




119869119894119895119865 (119909) asymp

119865119894(119909 + ℎ119890

119895) minus 119865

119894(119909 minus ℎ119890

119895)

2ℎ

119895 = 1 119873 + 2

(91)




1199060(119909) = cos(120587119909

119897)

V0(119909) = cos(120587119909

119897)

(92)


V = minus(119868 minus120573

21205972120585)Φ

119905+ 120573120590V

120585120585minus120572

2Φ2120585

= minusΦ119905+ 119874 (120572 120573 120590)

(93)


minus Φ119905119905+ Φ

120585120585+ 120573 (1 minus 120590)Φ

120585120585119905119905minus2120573

3Φ120585120585120585120585minus 2120572Φ

120585Φ120585119905

minus 120572Φ119905Φ120585120585= 0

(94)





3)Φ1015840101584010158401015840

+3120572119888

2(Φ10158402)

1015840

= 0

(96)



3)Φ101584010158401015840 +

3120572119888

2(Φ1015840)

2

= 0

(97)


(1 minus 1198882) (Φ1015840)2

+ 120573((1 minus 120590) 1198882 minus2

3) (Φ10158401015840)

2

+ 120572119888 (Φ1015840)3

= 0

(98)


V = 119888119906 minus120573119888

211990610158401015840 + 12057312059011988811990610158401015840 minus

120572

21199062 (99)



119860 =1198882 minus 1

120572119888

119861 = radic1198882 minus 1

4120573 ((1 minus 120590) 1198882 minus 23)

(101)



2)

sdot [41198601198612sech2 (119861119909) tanh2 (119861119909)

minus 21198601198612sech4 (119861119909)]

(102)






119906 (120585 0) = V (120585 0) = 119890minus20(119909minus28)2

(103)




v u

2 4 6 8 100x

0

02

04

06

08

1

12

14

2 4 6 8 100x

0

05

1

15


v u

minus1

minus05

0

05

2 4 6 8 100x

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

02

2 4 6 8 100x






1198622120590=1199082

1198962

=1 + (21205733) 1198962

(1 + (1205732) 1198962)2+1205731205901198962 (1 + (21205733) 1198962)

(1 + (1205732) 1198962)2

gt 11986220

(104)


0119890119894(119896120585minus119908119905)

V = V0119890119894(119896120585minus119908119905) where 119906

0 V0are constants


v u

minus02

0

02

04

06

08

1

12

14

16

2 4 6 8 100

x

0

02

04

06

08

1

12

14

2 4 6 8 100

x


v

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

2 4 6 8 100

x

u

minus25

minus2

minus15

minus1

minus05

0

2 4 6 8 100

x





120590is larger







at t = 10

0

05

1

15u at t = 10

0

02

04

06

08

1

12

14

2 4 6 8 100

120585

2 4 6 8 100

120585


v

minus01

0

01

03

02

04

05

06

07

45 50 55 6040

x

u

45 50 55 6040

x

minus01

0

01

03

02

04

05

06

07





119906 (120585 0) = V (120585 0) = 119890minus20(119909minus198)2 (105)


v

0

005

01

015

02

025

03

035

04

045

45 50 55 6040

x

u

0

005

01

015

02

025

03

035

04

45 50 55 6040

x


v

minus05

0

05

1

15

2

25

45 50 55 6040

x

u

minus05

0

05

1

15

2

45 50 55 6040

x




7 Conclusions



at t = 10 u at t = 10

minus01

0

01

02

03

04

05

06

07

minus01

0

01

02

03

04

05

06

07

60 65 7055

120585

60 65 7055

120585


u at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

20 30 40 5010

120585

20 30 40 5010

120585






minus04

minus03

minus02

minus01

0

01

02

03

04

05u at t = 20

35 40 45 50 55 6030120585

v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 55 6030120585


0 5 10 15 20 25 30 35 40 45 50

C2120590

C20

81205903

radic120573k

0

02

04

06

08

1

12

14

16





05060708091

1112131415

30 32 34 36 38 40 4228

x





u at t = 20

35 40 45 50 5530120585

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 5530120585


v at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

290 300 310 320280

120585

u at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

280 300 310 320290

120585




Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Given that (119878(119905))119905ge0




10038171003817100381710038171003817100381710038171003817int119905

0

119878 (119905 minus 1199051015840)G (119880) (1199051015840) 1198891199051015840 minus int

120591

0

119878 (120591 minus 1199051015840)G (119880)

sdot (1199051015840) 119889119905101584010038171003817100381710038171003817100381710038171003817119867119904minus1times119867119904

le int119905

0



1015840 + int120591

119905

10038171003817100381710038171003817119878 (120591 minus 1199051015840)G (119880)

sdot (1199051015840)10038171003817100381710038171003817119867119904minus1times119867119904 119889119905

1015840 = 1198681+ 1198682

(69)


119894119895(119910 119905minus1199051015840) as 120591 rarr 119905 119894 119895 =


1rarr 0 as 120591 rarr 119905


1198682= int

120591

119905

10038171003817100381710038171003817119878 (120591 minus 1199051015840)G (119880) (119905

1015840)10038171003817100381710038171003817119867119904minus1times119867119904 119889119905

1015840

le 119862int120591

119905

10038171003817100381710038171003817G (119880) (1199051015840)10038171003817100381710038171003817119867119904minus1times119867119904 119889119905

1015840

le 119862int120591

119905

119880119867119904minus1times119867119904 1198891199051015840


(70)



Ψ(X119872(119879)) sub X

119872(119879) Let 119880 = (V Φ)119879 isin X

119872(119879) For any

119905 isin [0 119879]


le int119905

0

10038171003817100381710038171003817119878 (119905 minus 1199051015840)G (119880) (119905

1015840)10038171003817100381710038171003817119867119904minus1times119867119904

le 119862int119905

0

10038171003817100381710038171003817G (119880) (1199051015840)10038171003817100381710038171003817119867119904minus1times119867119904

le 119862119879 sup119905isin[0119879]

119880 (119905)119867119904minus1times119867119904 le 119862119879 (119872 +

10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904)

le 119872

(71)



119872() Let 119880

1= (V

1 Φ1)119879

1198802= (V

2 Φ2)119879 isin X



119905

0

10038171003817100381710038171003817119878 (119905 minus 1199051015840)

sdot (G (1198801) (1199051015840) minusG (119880

2) (1199051015840))

10038171003817100381710038171003817119867119904minus1times119867119904 1198891199051015840

le 119862int119905

0


1015840 le int119905

0

(1

+1003817100381710038171003817Φ11003817100381710038171003817119867119904)1003817100381710038171003817V1 minus V2

1003817100381710038171003817119867119904minus1 + (1003817100381710038171003817V21003817100381710038171003817119867119904minus1

+1003817100381710038171003817Φ1 + Φ2

1003817100381710038171003817119867119904)1003817100381710038171003817Φ1 minus Φ2

1003817100381710038171003817119867119904 1198891199051015840 le 119862119879 [(1 +119872

+10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904)

1003817100381710038171003817V1 minus V21003817100381710038171003817119867119904minus1 + (3119872 + 3

10038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904)

sdot1003817100381710038171003817Φ1 minus Φ2

1003817100381710038171003817119867119904] le 119862119879 [3119872 + 310038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904]



(72)


119862 [3119872 + 310038171003817100381710038171206011003817100381710038171003817119867119904minus1times119867119904] lt 119872 (73)



119880 isin 119862 ([0 ] 119867119904minus1 times 119867119904) (74)


5 Numerical Schemes


(119868 minus120573

21205972120585) V119905=2120573

31205973120585119906 minus 120597

120585((1 +

120572V1198722) 119906)

(119868 minus120573

21205972120585)119906

119905

= 1205731205901205973120585V + 120573120590(minus

31198721015840

1198724V120585+ 3(1198721015840)

2

1198725V minus11987210158401015840

1198724V)

minus 120597120585(1

119872V) minus

120572

2120597120585(1

11987221199062)

(75)



120585 In




V (120585 119905) = sum119895

V119895(119905) 119890

119894119908119895120585

119906 (120585 119905) = sum119895

119895(119905) 119890

119894119908119895120585(76)

with

119908119895=2120587119895

119871 119895 = minus

119873

2+ 1 0

119873

2 (77)



V119895(119905) =

1

119871int119871

0

V (120585 119905) 119890minus119894119908119895120585119889120585 (78)




⟨119891 119892⟩ = int119871

0

119891 (120585) 119892 (120585) 119889120585 (79)

it follows that

V1015840119895(119905) = 120574

1119895+ 119873

1

1015840119895(119905) = 120574

2V119895+ 119873

2

(80)

where

1205741=minus (21205733) 1198941199083

119895

1 + (1205732)1199082119895

1205742=minus1205731205901198941199083

119895

1 + (1205732)1199082119895

1198731= minus119894119908119895119875119895[(1 + 120572V1198722) 119906]1 + (1205732)1199082

119895

1198732=119894119908119895119875119895[120573120590 (minus (311987210158401198724) V

120585+ (3 (1198721015840)

2


1 + (1205732)1199082119895

(81)


119875119895[119892] =

1

119871int119871

0

119892 (120585) 119890minus119894119908119895120585119889120585 (82)



V119899+1119895minus V119899

119895

Δ119905=1205741(119899+1119895+ 119899

119895)

2+3

21198731198991minus1

2119873119899minus11

119899+1119895minus 119899

119895

Δ119905=1205742(V119899+1119895+ V119899

119895)

2+3

21198731198992minus1

2119873119899minus12

(83)






119895 119899+1119895




120585are


V120585(120585 119905) = sum

119895

119894119908119895119875119895[V (sdot 119905)] 119890119894119908119895120585 (84)





V (120585 119905) = V (120585 minus 119888119905)

119906 (120585 119905) = (120585 minus 119888119905) (85)



(V10158401015840

11990610158401015840)

=1

(1205734) 1198882 minus 21205731205903(

119888

2

2

3

120590119888

2

)(119888V minus ((1 + 120572V) 119906)

119888119906 minus V minus120572

21199062)

(86)


V (119909) asymp V0+1198732

sum119899=1

V119899cos(119899120587

119897119909)

119906 (119909) asymp 1199060+1198732

sum119899=1

119906119899cos(119899120587

119897119909)

(87)

where

V0=1

119897int119897

0

V (119909) 119889119909 =1

2119897int2119897

0

V (119909) 119889119909

V119899=2

119897int119897

0

V (119909) cos(119899120587119909119897) 119889119909

=2

2119897int2119897

0

V (119909) cos(119899120587119909119897) 119889119909

(88)


119909119895=2119897 (119895 minus 1)

119873 119895 = 1

119873

2+ 1 (89)


119865 (V0 V1 V

1198732 1199060 1199061 119906

1198732) = 0 (90)




119869119894119895119865 (119909) asymp

119865119894(119909 + ℎ119890

119895) minus 119865

119894(119909 minus ℎ119890

119895)

2ℎ

119895 = 1 119873 + 2

(91)




1199060(119909) = cos(120587119909

119897)

V0(119909) = cos(120587119909

119897)

(92)


V = minus(119868 minus120573

21205972120585)Φ

119905+ 120573120590V

120585120585minus120572

2Φ2120585

= minusΦ119905+ 119874 (120572 120573 120590)

(93)


minus Φ119905119905+ Φ

120585120585+ 120573 (1 minus 120590)Φ

120585120585119905119905minus2120573

3Φ120585120585120585120585minus 2120572Φ

120585Φ120585119905

minus 120572Φ119905Φ120585120585= 0

(94)





3)Φ1015840101584010158401015840

+3120572119888

2(Φ10158402)

1015840

= 0

(96)



3)Φ101584010158401015840 +

3120572119888

2(Φ1015840)

2

= 0

(97)


(1 minus 1198882) (Φ1015840)2

+ 120573((1 minus 120590) 1198882 minus2

3) (Φ10158401015840)

2

+ 120572119888 (Φ1015840)3

= 0

(98)


V = 119888119906 minus120573119888

211990610158401015840 + 12057312059011988811990610158401015840 minus

120572

21199062 (99)



119860 =1198882 minus 1

120572119888

119861 = radic1198882 minus 1

4120573 ((1 minus 120590) 1198882 minus 23)

(101)



2)

sdot [41198601198612sech2 (119861119909) tanh2 (119861119909)

minus 21198601198612sech4 (119861119909)]

(102)






119906 (120585 0) = V (120585 0) = 119890minus20(119909minus28)2

(103)




v u

2 4 6 8 100x

0

02

04

06

08

1

12

14

2 4 6 8 100x

0

05

1

15


v u

minus1

minus05

0

05

2 4 6 8 100x

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

02

2 4 6 8 100x






1198622120590=1199082

1198962

=1 + (21205733) 1198962

(1 + (1205732) 1198962)2+1205731205901198962 (1 + (21205733) 1198962)

(1 + (1205732) 1198962)2

gt 11986220

(104)


0119890119894(119896120585minus119908119905)

V = V0119890119894(119896120585minus119908119905) where 119906

0 V0are constants


v u

minus02

0

02

04

06

08

1

12

14

16

2 4 6 8 100

x

0

02

04

06

08

1

12

14

2 4 6 8 100

x


v

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

2 4 6 8 100

x

u

minus25

minus2

minus15

minus1

minus05

0

2 4 6 8 100

x





120590is larger







at t = 10

0

05

1

15u at t = 10

0

02

04

06

08

1

12

14

2 4 6 8 100

120585

2 4 6 8 100

120585


v

minus01

0

01

03

02

04

05

06

07

45 50 55 6040

x

u

45 50 55 6040

x

minus01

0

01

03

02

04

05

06

07





119906 (120585 0) = V (120585 0) = 119890minus20(119909minus198)2 (105)


v

0

005

01

015

02

025

03

035

04

045

45 50 55 6040

x

u

0

005

01

015

02

025

03

035

04

45 50 55 6040

x


v

minus05

0

05

1

15

2

25

45 50 55 6040

x

u

minus05

0

05

1

15

2

45 50 55 6040

x




7 Conclusions



at t = 10 u at t = 10

minus01

0

01

02

03

04

05

06

07

minus01

0

01

02

03

04

05

06

07

60 65 7055

120585

60 65 7055

120585


u at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

20 30 40 5010

120585

20 30 40 5010

120585






minus04

minus03

minus02

minus01

0

01

02

03

04

05u at t = 20

35 40 45 50 55 6030120585

v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 55 6030120585


0 5 10 15 20 25 30 35 40 45 50

C2120590

C20

81205903

radic120573k

0

02

04

06

08

1

12

14

16





05060708091

1112131415

30 32 34 36 38 40 4228

x





u at t = 20

35 40 45 50 5530120585

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 5530120585


v at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

290 300 310 320280

120585

u at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

280 300 310 320290

120585




Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











V (120585 119905) = sum119895

V119895(119905) 119890

119894119908119895120585

119906 (120585 119905) = sum119895

119895(119905) 119890

119894119908119895120585(76)

with

119908119895=2120587119895

119871 119895 = minus

119873

2+ 1 0

119873

2 (77)



V119895(119905) =

1

119871int119871

0

V (120585 119905) 119890minus119894119908119895120585119889120585 (78)




⟨119891 119892⟩ = int119871

0

119891 (120585) 119892 (120585) 119889120585 (79)

it follows that

V1015840119895(119905) = 120574

1119895+ 119873

1

1015840119895(119905) = 120574

2V119895+ 119873

2

(80)

where

1205741=minus (21205733) 1198941199083

119895

1 + (1205732)1199082119895

1205742=minus1205731205901198941199083

119895

1 + (1205732)1199082119895

1198731= minus119894119908119895119875119895[(1 + 120572V1198722) 119906]1 + (1205732)1199082

119895

1198732=119894119908119895119875119895[120573120590 (minus (311987210158401198724) V

120585+ (3 (1198721015840)

2


1 + (1205732)1199082119895

(81)


119875119895[119892] =

1

119871int119871

0

119892 (120585) 119890minus119894119908119895120585119889120585 (82)



V119899+1119895minus V119899

119895

Δ119905=1205741(119899+1119895+ 119899

119895)

2+3

21198731198991minus1

2119873119899minus11

119899+1119895minus 119899

119895

Δ119905=1205742(V119899+1119895+ V119899

119895)

2+3

21198731198992minus1

2119873119899minus12

(83)






119895 119899+1119895




120585are


V120585(120585 119905) = sum

119895

119894119908119895119875119895[V (sdot 119905)] 119890119894119908119895120585 (84)





V (120585 119905) = V (120585 minus 119888119905)

119906 (120585 119905) = (120585 minus 119888119905) (85)



(V10158401015840

11990610158401015840)

=1

(1205734) 1198882 minus 21205731205903(

119888

2

2

3

120590119888

2

)(119888V minus ((1 + 120572V) 119906)

119888119906 minus V minus120572

21199062)

(86)


V (119909) asymp V0+1198732

sum119899=1

V119899cos(119899120587

119897119909)

119906 (119909) asymp 1199060+1198732

sum119899=1

119906119899cos(119899120587

119897119909)

(87)

where

V0=1

119897int119897

0

V (119909) 119889119909 =1

2119897int2119897

0

V (119909) 119889119909

V119899=2

119897int119897

0

V (119909) cos(119899120587119909119897) 119889119909

=2

2119897int2119897

0

V (119909) cos(119899120587119909119897) 119889119909

(88)


119909119895=2119897 (119895 minus 1)

119873 119895 = 1

119873

2+ 1 (89)


119865 (V0 V1 V

1198732 1199060 1199061 119906

1198732) = 0 (90)




119869119894119895119865 (119909) asymp

119865119894(119909 + ℎ119890

119895) minus 119865

119894(119909 minus ℎ119890

119895)

2ℎ

119895 = 1 119873 + 2

(91)




1199060(119909) = cos(120587119909

119897)

V0(119909) = cos(120587119909

119897)

(92)


V = minus(119868 minus120573

21205972120585)Φ

119905+ 120573120590V

120585120585minus120572

2Φ2120585

= minusΦ119905+ 119874 (120572 120573 120590)

(93)


minus Φ119905119905+ Φ

120585120585+ 120573 (1 minus 120590)Φ

120585120585119905119905minus2120573

3Φ120585120585120585120585minus 2120572Φ

120585Φ120585119905

minus 120572Φ119905Φ120585120585= 0

(94)





3)Φ1015840101584010158401015840

+3120572119888

2(Φ10158402)

1015840

= 0

(96)



3)Φ101584010158401015840 +

3120572119888

2(Φ1015840)

2

= 0

(97)


(1 minus 1198882) (Φ1015840)2

+ 120573((1 minus 120590) 1198882 minus2

3) (Φ10158401015840)

2

+ 120572119888 (Φ1015840)3

= 0

(98)


V = 119888119906 minus120573119888

211990610158401015840 + 12057312059011988811990610158401015840 minus

120572

21199062 (99)



119860 =1198882 minus 1

120572119888

119861 = radic1198882 minus 1

4120573 ((1 minus 120590) 1198882 minus 23)

(101)



2)

sdot [41198601198612sech2 (119861119909) tanh2 (119861119909)

minus 21198601198612sech4 (119861119909)]

(102)






119906 (120585 0) = V (120585 0) = 119890minus20(119909minus28)2

(103)




v u

2 4 6 8 100x

0

02

04

06

08

1

12

14

2 4 6 8 100x

0

05

1

15


v u

minus1

minus05

0

05

2 4 6 8 100x

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

02

2 4 6 8 100x






1198622120590=1199082

1198962

=1 + (21205733) 1198962

(1 + (1205732) 1198962)2+1205731205901198962 (1 + (21205733) 1198962)

(1 + (1205732) 1198962)2

gt 11986220

(104)


0119890119894(119896120585minus119908119905)

V = V0119890119894(119896120585minus119908119905) where 119906

0 V0are constants


v u

minus02

0

02

04

06

08

1

12

14

16

2 4 6 8 100

x

0

02

04

06

08

1

12

14

2 4 6 8 100

x


v

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

2 4 6 8 100

x

u

minus25

minus2

minus15

minus1

minus05

0

2 4 6 8 100

x





120590is larger







at t = 10

0

05

1

15u at t = 10

0

02

04

06

08

1

12

14

2 4 6 8 100

120585

2 4 6 8 100

120585


v

minus01

0

01

03

02

04

05

06

07

45 50 55 6040

x

u

45 50 55 6040

x

minus01

0

01

03

02

04

05

06

07





119906 (120585 0) = V (120585 0) = 119890minus20(119909minus198)2 (105)


v

0

005

01

015

02

025

03

035

04

045

45 50 55 6040

x

u

0

005

01

015

02

025

03

035

04

45 50 55 6040

x


v

minus05

0

05

1

15

2

25

45 50 55 6040

x

u

minus05

0

05

1

15

2

45 50 55 6040

x




7 Conclusions



at t = 10 u at t = 10

minus01

0

01

02

03

04

05

06

07

minus01

0

01

02

03

04

05

06

07

60 65 7055

120585

60 65 7055

120585


u at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

20 30 40 5010

120585

20 30 40 5010

120585






minus04

minus03

minus02

minus01

0

01

02

03

04

05u at t = 20

35 40 45 50 55 6030120585

v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 55 6030120585


0 5 10 15 20 25 30 35 40 45 50

C2120590

C20

81205903

radic120573k

0

02

04

06

08

1

12

14

16





05060708091

1112131415

30 32 34 36 38 40 4228

x





u at t = 20

35 40 45 50 5530120585

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 5530120585


v at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

290 300 310 320280

120585

u at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

280 300 310 320290

120585




Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120585are


V120585(120585 119905) = sum

119895

119894119908119895119875119895[V (sdot 119905)] 119890119894119908119895120585 (84)





V (120585 119905) = V (120585 minus 119888119905)

119906 (120585 119905) = (120585 minus 119888119905) (85)



(V10158401015840

11990610158401015840)

=1

(1205734) 1198882 minus 21205731205903(

119888

2

2

3

120590119888

2

)(119888V minus ((1 + 120572V) 119906)

119888119906 minus V minus120572

21199062)

(86)


V (119909) asymp V0+1198732

sum119899=1

V119899cos(119899120587

119897119909)

119906 (119909) asymp 1199060+1198732

sum119899=1

119906119899cos(119899120587

119897119909)

(87)

where

V0=1

119897int119897

0

V (119909) 119889119909 =1

2119897int2119897

0

V (119909) 119889119909

V119899=2

119897int119897

0

V (119909) cos(119899120587119909119897) 119889119909

=2

2119897int2119897

0

V (119909) cos(119899120587119909119897) 119889119909

(88)


119909119895=2119897 (119895 minus 1)

119873 119895 = 1

119873

2+ 1 (89)


119865 (V0 V1 V

1198732 1199060 1199061 119906

1198732) = 0 (90)




119869119894119895119865 (119909) asymp

119865119894(119909 + ℎ119890

119895) minus 119865

119894(119909 minus ℎ119890

119895)

2ℎ

119895 = 1 119873 + 2

(91)




1199060(119909) = cos(120587119909

119897)

V0(119909) = cos(120587119909

119897)

(92)


V = minus(119868 minus120573

21205972120585)Φ

119905+ 120573120590V

120585120585minus120572

2Φ2120585

= minusΦ119905+ 119874 (120572 120573 120590)

(93)


minus Φ119905119905+ Φ

120585120585+ 120573 (1 minus 120590)Φ

120585120585119905119905minus2120573

3Φ120585120585120585120585minus 2120572Φ

120585Φ120585119905

minus 120572Φ119905Φ120585120585= 0

(94)





3)Φ1015840101584010158401015840

+3120572119888

2(Φ10158402)

1015840

= 0

(96)



3)Φ101584010158401015840 +

3120572119888

2(Φ1015840)

2

= 0

(97)


(1 minus 1198882) (Φ1015840)2

+ 120573((1 minus 120590) 1198882 minus2

3) (Φ10158401015840)

2

+ 120572119888 (Φ1015840)3

= 0

(98)


V = 119888119906 minus120573119888

211990610158401015840 + 12057312059011988811990610158401015840 minus

120572

21199062 (99)



119860 =1198882 minus 1

120572119888

119861 = radic1198882 minus 1

4120573 ((1 minus 120590) 1198882 minus 23)

(101)



2)

sdot [41198601198612sech2 (119861119909) tanh2 (119861119909)

minus 21198601198612sech4 (119861119909)]

(102)






119906 (120585 0) = V (120585 0) = 119890minus20(119909minus28)2

(103)




v u

2 4 6 8 100x

0

02

04

06

08

1

12

14

2 4 6 8 100x

0

05

1

15


v u

minus1

minus05

0

05

2 4 6 8 100x

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

02

2 4 6 8 100x






1198622120590=1199082

1198962

=1 + (21205733) 1198962

(1 + (1205732) 1198962)2+1205731205901198962 (1 + (21205733) 1198962)

(1 + (1205732) 1198962)2

gt 11986220

(104)


0119890119894(119896120585minus119908119905)

V = V0119890119894(119896120585minus119908119905) where 119906

0 V0are constants


v u

minus02

0

02

04

06

08

1

12

14

16

2 4 6 8 100

x

0

02

04

06

08

1

12

14

2 4 6 8 100

x


v

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

2 4 6 8 100

x

u

minus25

minus2

minus15

minus1

minus05

0

2 4 6 8 100

x





120590is larger







at t = 10

0

05

1

15u at t = 10

0

02

04

06

08

1

12

14

2 4 6 8 100

120585

2 4 6 8 100

120585


v

minus01

0

01

03

02

04

05

06

07

45 50 55 6040

x

u

45 50 55 6040

x

minus01

0

01

03

02

04

05

06

07





119906 (120585 0) = V (120585 0) = 119890minus20(119909minus198)2 (105)


v

0

005

01

015

02

025

03

035

04

045

45 50 55 6040

x

u

0

005

01

015

02

025

03

035

04

45 50 55 6040

x


v

minus05

0

05

1

15

2

25

45 50 55 6040

x

u

minus05

0

05

1

15

2

45 50 55 6040

x




7 Conclusions



at t = 10 u at t = 10

minus01

0

01

02

03

04

05

06

07

minus01

0

01

02

03

04

05

06

07

60 65 7055

120585

60 65 7055

120585


u at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

20 30 40 5010

120585

20 30 40 5010

120585






minus04

minus03

minus02

minus01

0

01

02

03

04

05u at t = 20

35 40 45 50 55 6030120585

v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 55 6030120585


0 5 10 15 20 25 30 35 40 45 50

C2120590

C20

81205903

radic120573k

0

02

04

06

08

1

12

14

16





05060708091

1112131415

30 32 34 36 38 40 4228

x





u at t = 20

35 40 45 50 5530120585

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 5530120585


v at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

290 300 310 320280

120585

u at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

280 300 310 320290

120585




Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













3)Φ1015840101584010158401015840

+3120572119888

2(Φ10158402)

1015840

= 0

(96)



3)Φ101584010158401015840 +

3120572119888

2(Φ1015840)

2

= 0

(97)


(1 minus 1198882) (Φ1015840)2

+ 120573((1 minus 120590) 1198882 minus2

3) (Φ10158401015840)

2

+ 120572119888 (Φ1015840)3

= 0

(98)


V = 119888119906 minus120573119888

211990610158401015840 + 12057312059011988811990610158401015840 minus

120572

21199062 (99)



119860 =1198882 minus 1

120572119888

119861 = radic1198882 minus 1

4120573 ((1 minus 120590) 1198882 minus 23)

(101)



2)

sdot [41198601198612sech2 (119861119909) tanh2 (119861119909)

minus 21198601198612sech4 (119861119909)]

(102)






119906 (120585 0) = V (120585 0) = 119890minus20(119909minus28)2

(103)




v u

2 4 6 8 100x

0

02

04

06

08

1

12

14

2 4 6 8 100x

0

05

1

15


v u

minus1

minus05

0

05

2 4 6 8 100x

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

02

2 4 6 8 100x






1198622120590=1199082

1198962

=1 + (21205733) 1198962

(1 + (1205732) 1198962)2+1205731205901198962 (1 + (21205733) 1198962)

(1 + (1205732) 1198962)2

gt 11986220

(104)


0119890119894(119896120585minus119908119905)

V = V0119890119894(119896120585minus119908119905) where 119906

0 V0are constants


v u

minus02

0

02

04

06

08

1

12

14

16

2 4 6 8 100

x

0

02

04

06

08

1

12

14

2 4 6 8 100

x


v

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

2 4 6 8 100

x

u

minus25

minus2

minus15

minus1

minus05

0

2 4 6 8 100

x





120590is larger







at t = 10

0

05

1

15u at t = 10

0

02

04

06

08

1

12

14

2 4 6 8 100

120585

2 4 6 8 100

120585


v

minus01

0

01

03

02

04

05

06

07

45 50 55 6040

x

u

45 50 55 6040

x

minus01

0

01

03

02

04

05

06

07





119906 (120585 0) = V (120585 0) = 119890minus20(119909minus198)2 (105)


v

0

005

01

015

02

025

03

035

04

045

45 50 55 6040

x

u

0

005

01

015

02

025

03

035

04

45 50 55 6040

x


v

minus05

0

05

1

15

2

25

45 50 55 6040

x

u

minus05

0

05

1

15

2

45 50 55 6040

x




7 Conclusions



at t = 10 u at t = 10

minus01

0

01

02

03

04

05

06

07

minus01

0

01

02

03

04

05

06

07

60 65 7055

120585

60 65 7055

120585


u at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

20 30 40 5010

120585

20 30 40 5010

120585






minus04

minus03

minus02

minus01

0

01

02

03

04

05u at t = 20

35 40 45 50 55 6030120585

v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 55 6030120585


0 5 10 15 20 25 30 35 40 45 50

C2120590

C20

81205903

radic120573k

0

02

04

06

08

1

12

14

16





05060708091

1112131415

30 32 34 36 38 40 4228

x





u at t = 20

35 40 45 50 5530120585

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 5530120585


v at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

290 300 310 320280

120585

u at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

280 300 310 320290

120585




Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










v u

2 4 6 8 100x

0

02

04

06

08

1

12

14

2 4 6 8 100x

0

05

1

15


v u

minus1

minus05

0

05

2 4 6 8 100x

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

02

2 4 6 8 100x






1198622120590=1199082

1198962

=1 + (21205733) 1198962

(1 + (1205732) 1198962)2+1205731205901198962 (1 + (21205733) 1198962)

(1 + (1205732) 1198962)2

gt 11986220

(104)


0119890119894(119896120585minus119908119905)

V = V0119890119894(119896120585minus119908119905) where 119906

0 V0are constants


v u

minus02

0

02

04

06

08

1

12

14

16

2 4 6 8 100

x

0

02

04

06

08

1

12

14

2 4 6 8 100

x


v

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

2 4 6 8 100

x

u

minus25

minus2

minus15

minus1

minus05

0

2 4 6 8 100

x





120590is larger







at t = 10

0

05

1

15u at t = 10

0

02

04

06

08

1

12

14

2 4 6 8 100

120585

2 4 6 8 100

120585


v

minus01

0

01

03

02

04

05

06

07

45 50 55 6040

x

u

45 50 55 6040

x

minus01

0

01

03

02

04

05

06

07





119906 (120585 0) = V (120585 0) = 119890minus20(119909minus198)2 (105)


v

0

005

01

015

02

025

03

035

04

045

45 50 55 6040

x

u

0

005

01

015

02

025

03

035

04

45 50 55 6040

x


v

minus05

0

05

1

15

2

25

45 50 55 6040

x

u

minus05

0

05

1

15

2

45 50 55 6040

x




7 Conclusions



at t = 10 u at t = 10

minus01

0

01

02

03

04

05

06

07

minus01

0

01

02

03

04

05

06

07

60 65 7055

120585

60 65 7055

120585


u at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

20 30 40 5010

120585

20 30 40 5010

120585






minus04

minus03

minus02

minus01

0

01

02

03

04

05u at t = 20

35 40 45 50 55 6030120585

v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 55 6030120585


0 5 10 15 20 25 30 35 40 45 50

C2120590

C20

81205903

radic120573k

0

02

04

06

08

1

12

14

16





05060708091

1112131415

30 32 34 36 38 40 4228

x





u at t = 20

35 40 45 50 5530120585

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 5530120585


v at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

290 300 310 320280

120585

u at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

280 300 310 320290

120585




Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










v u

minus02

0

02

04

06

08

1

12

14

16

2 4 6 8 100

x

0

02

04

06

08

1

12

14

2 4 6 8 100

x


v

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

2 4 6 8 100

x

u

minus25

minus2

minus15

minus1

minus05

0

2 4 6 8 100

x





120590is larger







at t = 10

0

05

1

15u at t = 10

0

02

04

06

08

1

12

14

2 4 6 8 100

120585

2 4 6 8 100

120585


v

minus01

0

01

03

02

04

05

06

07

45 50 55 6040

x

u

45 50 55 6040

x

minus01

0

01

03

02

04

05

06

07





119906 (120585 0) = V (120585 0) = 119890minus20(119909minus198)2 (105)


v

0

005

01

015

02

025

03

035

04

045

45 50 55 6040

x

u

0

005

01

015

02

025

03

035

04

45 50 55 6040

x


v

minus05

0

05

1

15

2

25

45 50 55 6040

x

u

minus05

0

05

1

15

2

45 50 55 6040

x




7 Conclusions



at t = 10 u at t = 10

minus01

0

01

02

03

04

05

06

07

minus01

0

01

02

03

04

05

06

07

60 65 7055

120585

60 65 7055

120585


u at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

20 30 40 5010

120585

20 30 40 5010

120585






minus04

minus03

minus02

minus01

0

01

02

03

04

05u at t = 20

35 40 45 50 55 6030120585

v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 55 6030120585


0 5 10 15 20 25 30 35 40 45 50

C2120590

C20

81205903

radic120573k

0

02

04

06

08

1

12

14

16





05060708091

1112131415

30 32 34 36 38 40 4228

x





u at t = 20

35 40 45 50 5530120585

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 5530120585


v at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

290 300 310 320280

120585

u at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

280 300 310 320290

120585




Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










at t = 10

0

05

1

15u at t = 10

0

02

04

06

08

1

12

14

2 4 6 8 100

120585

2 4 6 8 100

120585


v

minus01

0

01

03

02

04

05

06

07

45 50 55 6040

x

u

45 50 55 6040

x

minus01

0

01

03

02

04

05

06

07





119906 (120585 0) = V (120585 0) = 119890minus20(119909minus198)2 (105)


v

0

005

01

015

02

025

03

035

04

045

45 50 55 6040

x

u

0

005

01

015

02

025

03

035

04

45 50 55 6040

x


v

minus05

0

05

1

15

2

25

45 50 55 6040

x

u

minus05

0

05

1

15

2

45 50 55 6040

x




7 Conclusions



at t = 10 u at t = 10

minus01

0

01

02

03

04

05

06

07

minus01

0

01

02

03

04

05

06

07

60 65 7055

120585

60 65 7055

120585


u at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

20 30 40 5010

120585

20 30 40 5010

120585






minus04

minus03

minus02

minus01

0

01

02

03

04

05u at t = 20

35 40 45 50 55 6030120585

v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 55 6030120585


0 5 10 15 20 25 30 35 40 45 50

C2120590

C20

81205903

radic120573k

0

02

04

06

08

1

12

14

16





05060708091

1112131415

30 32 34 36 38 40 4228

x





u at t = 20

35 40 45 50 5530120585

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 5530120585


v at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

290 300 310 320280

120585

u at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

280 300 310 320290

120585




Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










v

0

005

01

015

02

025

03

035

04

045

45 50 55 6040

x

u

0

005

01

015

02

025

03

035

04

45 50 55 6040

x


v

minus05

0

05

1

15

2

25

45 50 55 6040

x

u

minus05

0

05

1

15

2

45 50 55 6040

x




7 Conclusions



at t = 10 u at t = 10

minus01

0

01

02

03

04

05

06

07

minus01

0

01

02

03

04

05

06

07

60 65 7055

120585

60 65 7055

120585


u at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

20 30 40 5010

120585

20 30 40 5010

120585






minus04

minus03

minus02

minus01

0

01

02

03

04

05u at t = 20

35 40 45 50 55 6030120585

v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 55 6030120585


0 5 10 15 20 25 30 35 40 45 50

C2120590

C20

81205903

radic120573k

0

02

04

06

08

1

12

14

16





05060708091

1112131415

30 32 34 36 38 40 4228

x





u at t = 20

35 40 45 50 5530120585

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 5530120585


v at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

290 300 310 320280

120585

u at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

280 300 310 320290

120585




Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










at t = 10 u at t = 10

minus01

0

01

02

03

04

05

06

07

minus01

0

01

02

03

04

05

06

07

60 65 7055

120585

60 65 7055

120585


u at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

20 30 40 5010

120585

20 30 40 5010

120585






minus04

minus03

minus02

minus01

0

01

02

03

04

05u at t = 20

35 40 45 50 55 6030120585

v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 55 6030120585


0 5 10 15 20 25 30 35 40 45 50

C2120590

C20

81205903

radic120573k

0

02

04

06

08

1

12

14

16





05060708091

1112131415

30 32 34 36 38 40 4228

x





u at t = 20

35 40 45 50 5530120585

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 5530120585


v at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

290 300 310 320280

120585

u at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

280 300 310 320290

120585




Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus04

minus03

minus02

minus01

0

01

02

03

04

05u at t = 20

35 40 45 50 55 6030120585

v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 55 6030120585


0 5 10 15 20 25 30 35 40 45 50

C2120590

C20

81205903

radic120573k

0

02

04

06

08

1

12

14

16





05060708091

1112131415

30 32 34 36 38 40 4228

x





u at t = 20

35 40 45 50 5530120585

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 5530120585


v at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

290 300 310 320280

120585

u at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

280 300 310 320290

120585




Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










u at t = 20

35 40 45 50 5530120585

minus04

minus03

minus02

minus01

0

01

02

03

04

05v at t = 20

minus04

minus03

minus02

minus01

0

01

02

03

04

05

35 40 45 50 5530120585


v at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

290 300 310 320280

120585

u at t = 100

minus02

minus015

minus01

minus005

0

005

01

015

02

280 300 310 320290

120585




Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Propagation of Water Waves over Uneven …downloads.hindawi.com/journals/ijde/2015/805625.pdf · 2019-07-31 · Research Article Propagation of Water Waves over Uneven