Research Article A Strongly A-Stable Time Integration Method for … · 2019. 7. 31. · erential equation supplemented with Dirichlet or Neumann boundary condition. We successfully

Research ArticleA Strongly A-Stable Time Integration Method for Solving theNonlinear Reaction-Diffusion Equation

Wenyuan Liao

Department of Mathematics and Statistics University of Calgary 2500 University Drive NW Calgary AB Canada T2N 1N4

Correspondence should be addressed to Wenyuan Liao wliaoucalgaryca

Received 28 July 2014 Accepted 17 October 2014

Academic Editor Santanu Saha Ray

Copyright copy 2015 Wenyuan LiaoThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The semidiscrete ordinary differential equation (ODE) system resulting from compact higher-order finite difference spatialdiscretization of a nonlinear parabolic partial differential equation for instance the reaction-diffusion equation is highly stiffTherefore numerical time integration methods with stiff stability such as implicit Runge-Kutta methods and implicit multistepmethods are required to solve the large-scale stiff ODE system However those methods are computationally expensive especiallyfor nonlinear cases Rosenbrock method is efficient since it is iteration-free however it suffers from order reduction when it is usedfor nonlinear parabolic partial differential equation In this work we construct a new fourth-order Rosenbrock method to solve thenonlinear parabolic partial differential equation supplemented with Dirichlet or Neumann boundary condition We successfullyresolved the phenomena of order reduction so the new method is fourth-order in time when it is used for nonlinear parabolicpartial differential equations Moreover it has been shown that the Rosenbrock method is strongly A-stable hence suitable for thestiff ODE system obtained from compact finite difference discretization of the nonlinear parabolic partial differential equationSeveral numerical experiments have been conducted to demonstrate the efficiency stability and accuracy of the new method

1 Introduction

Let us consider the following parabolic partial differentialequation

119906119905= 119863119906119909119909+ 119891 (119906 119909 119905) (119909 119905) isin (119886 119887) times (0 119879] (1)

with the initial condition

119906 (119909 0) = 1199060(119909) 119909 isin [119886 119887] (2)

where 119863 is a positive constant describing the diffusionproperty and119891(119906 119909 119905) is a function representing the reactionterm which is nonlinear on 119906 The unknown function 119906represents depending on the applications variables such asmass concentration in chemical reaction process tempera-ture in heat conduction neutron flux in nuclear reactorsand population density in population dynamics On theboundary either Dirichlet condition

119906 (119886 119905) = 1198921(119905) 119906 (119887 119905) = 119892

2(119905) 0 le 119905 le 119879 (3)

or Neumann condition

119906119909(119886 119905) = 119892

3(119905) 119906

119909(119887 119905) = 119892

4(119905) 0 le 119905 le 119879 (4)

is specified where 1198921 1198922 1198923 and 119892

4are sufficiently smooth

functions Here in this paper we restrict our attention onDirichlet and Neumann boundary conditions while thedeveloped techniques can be easily extended toRobin bound-ary condition

Efficient and accurate numerical methods for solving (1)had attracted great attentions from scientists and engineersas for many application problems in science and engineer-ing it is preferable to use high-order compact numericalalgorithms to compute accurate solutions In the past severaldecades a great deal of work has been done in the develop-ment of efficient accurate and robust numerical algorithmfor solving such problem For more details the reader isreferred to [1ndash4]

Since both temporal and spatial derivatives are involvedin the equation we discuss the numerical treatments in

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2015 Article ID 539652 12 pageshttpdxdoiorg1011552015539652

2 Abstract and Applied Analysis

time and space separately Here we first apply the high-order compact finite difference approximation to the spatialderivative so a semidiscrete ODE system is obtained whichis then solved by a fourth-order Rosenbrockmethod that willbe discussed later

Recently there have been attempts to develop high-ordercompact scheme for the spatial derivative In [5] a three-point combined compact difference scheme was proposedto approximate the first and second derivatives for problemwith periodic boundary condition The resulting scheme hasup to sixth-order accuracy at all grid points including theboundary nodes for periodic boundaries however it is onlyfourth-order accurate for nonperiodic boundary condition

Because the semidiscrete ODE system obtained fromspatial discretization such as method of lines of the nonlin-ear parabolic partial differential equation is highly stiff thechoices of time integration methods are limited to implicitmethods only Explicit algorithm is efficient in a single timestep but suffers from strict step size restriction which makesit less efficient Implicit method on the other side is lessefficient in a single step but the unconditional stability allowsthe use of larger time step hence the overall computationalefficiency can be significantly improved One issue howeveris that the iteration is usually slow when large step size Δ119905is used Also due to the stiffness of the ODE system onlyNewton-type iterative methods are applicable to solve thenonlinear algebraic system Furthermore strong A-stabilityor L-stability of the time integration method is necessaryfor error damping A great deal of work has been done inthe development of efficient time stepping methods for thestiff ODE system In [6] explicit exponential Rosenbrockmethods of order five have been constructed to solve thelarge-scale stiff ODE system Through the derivation of stifforder condition new pairs of embedded methods of higher-order can be obtained Similarly fifth-order explicit expo-nential Runge-Kutta methods were constructed to efficientlyintegrate the semilinear stiff problems in [7]The authors havealso shown that there does not exist an explicit exponentialRunge-Kutta method of order 5 with less than or equal to 6stages therefore the resultant methods are 8-stage methodsIn [8] a fourth-order time stepping method which is amodification of the exponential time-differencing fourth-order Runge-Kutta method has been developed for stiffODEs These methods are efficient and accurate HoweverA-stability of the time stepping method is not sufficient forhighly stiff problem To overcome these difficulties it isdesirable to construct new algorithms with strong A-stabilityor L-stability that are free of solving nonlinear equationsIt turns out that the Rosenbrock method which was firstlyreported by Rosenbrock [9] and then improved by Haines[10] responded to these issues with considerably satisfyingand promising results

The objective herein is to develop a strongly A-stableRosenbrock method to solve the semidiscrete stiff ODEsresulting from compact high-order finite difference approx-imation of a semilinear parabolic partial differential equa-tion The rest of the paper is organized as the followingIn Section 2 we discretize the spatial derivatives of thesemilinear parabolic partial differential equation using a

fourth-order compact finite difference scheme which is thencombined with a newly proposed compact fourth-orderboundary condition treatment to form the semidiscrete ODEsystem In Section 3 we focus on the development of a fourth-order strongly A-stable Rosenbrock method and the stabilityanalysis Several numerical examples are used to demonstratethe accuracy and efficiency of the new algorithm in Section 4which is followed by conclusions and possible future work

2 Compact Fourth-OrderSpatial Discretization

For the sake of simplicity we assume that the 1D spatialdomain Ω = [119886 119887] is divided into119872 subintervals with equallength ℎ = (119887 minus 119886)119872 Let 119909

119894= 119886 + 119894 sdot ℎ 119894 = 0 1 119872 be the

grid points A variety of compact high-order discretizationscan be utilized to approximate the second derivative 119906

119909119909in

(1)Here we introduce a compact finite difference scheme to

approximate 119906119909119909 such that the resulting semidiscrete ODE

system is an accurate and compact approximation to theoriginal semilinear parabolic partial differential equationThis operator-approximation based method has been widelyused to solve various multidimensional problems We firstdefine the central finite difference operator 1205752

119909as

1205752

119909119906119894= 119906119894+1minus 2119906119894+ 119906119894minus1 (5)

then1205752119909ℎ2 gives second-order accurate approximation to119906

119909119909

Using Taylor series to expand all terms on the right-hand sideof (5) under the assumption that 119906(119909) is sufficiently smoothwe have

1205752

119909

ℎ2119906119894= 119906119909119909(119909119894) +

ℎ2

12

119906119909119909119909119909(119909119894) + 119874 (ℎ

4) 1 le 119894 le 119872 minus 1

(6)

To improve the above finite difference approximation tofourth-order accurate one just needs to eliminate the second-order error term Applying 1205752

11990912 to both sides of (6) we have

1205752

119909

ℎ2

1205752

119909

12

119906119894= 119906119909119909119909119909(119909119894) + 119874 (ℎ

4) 1 le 119894 le 119872 minus 1 (7)

Combining (6) with (7) neglecting 119874(ℎ4) we obtain thefollowing fourth-order accurate approximation to119906

119909119909at node

119909119894

119906119909119909(119909119894) asymp

1

ℎ21205752

119909(1 minus

1205752

119909

12

)119906119894 1 le 119894 le 119872 minus 1 (8)

The drawback is that a five-point stencil is required thereforethe compactness is destroyed so the method becomes lessefficient Further investigation shows that the differencebetween 1 minus 1205752

11990912 and (1 + 1205752

11990912)minus1 is 119874(ℎ4) so a natural

way is to approximate 119906119909119909(119909119894) as 1205752119909(1 + 120575

2

11990912)minus1119906119894 which is

fourth-order accurate and compact

Abstract and Applied Analysis 3

Applying the fourth-order Pade approximation to 119906119909119909

in(1) we obtain the following ODE system

1199061015840

119894(119905) =

119863

ℎ2

1205752

119909

1 + 1205752

11990912

119906119894(119905) + 119891 (119906

119894(119905) 119909119894 119905)

1 le 119894 le 119872 minus 1

(9)

which is a fourth-order accurate approximation (in space) tothe original semilinear parabolic partial differential equationdefined in (1)

However the above algorithm is difficult to implementso we multiply 1+1205752

11990912 to both sides to obtain the following

implicit ODE system

(1 +

1205752

119909

12

)1199061015840

119894(119905) =

119863

ℎ21205752

119909119906119894(119905) + (1 +

1205752

119909

12

)119891 (119906119894(119905) 119909119894 119905)

1 le 119894 le 119872 minus 1 0 lt 119905 le 119879

(10)

which can be written in vector form as

1198601198801015840(119905) = 119865 (119880119883 119905) 0 lt 119905 le 119879 (11)

where 119880(119905) = (1199061(119905) 1199062(119905) 119906

119872minus1(119905)) is the discrete

solution of (1) at time 119905 with 119906119894(119905) = 119906(119909

119894 119905)119860 is an (119872minus1)times

(119872minus 1) tridiagonal matrix and 119865 is a vector-valued functiondefined through (10) To complete the ODE system we needthe boundary conditions at 119909 = 119909

0and 119909 = 119909

119872 which can be

derived from the original boundary conditions defined in (3)or (4)

First if the Dirichlet boundary condition (3) is specifiedone can add the following two ODEs to (9)

1199061015840

0(119905) = 119892

1015840

1(119905) 119906

1015840

119872(119905) = 119892

1015840

2(119905) (12)

Consequently the matrix is modified as

119860 =

(

(

(

(

1 0 0 0 sdot sdot sdot 0 0 0

1

12

5

6

1

12

0 sdot sdot sdot 0 0 0

sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot

0 0 0 0 sdot sdot sdot

1

12

5

6

1

12

0 0 0 0 sdot sdot sdot sdot sdot sdot sdot sdot sdot 1

)

)

)

)

(13)

while the vector-valued function 119865(119880119883 119905) after modifica-tions is defined as

1198650= 1198921015840

1(119905)

119865119894=

119863

ℎ21205752

119909119906119894(119905) + (1 +

1205752

119909

12

)119891 (119906119894(119905) 119909119894 119905)

1 le 119894 le 119872 minus 1

119865119872= 1198921015840

2(119905)

(14)

Alternatively we can incorporate the boundary conditionby replacing 119906

0(119905) and 119906

119872(119905) in (10) with 119892

1(119905) and 119892

2(119905)

respectively so the ODE system equation (11) has only119872minus 2equations

As one can imagine the situation is more complicatedwhen the Neumann boundary condition (4) is specified Tocomplete the ODE system and maintain the higher-orderoverall accuracy a compact fourth-order approximation ofthe Neumann boundary condition is needed Let us use theboundary condition at 119909 = 119886 as the example to demonstratethe idea of the new algorithm

Unlike the Dirichlet boundary condition which specifiesthe solution 119906 on the boundary point explicitly the Neumannboundary condition defines 119906

119909at the boundary points thus

1199060(119905) and 119906

119872(119905) need to be calculated along with solution

at the interior grid points Consequently the range for 119894 in(10) should be changed to 0 le 119894 le 119872 so 119860 is an (119872 +1) times (119872 + 1) matrix To approximate the derivative at 119909

0

we introduce a ghost point 119909minus1= 119886 minus ℎ and assume that

(1) holds and the solution 119906 is sufficiently smooth on theextended domain [119886 minus ℎ 119887] Let 119906

minus1(119905) denote the solution at

119909minus1= 119886 minus ℎ and then apply the second-order central finite

difference approximation to 119906119909(119886 119905)

1199061(119905) minus 119906

minus1(119905)

2ℎ

= 119906119909(119886 119905) +

ℎ2

6

119906119909119909119909(119886 119905) + 119874 (ℎ

4) (15)

Taking partial derivative with respect to 119909 on both sidesof (1) we have

119906119909119909119909=

1

119863

(119906119909119905minus 119891119909minus 119891119906sdot 119906119909) (16)

Letting 119909 rarr 119886 in (16) and then applying the Neumannboundary condition (4) we obtain

119906119909119909119909(119886)

=

1

119863

(1198921015840

3(119905) minus 119891

119909(1199060(119905) 119886 119905) minus 119891

119906(1199060(119905) 119886 119905) sdot 119892

3(119905))

(17)

Combining (15) with (17) we obtain the following fourth-order compact approximation for 119906

minus1(119905)

119906minus1(119905) = 119906

1(119905) minus 2ℎ119892

3(119905) minus

ℎ3

3119863

(1198921015840

3(119905) minus 119891

119909(1199060(119905) 119886 119905)

minus 119891119906(1199060(119905) 119886 119905) sdot 119892

3(119905))

(18)

which involves 1199060(119905) and 119906

1(119905) only so the compact structure

is preserved


Similarly the fourth-order compact approximation for119906119872+1(119905) can be derived as

119906119872+1(119905) = 119906

119872minus1(119905) + 2ℎ119892

4(119905)

+

ℎ3

3119863

(1198921015840

4(119905) minus 119891

119909(119906119872(119905) 119887 119905)

minus 119891119906(119906119872(119905) 119887 119905) sdot 119892

4(119905))

(19)

Now the matrix 119860 involves 119905 and 119880 and the first and lastrows are modified as

11986000sdotsdotsdot119872= (

10

12

+

ℎ3

36119863

(119891119909119906(1199060 119886 119905)

+119891119906119906(1199060 119886 119905) sdot 119892

3(119905) )

1

6

0)

1198601198720sdotsdotsdot119872= (0

1

6

10

12

minus

ℎ3

36119863

(119891119909119906(119906119872 119887 119905)

+119891119906119906(119906119872 119887 119905) sdot 119892

4(119905)) )

(20)

Consequently the first and last components of 119865 are

1198650=

119863

ℎ2(21199061minus 21199060minus 2ℎ119892

3)

+

1

12

(10119891 (1199060 119886 119905) + 119891 (119906

1 119886 + ℎ 119905))

minus

ℎ

3

(1198921015840

3minus 119891119909(1199060 119886 119905) minus 119891

119906(1199060 119886 119905) sdot 119892

3) +

ℎ

6

1198921015840

3

+

ℎ3

36119863

(11989210158401015840

3minus 119891119909119905(1199060 119886 119905) minus 119891

119906(1199060 119886 119905) sdot 119892

1015840

3

minus119891119906119905(1199060 119886 119905) sdot 119892

3)

+

1

12

119891(1199061minus 2ℎ119892

3minus

ℎ3

3119863

times (1198921015840

3minus 119891119909(1199060 119886 119905) minus 119891

119906(1199060 119886 119905) sdot 119892

3)

119886 minus ℎ 119905)

119865119872=

2119863

ℎ2(119906119872minus1minus 119906119872+ ℎ1198924)

+

1

12

(10119891 (119906119872 119887 119905) + 119891 (119906

119872minus1 119887 minus ℎ 119905))

minus

ℎ

3

(1198921015840

4minus 119891119909(119906119872 119887 119905) minus 119891

119906(119906119872 119887 119905) sdot 119892

4) minus

ℎ

6

1198921015840

4

minus

ℎ3

36119863

(11989210158401015840

4minus 119891119909119905(119906119872 119887 119905) minus 119891

119906(119906119872 119887 119905) sdot 119892

1015840

4

minus119891119906119905(119906119872 119887 119905) sdot 119892

4)

+

1

12

119891(119906119872minus1+ 2ℎ119892

4+

ℎ3

3119863

times (1198921015840

4minus 119891119909(119906119872 119887 119905) minus 119891

119906(119906119872 119887 119905) sdot 119892

4)

119887 + ℎ 119905)

(21)

Finally the ODE system is written in the form of119860(119905 119880)1198801015840 =119865(119880 119905) Apparently the matrix119860 preserves tridiagonal struc-ture but it depends on 119905 and 119880 hence the development ofRosenbrockmethod becomes difficult Fortunately we noticethat 119905 and119880 are involved in two entries 119860

00and 119860

119872119872only

Further investigation shows that the extra terms in (20) are

ℎ3

36119863

(119891119909119906(1199060 119886 119905) + 119891

119906119906(1199060 119886 119905) sdot 119892

1(119905)) (22)

minus

ℎ3

36119863

(119891119909119906(119906119872 119887 119905) + 119891

119906119906(119906119872 119887 119905) sdot 119892

4(119905)) (23)

respectively We can eliminate these two extra terms byincorporating them into vector 119865 therefore 119865

0and 119865

119872are

modified as

1198650= 1198650

minus

ℎ3

36119863

[(119891119909119906(1199060 119886 119905) + 119891

119906119906(1199060 119886 119905) sdot 119892

3(119905))] sdot 119906

1015840

0(119905)

(24)

119865119872= 119865119872

+

ℎ3

36119863

[(119891119909119906(119906119872 119887 119905) + 119891

119906119906(119906119872 119887 119905) sdot 119892

4(119905))]

sdot 1199061015840

119872(119905)

(25)

Using (1) we have

1199061015840

0(119905) =

2119863

ℎ2(1199061(119905) minus 119906

0(119905) minus ℎ119892

3(119905))

+ 119891 (1199060(119905) 119886 119905) + 119874 (ℎ)

(26)

1199061015840

119872(119905) =

2119863

ℎ2(119906119872minus1(119905) minus 119906

119872(119905) + ℎ119892

4(119905))

+ 119891 (119906119872(119905) 119887 119905) + 119874 (ℎ)

(27)


Inserting (26) into (24) and then ignoring the fourth-order error term 119874(ℎ4) we obtain

1198650= 1198650minus

ℎ3

36119863

[(119891119909119906(1199060 119886 119905) + 119891

119906119906(1199060 119886 119905) sdot 119892

3(119905))]

times (

2119863

ℎ2(1199061minus 1199060minus ℎ1198923(119905)) + 119891 (119906

0 119886 119905))

(28)

Similarly inserting (27) into (25) and then ignoring thefourth-order error term we obtain

119865119872= 119865119872minus

ℎ3

36119863

[(119891119909119906(119906119872 119887 119905) + 119891

119906119906(119906119872 119887 119905) sdot 119892

4(119905))]

times (

2119863

ℎ2(119906119872minus1(119905) minus 119906

119872(119905) + ℎ119892

4(119905))

+119891 (119906119872(119905) 119887 119905) )

(29)

We then obtain the closed ODE system 1198601198801015840(119905) = 119865(119880 119905)where the vector-valued function is given as 119865 =(1198650 1198651 119865

119872minus1 119865119872) and119860 is a nonsingular (119872+1)times(119872+1)

constant matrix given as

119860 =

(

(

(

(

(

(

(

(

5

6

1

12

0 0 sdot sdot sdot 0 0 0

1

12

5

6

1

12




1

12

5

6

1

12

0 0 0 0 sdot sdot sdot sdot sdot sdot

1

12

5

6

)

)

)

)

)

)

)

)

(30)

If the Robin boundary condition is specified a similarnumerical technique can be used to derive the semidiscreteODE system

Here we mention without theoretical proof that theresulting semidiscrete ODE system (9) is a fourth-orderaccurate approximation to the original semilinear parabolicpartial differential equation given in (1) supplemented witheitherDirichlet orNeumann boundary conditions Interestedreaders can find a similar theorem and proof in [11]

3 Fourth-Order Strongly A-StableRosenbrock Method

Various numerical methods can be used to solve the ODEsystem in (11) However due to the stiffness of the problemonly stiffly stable methods are applicable thus the choices arelimited to the subclass of implicit methods such as implicitlinear multistepmethods and implicit Runge-Kutta methodsIt is known that A-stability is necessary for stiff problemand in general strong A-stable or even L-stable methods arepreferred The A-stability was firstly introduced and definedby Dahlquist [12] as the following

Definition 1 A numerical method is called A-stable if there isno restriction on the step size when it is applied to solve thetest equation 1199101015840 = 120582119910 where Re(120582) lt 0

For a single-step method such as Runge-Kutta methodthat can be written as 119910

119899+1= 119877(119910

119899) the A-stability is equi-

valent to the condition |119877(119911)| le 1 for any 119911 isin Cminus1 where119877(119911) is called the stability function of the method Althoughsuccessfully used in various applications an A-stable linearmultistep method has the highest order of 2 In fact thesecond-order A-stable linear multistep method with optimalerror constant is the Trapezium rule [12] Further Gourlay[13] pointed out that an A-stable method is necessary butnot sufficient for very stiff system as it has the incorrectdamping rate For example the widely used Trapezium rulehas stability function119877(119911) = (1+119911)(1minus119911) satisfying |119877(119911)| lt1 for any 119911 isin Cminus1 but its damping rate converges to minus1 when119911 rarr minusinfin To overcome this difficulty strong A-stability wasintroduced

Definition 2 A numerical method is called strongly A-stableif it is A-stable and |119877(minusinfin)| lt 1

It has been shown that a numerical method with strongA-stability is effective in damping numerical oscillations forhighly stiff system Formore details about the description andcomparison of A-stability and strong A-stability the readersare referred to [14]

Implicit Runge-Kutta method is usually unconditionallystable but suffers from the issue of high computational com-plexity especially for nonlinear ODE system For exampleduring each time step an algebraic system with 119904 times 119872unknown variables needs to be solved if an 119904-stage implicitRunge-Kutta method is used to solve an ODE system with119872 equations Therefore fully implicit Runge-Kutta methodsare too computationally expensive to be useful for large-scale problems In the past several decades efforts havebeen made to reduce the computational cost which resultsin various modified implicit Runge-Kutta methods such asdiagonally implicit Runge-Kutta method singly diagonallyimplicit Runge-Kutta method explicit-implicit Runge-Kuttamethod to name a few For more details of these methodsthe reader is referred to [15ndash19]

To completely avoid solving a nonlinear algebraic systemRosenbrock method which is a special class of Runge-Kuttamethod had been proposed Since the spatial discretization isfourth-order our aim herein is to develop a strongly A-stablefourth-order Rosenbrockmethod for solving theODE system(11) so the new algorithm is fourth-order accurate in bothtemporal and spatial dimensions

31 Rosenbrock Method for Scalar Equation We first derivethe Rosenbrock method based on an autonomous scalarequation 1199101015840 = 119891(119910) for which the initial condition is givenas 119910(119905

0) = 119910

0 Nonautonomous equations can be converted

to autonomous form by adding an extra equation to thesystem Some previous research [20] suggested that it isunlikely to find a 3-stage fourth-order Rosenbrock methodwith strong A-stability or L-stability so herein we focus on


the development of a 4-stage Rosenbrock method Supposethe numerical solution at time 119905

119899is known as 119910

119899 the 4-stage

Rosenbrock method calculates the numerical solution at 119905119899+1

as

119910119899+1= 119910119899+ 11988711198961+ 11988721198962+ 11988731198963+ 11988741198964 (31)

119896119894= Δ119905119891(119910

119899+

119894minus1

sum

119895=1

120572119894119895119896119895) + Δ119905119869 (119910

119899)

119894

sum

119895=1

120574119894119895119896119895

119894 = 1 2 3 4

(32)

where 119887119894 120572119894119895 and 120574

119894119895are coefficients to be determined and

119869(119910119899) = 119891119910(119910119899)

To extend the above algorithm to the nonautonomousproblems 1199101015840 = 119891(119910 119905) we first convert it to autonomous formby adding a new equation 1199051015840 = 1 and then apply the algorithm(31) to the augmented system Note that the componentscorresponding to the last variable can be computed explicitlythus we can derive the modified algorithm as the following

119910119899+1= 119910119899+ 11988711198961+ 11988721198962+ 11988731198963+ 11988741198964

119896119894= Δ119905119891(119910

119899+

119894minus1

sum

119895=1

120572119894119895119896119895 119905119899+ 120572119894Δ119905)

+ 120574119894Δ1199052119891119905(119910119899 119905119899) + Δ119905119891

119910(119910119899 119905119899)

119894

sum

119895=1

120574119894119895119896119895

119894 = 1 2 3 4

(33)

where

120572119894=

119894minus1

sum

119895=1

120572119894119895 120574

119894=

119894

sum

119895=1

120574119894119895 (34)

A Rosenbrock method of order 119901 is obtained throughchoosing coefficients in (31)-(32) so that the local errorsatisfies 119910(119905

119899+ Δ119905) minus 119910

119899+1= 119874(Δ119905

119901+1) This can be done

either by solving the so-called Butcher Series [14] or bystraightforward differentiation Here we derive the orderconditions for the fourth-order Rosenbrock method in adifferent way using Taylor series First both 119910(119905

119899+Δ119905) and 119896

119894

are expanded as Taylor series so that the difference119910(119905119899+Δ119905)minus

119910119899+1

can be expressed as a Taylor series and its coefficients ofthe terms up to 119874(Δ1199054) are set to 0 which results in a set ofequations involving these coefficients

Given 119910(119905119899) = 119910

119899 the Taylor series of 119910(119905

119899+ Δ119905) at 119905

119899is

expanded as

119910 (119905119899+ Δ119905) = 119910

119899+ Δ119905119891119899+

Δ1199052

2

119869119899119891119899+

Δ1199053

6

(1198692

119899119891119899+ 1198691015840

1198991198912

119899)

+

Δ1199054

24

(11986910158401015840

1198991198913

119899+ 21198691015840

1198991198691198991198912

119899+ 21198691198991198691015840

1198991198912

119899+ 1198693

119899119891119899)

+ 119874 (Δ1199055)

(35)

where 119891119899= 119891(119910

119899) 119869119899= (120597119891120597119910)(119910

119899) 1198691015840119899= (12059721198911205971199102)(119910119899) and

11986910158401015840

119899= (12059731198911205971199103)(119910119899) For the sake of simplicity let120573

119894119895= 120572119894119895+120574119894119895

and 120572119894119894= 0

Letting 119894 = 1 in (32) we have

1198961= Δ119905 (1 minus Δ119905120574

11119869119899)minus1

119891119899

= Δ119905119891119899+ Δ119905212057411119869119899119891119899+ Δ11990531205742

111198692

119899119891119899

+ Δ11990541205743

111198693

119899119891119899+ 119874 (Δ119905

5)

(36)

Similarly letting 119894 = 2 in (32) we have

1198962= Δ119905 (1 minus Δ119905120574

22119869119899)minus1

(119891 (119910119899+ 120572211198961) + 120574211198691198991198961) (37)

Combining it with the Taylor series of 1198961in (36) we

obtain

1198962= Δ119905119891

119899+ Δ1199052(12057221+ 12057421+ 12057422) 119869119899119891119899

+ Δ1199053[(12057411+ 12057422) (12057221+ 12057421) + 1205742

22] 1198692

119899119891119899+

Δ1199053

2

1205722

211198691015840

1198991198912

119899

+ Δ1199054[(1205741112057422+ 1205742

11+ 1205742

22) (12057221+ 12057421) + 1205743

22] 1198693

119899119891119899

+ Δ1199054120574111205722

211198691015840

1198991198691198991198912

119899+

Δ1199054

2

120574221205722

211198691198991198691015840

1198991198912

119899

+

Δ1199054

6

1205723

2111986910158401015840

1198991198913

119899+ 119874 (Δ119905

5)

(38)

Following the same way we have

1198963= Δ119905119891

119899+ Δ1199052(12057331+ 12057332+ 12057433) 119869119899119891119899

+ Δ1199053[

[

3

sum

119895=1

((1205723119895+ 1205743119895) sdot

119895

sum

119897=1

(120572119895119897+ 120574119895119897))]

]

1198692

119899119891119899

+

Δ1199053

2

1205722

31198691015840

1198991198912

119899

+ Δ1199054[

[

3

sum

119894=1

((1205723119894+ 1205743119894) sdot

119894

sum

119895=1

((120572119894119895+ 120574119894119895)

sdot

119895

sum

119897=1

(120572119895119897+ 120574119895119897)))]

]

1198693

119899119891119899

+

Δ1199054

2

[1205722

212057332+ 120574331205722

3] 1198691198991198691015840

1198991198912

119899

+ Δ1199054[1205723((12057321+ 12057422) 12057232+ 1205741112057231)] 1198691015840

1198991198691198991198912

119899

+

Δ1199054

6

(12057231+ 12057232)3

11986910158401015840

1198991198913

119899+ 119874 (Δ119905

5)


1198964= Δ119905119891

119899+ Δ1199052(12057341+ 12057342+ 12057342+ 12057444) 119869119899119891119899

+ Δ1199053[

[

4

sum

119895=1

((1205724119895+ 1205744119895) sdot

119895

sum

119897=1

(120572119895119897+ 120574119895119897))]

]

1198692

119899119891119899

+

Δ1199053

2

1205722

41198691015840

1198991198912

119899

+ Δ1199054[

[

4

sum

119894=1

((1205724119894+ 1205744119894) sdot

119894

sum

119895=1

((120572119894119895+ 120574119894119895)

sdot

119895

sum

119897=1

(120572119895119897+ 120574119895119897)))]

]

1198693

119899119891119899

+ Δ1199054[1205724((12057433+ 12057332+ 12057331) 12057243

+ (12057422+ 12057321) 12057242+ 1205741112057241)] 1198691015840

1198991198691198991198912

119899

+

Δ1199054

2

[120574441205722

4+ 120573431205722

3+ 120573421205722

2] 1198691198991198691015840

1198991198912

119899

+

Δ1199054

6

(12057241+ 12057242+ 12057243)3

11986910158401015840

1198991198913

119899+ 119874 (Δ119905

5)

(39)

Inserting the four Taylor series into (35) andmatching thecoefficients of Δ119905119901 for 119901 = 0 1 2 3 4 on both sides of (31) weobtain the following order conditions

1 =

4

sum

119894=1

119887119894 (40)

1

2

=

4

sum

119894=1

119887119894sdot (120574119894119894+

119894minus1

sum

119895=1

120573119894119895) (41)

1

6

=

4

sum

119894=1

119887119894sdot (

119894

sum

119895=1

120573119894119895sdot

119895

sum

119897=1

120573119895119897) (42)

1

3

=

4

sum

119894=1

119887119894sdot (

119894minus1

sum

119895=1

120572119894119895)

2

(43)

1

4

=

4

sum

119894=1

119887119894sdot (

119894minus1

sum

119895=1

120572119894119895)

3

(44)

1

24

=

4

sum

119894=1

119887119894sdot (120572119894

119894minus1

sum

119895=1

120572119894119895sdot (120574119895119895+

119895minus1

sum

119897=1

120573119895119897)) (45)

1

12

=

4

sum

119894=1

119887119894sdot (

119894

sum

119895=1

120573119894119895sdot (

119895minus1

sum

119897=1

120572119895119897)

2

) (46)

1

24

=

4

sum

119894=1

119887119894sdot (

119894

sum

119895=1

(120573119894119895sdot

119895

sum

119897=1

(120573119895119897sdot

119897

sum

119896=1

120573119897119896))) (47)

Note that the set of order conditions obtained by usingButcher series [14 page 108] is a special case herewhen 120574

119894119894= 120574

Also it is worthy to point out that if 119891(119910) is a scalarfunction of a single variable 119869

1198991198691015840

119899= 1198691015840

119899119869119899 thus conditions (45)

and (46) can be combined to one single condition Howeverhere since the method is developed for ODE system bothconditions should be satisfied individually

Apparently there are 12 degrees of freedom to determinethe method defined in (32) since there are 20 parameterswhile only 8 constraints are given by (40)ndash(47) To simplifythe procedure of determining these parameters and reducethe number of matrix inversion we assume that 120573

119894119894= 120574 for

119894 = 1 2 3 4 so that only one matrix inversion is required tosolve all 119896

119894during each time step After this simplification

there are 17 parameters left and the order conditions aresimplified as well

It is known that the Rosenbrock method suffers fromorder reduction when applied to nonlinear parabolic partialdifferential equations see [20 21] In order to avoid suchreduction of accuracy the following two extra order condi-tions simplified after taking into account the previous orderconditions should be satisfied [22]

119887412057343120573321205722

2= minus2120574

4+ 41205743minus

5

3

1205742+

120574

6

(48)

119887412057343120573321205722

2= minus

81205744

3

+ 51205743minus 21205742+

7120574

36

(49)

which imply

1205743minus

3

2

1205742+

120574

2

minus

1

24

= 0 (50)

Solving (50) results in three distinct real roots while stabilityanalysis shows that only one (120574 = 106857902130162885) canensure A-stability for the Rosenbrock method More detailsregarding stability will be provided later

To perform step size control and error estimation weneed a third-order embedded formula [23] defined as 119910

119899+1=

119910119899+sum119904

119895=1

119887119895119896119895which uses the same values of 119896

119895rsquos as the Rosen-

brock method in (31) but coefficients of 119896119895rsquos are different

To ensure the existence of such embedded formula we needDet(119860) = 0 where 119860 is a 4 times 4 square matrix defined as

119860 =(

1 1 1 1

12057311205732

1205733

1205734

1205732

1

2

sum

119894=1

1205732119894sdot 120573119894

3

sum

119894=1

1205733119894sdot 120573119894

4

sum

119894=1

1205734119894sdot 120573119894

0 1205722

21205722

31205722

4

) (51)

and 120573119894= sum119894

119895=1120573119894119895

Taking into account the order conditions we obtain thesimplified condition as 120573

2

31205722

212057343= 0 which unfortunately

contradicts the order condition given in (47) Thus one caneither sacrifices efficiency by relaxing the condition 120574

119894119894=

120574 for the existence of a third-order embedded formula orsimply skip the condition and then ignore step size controland error estimation In this work we are interested in the


Table 1 Coefficients of the 4th-order Rosenbrock method

1198871= 04074074074074 119887

2= minus02568608534470 119887

3= 02

1198874= 06494534460396 120572

21= 075 120572

31= 075

12057232= 0 120572

41= 29193596398302 120572

42= 04

12057243= minus25693596398302 120574

21= minus075 120574

31= minus13152686912402

12057432= 075 120574

41= minus28738466294648 120574

42= minus33778743470341

12057443= 45693596398302 120574 = 1068579021301629

efficiency and order reduction of the Rosenbrock method sothe condition for step size control is bypassed

Upon the determination of 120574 there are 17 free parametersremaining while the order conditions are simplified as thefollowing

1198871+ 1198872+ 1198873+ 1198874= 1 (52)

11988721205732+ 11988731205733+ 11988741205734=

1

2

minus 120574 (53)

1198873120573321205732+ 1198874(120573421205732+ 120573431205733) =

1

6

minus 120574 + 1205742 (54)

11988721205722

21+ 11988731205722

3+ 11988741205722

4=

1

3

(55)

11988721205723

21+ 11988731205723

3+ 11988741205723

4=

1

4

(56)

11988731205723120572321205732+ 11988741205724(120572421205732+ 120572431205733) =

1

8

minus

120574

3

(57)

1198873120573321205722

2+ 1198874(120573421205722

2+ 120573431205722

3) =

1

12

minus

120574

3

(58)

119887412057343120573321205732=

1

24

minus

120574

2

+

3

2

1205742minus 1205743 (59)

which are equivalent to those given in [14 page 108] where120573119894= sum119894minus1

119895=1120573119894119895

Since 120574 is a root of (50) 119887412057343120573321205732= 0 but 119887

412057343120573321205722

2=

0 thus 12057321= 0Then the order conditions in (52)ndash(58) can be

further simplifiedWe now choose the coefficients 120572

119894119895with the purpose of

reducing the number of functional evaluations Particularlyletting 120572

2= 1205723= 1205724 (55)-(56) imply that 120572

21= 1205722= 34

which further implies 1198872+ 1198873+ 1198874= 1627 so 119887

1= 1127

Letting 12057231= 12057221and 120572

32= 0 we can reduce the number of

functional evaluation by 1We choose 120573

43as a free parameter so

11988741205733=

1 minus 6120574 + 61205742

612057343

12057243=

(1 minus 81205743) 12057343

1 minus 6120574 + 61205742

(60)

By choosing 12057242as a free parameter we have

12057241=

3

4

minus 12057242minus

(1 minus 81205743) 12057343

1 minus 6120574 + 61205742 (61)

Letting 12057332be a free parameter we obtain

1198874=

16 (minus21205744+ 41205743minus 512057423 + 1205746)

91205734312057332

12057331=

1 minus 6120574 + 61205742

6120573431198874

minus 12057332

(62)

Finally choose 1198873as free parameter so we have 119887

2= 1627 minus

1198873minus 1198874and

12057342=

(4 minus 16120574 minus 27119887312057332)

(271198874)

minus 12057343

12057341=

(12 minus 120574 minus 1198873(12057331+ 12057332))

1198874

minus 12057342minus 12057343

(63)

In summary the coefficients of the Rosenbrock methodare listed in Table 1

Note that one can also use these free parameters toeliminate some fifth-order truncation error terms so thetruncation error constant can be further optimized

32 Extend the Rosenbrock Method to ODE System Wenow extend the Rosenbrock algorithm to the ODE system1198601198801015840(119905) = 119865(119880 119905) Suppose that 119880119899 = (119906119899

0 119906119899

1 119906

119899

119872) is the

numerical solution of (10) at time 119905119899 then 119880119899+1 is calculated

as

119880119899+1= 119880119899+ 11988711198701+ 11988721198702+ 11988731198703+ 11988741198704 (64)

(119860 minus 120574Δ119905119869119899)119870119894= 119865(119880

119899+

119894minus1

sum

119895=1

120572119894119895119870119895 119905119899+ 120590119894Δ119905)

+ 120583119894Δ1199052 120597119865

120597119905

(119880119899 119905119899) + Δ119905J

119899(

119894minus1

sum

119895=1

120574119894119895119870119895)

119894 = 1 2 3 4

(65)

where 119869119899= (120597119865120597119880)(119880

119899 119905119899) is the Jacobi matrix 120590

119894and 120583

119894are

defined through (34)


Obviously each stage of this method consists of solvinga linear system of119872 + 1 equations Since 120574

11= sdot sdot sdot = 120574

44=

120574 only one LU-decomposition is required during each timestep

33 Stability of the Rosenbrock Method We now study thestability of the Rosenbrock method defined by (31)-(32)with coefficients given in Table 1 Applying the Rosenbrockmethod to the linear test equation 1199101015840 = 120582119910 120582 lt 0 and takinginto account the order condition equation (49) we obtain thestability function of the Rosenbrock method as

119877 (119911) = (1 + (1 minus 4120574) 119911 + (

1

2

minus 4120574 + 61205742) 1199112

minus(

51205742

4

minus

5120574

8

+

1

16

) 1199114)

times ((1 minus 120574119911)4)

minus1

(66)

where 119911 = 120582Δ119905In order for the Rosenbrock method to be strongly A-

stable we need

|119877 (119911)| le 1 forall119911 isin Cminus (67)

lim119911rarrminusinfin|119877 (119911)| lt 1 (68)

Apparently

lim119911rarrminusinfin

|119877 (119911)| =

100381610038161003816100381610038161003816100381610038161003816

512057424 minus 51205748 + 116

1205744

100381610038161003816100381610038161003816100381610038161003816

= 06304149382 lt 1

(69)

so (68) is satisfiedTo prove (67) we can substitute 119911 with 119909 + 119894119910 and then

show that |119877(119909 + 119894119910)| le 1 for any 119909 lt 0 However suchproof is technically tedious although possible Here we usethe Boundary-Locus method to draw the region of absolutestability of themethod which is shown in Figure 1 Obviouslythe region of absolute stability contains the whole half-planeCminus so the Rosenbrock method is A-stable Combined with(69) we can prove that the Rosenbrock method is stronglyA-stable

4 Numerical Examples and Discussions

We solve three numerical examples to demonstrate theaccuracy and efficiency of the new algorithm The first andsecond examples are reaction-diffusion equations supple-mented with Dirichlet and Neumann boundary conditionsrespectivelyThe two examples are solved by the newmethodto demonstrate the fourth-order accuracy in time and spaceThe third example is used to demonstrate that the newalgorithm is free of order reduction and more efficient thanseveral other existing fourth-order Rosenbrock methodsFor comparison the rate of convergence is compared withseveral other fourth-order methods which suffer from orderreduction In what follows we use HOC-ROSB4 to represent

0 1 2 3 4 5 6 7 8 9 10 11 12

0

1

2

3

4

5

6

7

8

x

y

minus4 minus3 minus2 minus1minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

Stable

Stable

Stable Unstable

Figure 1 Absolute stable region of the Rosenbrock method

the new algorithm developed in this paper andHOC-GRK4Ato represent the fourth-order method that combines high-order compact finite difference scheme and fourth-orderGRK4A [23] HOC-SHAMP represents the algorithm thatcombines high-order compact finite difference scheme withthe fourth-order Rosenbrock method Shampine [24] HOC-LSTAB represents the algorithm that combines the high-order compact finite difference scheme in space and the L-stable fourth-order Rosenbrock method [14] in time Finallywe useHOC-VELD to represent the algorithm that combinesthe high-order compact finite difference scheme in spacewithfourth-order Rosenbrock method proposed in [25]

41 Example 1 Consider the following reaction-diffusionequation

119906119905= 119906119909119909+ cos (119906) minus cos (119890minus119905 cos (119909))

(119909 119905) isin (0 2) times (0 1]

119906 (119909 0) = cos (119909) 119909 isin [0 2]

119906 (0 119905) = 119890minus119905 119906 (2 119905) = cos (2) 119890minus119905 119905 isin [0 1]

(70)

for which the analytical solution is 119906(119909 119905) = 119890minus119905 cos(119909)We first show that the compact finite difference scheme

for spatial discretization and the compact boundary condi-tion treatment are fourth-order accurate In order to do sowe fixed Δ119905 = 00001 so the truncation error from timediscretization is negligible The results included in Table 2clearly demonstrate the fourth-order convergence in space asthe maximal error reduced by a factor of 16 (roughly) whenthe grid size ℎ is halved

We then show the fourth-order accuracy in time byfixing ℎ = 0001 The data in Table 3 confirmed that theRosenbrock method is fourth-order accurate in time so thenew algorithm is free of order reduction

Since themethod is fourth-order accurate in both tempo-ral and spatial dimensions we can write the leading term of


Table 2 Numerical results of example 1 by HOC-ROSB4 with Δ119905 = 00001

ℎ 110 120 140 180 1160119864(ℎ) 738119890 minus 08 462119890 minus 09 289119890 minus 10 180119890 minus 11 106119890 minus 12

119864(ℎ)119864(ℎ2) mdash 159562 160049 160607 170153Convergence rate mdash 39960 40004 40055 40888

Table 3 Numerical results of example 1 by HOC-ROSB4 with ℎ = 0001

Δ119905 110 120 140 180 1160119864(Δ119905) 903119890 minus 06 616119890 minus 07 396119890 minus 08 245119890 minus 09 149119890 minus 10

119864(Δ119905)119864(Δ1199052) mdash 146679 155402 161607 164631Convergence rate mdash 38746 39579 40144 40412

Table 4 Numerical results of example 1 by HOC-ROSB4 with ℎΔ119905 = 32

ℎ 110 120 140 180 1160Δ119905 132 164 1128 1256 1512119864(Δ119905 ℎ) 594119890 minus 08 409119890 minus 09 273119890 minus 10 178119890 minus 11 115119890 minus 12

119864(Δ119905 ℎ)119864(Δ1199052 ℎ2) mdash 144970 149793 153189 154959Convergence rate mdash 38577 39049 39372 39538

the truncation error as119864(ℎ Δ119905) = 1198621ℎ4+1198622Δ1199054 where119862

1and

1198622are two constants To obtain optimal performance we can

adjust the ratio Δ119905ℎ to balance the two error terms To do sowe adjust Δ119905 and ℎ so that the two error terms are balancedwhich is utilized by letting |119862

2Δ11990541198621ℎ4| asymp 1 from which we

can estimate the optimal ratio as (11986211198622)14 To estimate 119862

1

we solve the example using small Δ119905 and then 1198621asymp 119864(ℎ)ℎ

4Similarly we can estimate 119862

2using the same method

Simple calculation based on numerical results fromTables 2 and 3 suggests that the optimal ratio for thisexample is ℎΔ119905 asymp 32 We then solve the reaction-diffusionequation using the optimal ratio to show that the newalgorithm is fourth-order accurate in both temporal andspatial dimensions The numerical results in Table 4 indicatethat the new algorithm is fourth-order accurate in bothtemporal and spatial dimensions as the maximal error isreduced by a factor of 16 (roughly) when Δ119905 and ℎ are halvedsimultaneously We also notice that the algorithm is veryefficient as themaximal error drops to the level of 10minus12 whenℎ and Δ119905 are still reasonably large

42 Example 2 We solve the following semilinear parabolicpartial differential equation with the Neumann boundaryconditions119906119905= 2119906119909119909+ 119906 + 119906

2minus 119890minus2119905cos2 (119909) (119909 119905) isin (0 2) times (0 1]

119906 (119909 0) = cos (119909) 119909 isin [0 2]

119906119909(0 119905) = 0 119906

119909(2 119905) = minus sin (2) 119890minus119905 119905 isin [0 1]

(71)

for which the analytical solution is 119906(119909 119905) = 119890minus119905 cos(119909)Notice that the boundary conditions are approximated by

the compact fourth-order boundary scheme given in (18)-(19) so the compactness of the resulting linear system is

preserved consequently in each time step only a tridiagonalsystem is solved for each 119896

119894 so the solution procedure is very

efficient However we point out that some tedious work isneeded to form the semidiscrete ODE system for instancethe first and last rows of the matrix119860 in (20) and the first andlast components of 119865 in (28)-(29)

To obtain the optimal performance we choose the opti-mal ratio as Δ119905 as ℎΔ119905 = 25 to balance the two error termsThe data in Table 5 shows that the method is fourth-orderaccurate in both time and space as the maximum error isreduced by a factor of 16 (roughly) when ℎ andΔ119905 are halved

It is worthy to mention that one can also use any otherone-sided formula to approximate the Neumann boundarycondition which apparently reduces the effort to derive thesemidiscrete ODE system but will destroy the tridiagonalstructure of the matrix 119860 and consequently affect the effi-ciency of the algorithm Another side-effect of using one-sided approximation on the boundary is the stability issuethat may arise because the modifications to matrix 119860 mayresult in positive eigenvalues of 119860

43 Example 3 In this example we compare the new algo-rithm with several other fourth-order Rosenbrock methodsin terms of the rate of convergence in time and efficiencyThenonlinear parabolic partial differential equation to be solvedis defined as

119906119905= 119906119909119909+ 1199063minus 119890minus3119905cos3 (119909) (119909 119905) isin (0 1) times (0 1]

119906 (119909 0) = cos (119909) 119909 isin [0 1]


(72)

for which the exact solution is 119906(119909 119905) = 119890minus119905 cos(119909)The data in Table 6 shows that HOC-ROSB4 is fourth-

order accurate and thus is free of order reduction while the



ℎ 110 120 140 180 1160Δ119905 125 150 1100 1200 1400119864(ℎ Δ119905) 627119890 minus 06 440119890 minus 07 295119890 minus 08 196119890 minus 09 131119890 minus 10

119864(ℎ Δ119905)119864(ℎ2 Δ1199052) mdash 142495 149178 150510 150135Convergence rate mdash 38328 38990 39118 39082

Table 6 Comparison of the rate of convergence among the five fourth-order Rosenbrock methods for example 3 with ℎ = 0001 119864(Δ119905)represents the maximum error of the numerical solution obtained by using Δ119905

Δ119905 110 120 140 180

HOC-ROSB4 119864(Δ119905) 959119890 minus 06 694119890 minus 0 458119890 minus 08 288119890 minus 09

Rate of Conv mdash 378757 39221 39896

HOC-GRK4A 119864(Δ119905) 588119890 minus 06 616119890 minus 07 683119890 minus 08 756119890 minus 09


HOC-LSTAB 119864(Δ119905) 396119890 minus 06 456119890 minus 07 523119890 minus 08 619119890 minus 09


HOC-VELD 119864(Δ119905) 841119890 minus 06 877119890 minus 07 977119890 minus 08 114119890 minus 08


HOC-SHAMP 119864(Δ119905) 635119890 minus 06 634119890 minus 07 692119890 minus 08 801119890 minus 09


Table 7 Comparison of the efficiency among the five fourth-order Rosenbrock methods for example 3

Method (Δ119905 ℎ) Max error CPU time (seconds)HOC-ROSB4 (1180 140) 772119890 minus 011 0107HOC-GRK4A (1256 140) 728119890 minus 011 0134HOC-LSTAB (1360 140) 783119890 minus 011 0174HOC-VELD (1400 140) 777119890 minus 011 0188HOC-SHAMP (1360 140) 760119890 minus 011 0170

other four fourth-order Rosenbrock methods which obvi-ously suffer from order reduction show rates of convergenceranging from 30 to 325 Note that ℎ = 0001 is fixed andthe same spatial discretization method is used for all fivemethods so the demonstrated rate of convergence is in timeOne can see that the HOC-LSTAB method has the mostoptimal error constant while the HOC-ROSB4 method hasthe largest error constant However due to the freedom indetermining those coefficients in Rosenbrock method bettererror constant can be accomplished by eliminating severalfifth-order truncation error terms

Finally we show that HOC-ROSB4 is the most efficientmethod For consistency we adjust Δ119905 and ℎ for each methodto reach the same error level and record the average CPUtime of 5 simulation runs Since all of the five methods usethe same spatial discretization the same ℎ is used for allmethods hence we adjust Δ119905 only The comparison resultincluded in Table 7 clearly indicates that the new methodis the most efficient one The higher efficiency apparently isobtained from the fact that the new method is free of orderreduction hence large time step can be used to reach the sameerror level

5 Conclusion

An efficient fourth-order numerical algorithm that combinesthe Pade approximation in space and fourth-order accurateRosenbrock method in time is proposed in this paper Ourinvestigation shows that many widely used fourth-orderRosenbrock methods (A-stable or L-stable) suffer from orderreduction when they are used to solve nonlinear parabolicpartial differential equation To avoid order reduction extraorder conditions are required which are implemented inthis paper to develop the new algorithm Also it has beenshown [26] that the extra condition to resolve order reductioncontradicts the L-stability of the Rosenbrock method sothere is no L-stable Rosenbrock method that is also freeof order reduction The new method can be used to solvenonlinear parabolic partial differential equation with alltypes of boundary conditions however for Neumann orRobin boundary condition extra efforts are needed to formthe semidiscrete ODE system Two numerical examples aresolved to demonstrate that the new method is fourth-orderaccurate in both time and space while the third exampleshows that the new method is free of order reduction and


is very efficient In the future we plan to extend the newmethod to multidimensional problems supplemented withvarious types of boundary conditions

Conflict of Interests

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

Acknowledgment

This work of was supported by the Natural Sciences amp Engi-neering Research Council of Canada (NSERC) through theindividual Discovery Grant program The author gratefullyacknowledges the financial support from NSERC

References

[1] Y Adam ldquoHighly accurate compact implicit methods andboundary conditionsrdquo Journal of Computational Physics vol 24no 1 pp 10ndash22 1977

[2] A R Mitchell and D F Griffths The Finite Difference Methodin Partial Differential Equations JohnWiley amp Sons New YorkNY USA 1980

[3] J I Ramos ldquoLinearization methods for reaction-diffusionequations multidimensional problemsrdquo Applied Mathematicsand Computation vol 88 no 2-3 pp 225ndash254 1997

[4] J I Ramos ldquoImplicit compact linearized 120579-methods with fac-torization for multidimensional reaction-diffusion equationsrdquoApplied Mathematics and Computation vol 94 no 1 pp 17ndash431998

[5] P C Chu and C Fan ldquoA three-point combined compact differ-ence schemerdquo Journal of Computational Physics vol 140 no 2pp 370ndash399 1998

[6] V Luan and A Ostermann ldquoExponential Rosenbrock methodsof order fivemdashconstruction analysis and numerical compar-isonsrdquo Journal of Computational and Applied Mathematics vol255 no 1 pp 417ndash431 2014

[7] V T Luan and A Ostermann ldquoExplicit exponential Runge-Kutta methods of high order for parabolic problemsrdquo Journal ofComputational and Applied Mathematics vol 256 pp 168ndash1792014

[8] A-K Kassam and L N Trefethen ldquoFourth-order time-steppingfor stiff PDEsrdquo SIAM Journal on Scientific Computing vol 26no 4 pp 1214ndash1233 2005

[9] HRosenbrock ldquoSome general implicit processes for the numer-ical solution of differential equationsrdquo The Computer Journalvol 5 pp 329ndash330 1963

[10] C F Haines ldquoImplicit integration processes with error estimatefor the numerical solution of differential equationsrdquo The Com-puter Journal vol 12 no 2 pp 183ndash188 1969

[11] W Liao and Y Yan ldquoSingly diagonally implicit Runge-Kuttamethod for time-dependent reaction-diffusion equationrdquoNumericalMethods for Partial Differential Equations vol 27 no6 pp 1423ndash1441 2011

[12] G G Dahlquist ldquoA special stability problem for linearmultistepmethodsrdquo BIT Numerical Mathematics vol 3 pp 27ndash43 1963

[13] A R Gourlay ldquoA note on trapezoidal methods for the solutionof initial value problemsrdquoMathematics of Computation vol 24pp 629ndash633 1970

[14] E Hairer and G Wanner Solving Ordinary Differential Equa-tions II Stiff and Algebraic Problems Springer Berlin Ger-many 2nd edition 1996

[15] R Alexander ldquoDiagonally implicit Runge-Kutta methods forstiff ODErsquosrdquo SIAM Journal on Numerical Analysis vol 14 no6 pp 1006ndash1021 1977

[16] K Burrage J C Butcher and F H Chipman ldquoAn implemen-tation of singly-implicit Runge-Kutta methodsrdquo BIT NumericalMathematics vol 20 no 3 pp 326ndash340 1980

[17] H Claus ldquoSingly-implicit Runge-Kutta methods for retardedand ordinary differential equationsrdquo Computing vol 43 no 3pp 209ndash222 1990

[18] W Liniger andR AWilloughby ldquoEfficient integrationmethodsfor stiff systems of ordinary differential equationsrdquo SIAMJournal on Numerical Analysis vol 7 pp 47ndash66 1970

[19] J G Verwer E J Spee J G Blom and W Hundsdorfer ldquoAsecond-order Rosenbrock method applied to photochemicaldispersion problemsrdquo SIAM Journal on Scientific Computingvol 20 no 4 pp 1456ndash1480 1999

[20] J Lang and J Verwer ldquoROS3Pmdashan accurate third-order Rosen-brock solver designed for parabolic problemsrdquo BIT NumericalMathematics vol 41 no 4 pp 731ndash738 2001

[21] W H Hundsdorfer ldquoStability and B-convergence of linearlyimplicit Runge-Kutta methodsrdquo Numerische Mathematik vol50 no 1 pp 83ndash95 1986

[22] C Lubich and A Ostermann ldquoLinearly implicit time dis-cretization of non-linear parabolic equationsrdquo IMA Journal ofNumerical Analysis vol 15 no 4 pp 555ndash583 1995

[23] P Kaps and P Rentrop ldquoGeneralized Runge-Kutta methods oforder four with stepsize control for stiff ordinary differentialequationsrdquo Numerische Mathematik vol 33 no 1 pp 55ndash681979

[24] L F Shampine ldquoImplementation of Rosenbrock methodsrdquoACM Transactions on Mathematical Software vol 8 no 2 pp93ndash113 1982

[25] M van Veldhuizen ldquo119863-stability and Kaps-Rentrop-methodsrdquoComputing Archives for Scientific Computing vol 32 no 3 pp229ndash237 1984

[26] T D Bui ldquoOn an 119871-stable method for stiff differential equa-tionsrdquo Information Processing Letters vol 6 no 5 pp 158ndash1611977

Submit your manuscripts athttpwwwhindawicom

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


time and space separately Here we first apply the high-order compact finite difference approximation to the spatialderivative so a semidiscrete ODE system is obtained whichis then solved by a fourth-order Rosenbrockmethod that willbe discussed later

Recently there have been attempts to develop high-ordercompact scheme for the spatial derivative In [5] a three-point combined compact difference scheme was proposedto approximate the first and second derivatives for problemwith periodic boundary condition The resulting scheme hasup to sixth-order accuracy at all grid points including theboundary nodes for periodic boundaries however it is onlyfourth-order accurate for nonperiodic boundary condition

Because the semidiscrete ODE system obtained fromspatial discretization such as method of lines of the nonlin-ear parabolic partial differential equation is highly stiff thechoices of time integration methods are limited to implicitmethods only Explicit algorithm is efficient in a single timestep but suffers from strict step size restriction which makesit less efficient Implicit method on the other side is lessefficient in a single step but the unconditional stability allowsthe use of larger time step hence the overall computationalefficiency can be significantly improved One issue howeveris that the iteration is usually slow when large step size Δ119905is used Also due to the stiffness of the ODE system onlyNewton-type iterative methods are applicable to solve thenonlinear algebraic system Furthermore strong A-stabilityor L-stability of the time integration method is necessaryfor error damping A great deal of work has been done inthe development of efficient time stepping methods for thestiff ODE system In [6] explicit exponential Rosenbrockmethods of order five have been constructed to solve thelarge-scale stiff ODE system Through the derivation of stifforder condition new pairs of embedded methods of higher-order can be obtained Similarly fifth-order explicit expo-nential Runge-Kutta methods were constructed to efficientlyintegrate the semilinear stiff problems in [7]The authors havealso shown that there does not exist an explicit exponentialRunge-Kutta method of order 5 with less than or equal to 6stages therefore the resultant methods are 8-stage methodsIn [8] a fourth-order time stepping method which is amodification of the exponential time-differencing fourth-order Runge-Kutta method has been developed for stiffODEs These methods are efficient and accurate HoweverA-stability of the time stepping method is not sufficient forhighly stiff problem To overcome these difficulties it isdesirable to construct new algorithms with strong A-stabilityor L-stability that are free of solving nonlinear equationsIt turns out that the Rosenbrock method which was firstlyreported by Rosenbrock [9] and then improved by Haines[10] responded to these issues with considerably satisfyingand promising results

The objective herein is to develop a strongly A-stableRosenbrock method to solve the semidiscrete stiff ODEsresulting from compact high-order finite difference approx-imation of a semilinear parabolic partial differential equa-tion The rest of the paper is organized as the followingIn Section 2 we discretize the spatial derivatives of thesemilinear parabolic partial differential equation using a

fourth-order compact finite difference scheme which is thencombined with a newly proposed compact fourth-orderboundary condition treatment to form the semidiscrete ODEsystem In Section 3 we focus on the development of a fourth-order strongly A-stable Rosenbrock method and the stabilityanalysis Several numerical examples are used to demonstratethe accuracy and efficiency of the new algorithm in Section 4which is followed by conclusions and possible future work

2 Compact Fourth-OrderSpatial Discretization

For the sake of simplicity we assume that the 1D spatialdomain Ω = [119886 119887] is divided into119872 subintervals with equallength ℎ = (119887 minus 119886)119872 Let 119909

119894= 119886 + 119894 sdot ℎ 119894 = 0 1 119872 be the

grid points A variety of compact high-order discretizationscan be utilized to approximate the second derivative 119906

119909119909in

(1)Here we introduce a compact finite difference scheme to

approximate 119906119909119909 such that the resulting semidiscrete ODE

system is an accurate and compact approximation to theoriginal semilinear parabolic partial differential equationThis operator-approximation based method has been widelyused to solve various multidimensional problems We firstdefine the central finite difference operator 1205752

119909as

1205752

119909119906119894= 119906119894+1minus 2119906119894+ 119906119894minus1 (5)

then1205752119909ℎ2 gives second-order accurate approximation to119906

119909119909

Using Taylor series to expand all terms on the right-hand sideof (5) under the assumption that 119906(119909) is sufficiently smoothwe have

1205752

119909

ℎ2119906119894= 119906119909119909(119909119894) +

ℎ2

12

119906119909119909119909119909(119909119894) + 119874 (ℎ

4) 1 le 119894 le 119872 minus 1

(6)

To improve the above finite difference approximation tofourth-order accurate one just needs to eliminate the second-order error term Applying 1205752

11990912 to both sides of (6) we have

1205752

119909

ℎ2

1205752

119909

12

119906119894= 119906119909119909119909119909(119909119894) + 119874 (ℎ

4) 1 le 119894 le 119872 minus 1 (7)

Combining (6) with (7) neglecting 119874(ℎ4) we obtain thefollowing fourth-order accurate approximation to119906

119909119909at node

119909119894

119906119909119909(119909119894) asymp

1

ℎ21205752

119909(1 minus

1205752

119909

12

)119906119894 1 le 119894 le 119872 minus 1 (8)

The drawback is that a five-point stencil is required thereforethe compactness is destroyed so the method becomes lessefficient Further investigation shows that the differencebetween 1 minus 1205752

11990912 and (1 + 1205752

11990912)minus1 is 119874(ℎ4) so a natural

way is to approximate 119906119909119909(119909119894) as 1205752119909(1 + 120575

2

11990912)minus1119906119894 which is

fourth-order accurate and compact




1199061015840

119894(119905) =

119863

ℎ2

1205752

119909

1 + 1205752

11990912

119906119894(119905) + 119891 (119906

119894(119905) 119909119894 119905)

1 le 119894 le 119872 minus 1

(9)




implicit ODE system

(1 +

1205752

119909

12

)1199061015840

119894(119905) =

119863

ℎ21205752

119909119906119894(119905) + (1 +

1205752

119909

12

)119891 (119906119894(119905) 119909119894 119905)

1 le 119894 le 119872 minus 1 0 lt 119905 le 119879

(10)


1198601198801015840(119905) = 119865 (119880119883 119905) 0 lt 119905 le 119879 (11)

where 119880(119905) = (1199061(119905) 1199062(119905) 119906



119894 119905)119860 is an (119872minus1)times


0and 119909 = 119909

119872 which can be



1199061015840

0(119905) = 119892

1015840

1(119905) 119906

1015840

119872(119905) = 119892

1015840

2(119905) (12)


119860 =

(

(

(

(


1

12

5

6

1

12




1

12

5

6

1

12


)

)

)

)

(13)


1198650= 1198921015840

1(119905)

119865119894=

119863

ℎ21205752

119909119906119894(119905) + (1 +

1205752

119909

12

)119891 (119906119894(119905) 119909119894 119905)

1 le 119894 le 119872 minus 1

119865119872= 1198921015840

2(119905)

(14)


0(119905) and 119906

119872(119905) in (10) with 119892

1(119905) and 119892

2(119905)





1199060(119905) and 119906



0






1199061(119905) minus 119906

minus1(119905)

2ℎ

= 119906119909(119886 119905) +

ℎ2

6

119906119909119909119909(119886 119905) + 119874 (ℎ

4) (15)


119906119909119909119909=

1

119863

(119906119909119905minus 119891119909minus 119891119906sdot 119906119909) (16)


119906119909119909119909(119886)

=

1

119863

(1198921015840

3(119905) minus 119891

119909(1199060(119905) 119886 119905) minus 119891

119906(1199060(119905) 119886 119905) sdot 119892

3(119905))

(17)


minus1(119905)

119906minus1(119905) = 119906

1(119905) minus 2ℎ119892

3(119905) minus

ℎ3

3119863

(1198921015840

3(119905) minus 119891

119909(1199060(119905) 119886 119905)

minus 119891119906(1199060(119905) 119886 119905) sdot 119892

3(119905))

(18)



is preserved



119906119872+1(119905) = 119906

119872minus1(119905) + 2ℎ119892

4(119905)

+

ℎ3

3119863

(1198921015840

4(119905) minus 119891

119909(119906119872(119905) 119887 119905)

minus 119891119906(119906119872(119905) 119887 119905) sdot 119892

4(119905))

(19)



10

12

+

ℎ3

36119863

(119891119909119906(1199060 119886 119905)

+119891119906119906(1199060 119886 119905) sdot 119892

3(119905) )

1

6

0)

1198601198720sdotsdotsdot119872= (0

1

6

10

12

minus

ℎ3

36119863

(119891119909119906(119906119872 119887 119905)

+119891119906119906(119906119872 119887 119905) sdot 119892

4(119905)) )

(20)


1198650=

119863

ℎ2(21199061minus 21199060minus 2ℎ119892

3)

+

1

12

(10119891 (1199060 119886 119905) + 119891 (119906

1 119886 + ℎ 119905))

minus

ℎ

3

(1198921015840

3minus 119891119909(1199060 119886 119905) minus 119891

119906(1199060 119886 119905) sdot 119892

3) +

ℎ

6

1198921015840

3

+

ℎ3

36119863

(11989210158401015840

3minus 119891119909119905(1199060 119886 119905) minus 119891

119906(1199060 119886 119905) sdot 119892

1015840

3

minus119891119906119905(1199060 119886 119905) sdot 119892

3)

+

1

12

119891(1199061minus 2ℎ119892

3minus

ℎ3

3119863

times (1198921015840

3minus 119891119909(1199060 119886 119905) minus 119891

119906(1199060 119886 119905) sdot 119892

3)

119886 minus ℎ 119905)

119865119872=

2119863

ℎ2(119906119872minus1minus 119906119872+ ℎ1198924)

+

1

12

(10119891 (119906119872 119887 119905) + 119891 (119906

119872minus1 119887 minus ℎ 119905))

minus

ℎ

3

(1198921015840

4minus 119891119909(119906119872 119887 119905) minus 119891

119906(119906119872 119887 119905) sdot 119892

4) minus

ℎ

6

1198921015840

4

minus

ℎ3

36119863

(11989210158401015840

4minus 119891119909119905(119906119872 119887 119905) minus 119891

119906(119906119872 119887 119905) sdot 119892

1015840

4

minus119891119906119905(119906119872 119887 119905) sdot 119892

4)

+

1

12

119891(119906119872minus1+ 2ℎ119892

4+

ℎ3

3119863

times (1198921015840

4minus 119891119909(119906119872 119887 119905) minus 119891

119906(119906119872 119887 119905) sdot 119892

4)

119887 + ℎ 119905)

(21)


00and 119860

119872119872only


ℎ3

36119863

(119891119909119906(1199060 119886 119905) + 119891

119906119906(1199060 119886 119905) sdot 119892

1(119905)) (22)

minus

ℎ3

36119863

(119891119909119906(119906119872 119887 119905) + 119891

119906119906(119906119872 119887 119905) sdot 119892

4(119905)) (23)


0and 119865

119872are

modified as

1198650= 1198650

minus

ℎ3

36119863

[(119891119909119906(1199060 119886 119905) + 119891

119906119906(1199060 119886 119905) sdot 119892

3(119905))] sdot 119906

1015840

0(119905)

(24)

119865119872= 119865119872

+

ℎ3

36119863

[(119891119909119906(119906119872 119887 119905) + 119891

119906119906(119906119872 119887 119905) sdot 119892

4(119905))]

sdot 1199061015840

119872(119905)

(25)

Using (1) we have

1199061015840

0(119905) =

2119863

ℎ2(1199061(119905) minus 119906

0(119905) minus ℎ119892

3(119905))

+ 119891 (1199060(119905) 119886 119905) + 119874 (ℎ)

(26)

1199061015840

119872(119905) =

2119863

ℎ2(119906119872minus1(119905) minus 119906

119872(119905) + ℎ119892

4(119905))

+ 119891 (119906119872(119905) 119887 119905) + 119874 (ℎ)

(27)



1198650= 1198650minus

ℎ3

36119863

[(119891119909119906(1199060 119886 119905) + 119891

119906119906(1199060 119886 119905) sdot 119892

3(119905))]

times (

2119863

ℎ2(1199061minus 1199060minus ℎ1198923(119905)) + 119891 (119906

0 119886 119905))

(28)


119865119872= 119865119872minus

ℎ3

36119863

[(119891119909119906(119906119872 119887 119905) + 119891

119906119906(119906119872 119887 119905) sdot 119892

4(119905))]

times (

2119863

ℎ2(119906119872minus1(119905) minus 119906

119872(119905) + ℎ119892

4(119905))

+119891 (119906119872(119905) 119887 119905) )

(29)




119860 =

(

(

(

(

(

(

(

(

5

6

1

12


1

12

5

6

1

12




1

12

5

6

1

12


1

12

5

6

)

)

)

)

)

)

)

)

(30)







119899+1= 119877(119910








0) = 119910





119899is known as 119910

119899 the 4-stage


as

119910119899+1= 119910119899+ 11988711198961+ 11988721198962+ 11988731198963+ 11988741198964 (31)

119896119894= Δ119905119891(119910

119899+

119894minus1

sum

119895=1

120572119894119895119896119895) + Δ119905119869 (119910

119899)

119894

sum

119895=1

120574119894119895119896119895

119894 = 1 2 3 4

(32)

where 119887119894 120572119894119895 and 120574


119869(119910119899) = 119891119910(119910119899)


119910119899+1= 119910119899+ 11988711198961+ 11988721198962+ 11988731198963+ 11988741198964

119896119894= Δ119905119891(119910

119899+

119894minus1

sum

119895=1

120572119894119895119896119895 119905119899+ 120572119894Δ119905)

+ 120574119894Δ1199052119891119905(119910119899 119905119899) + Δ119905119891

119910(119910119899 119905119899)

119894

sum

119895=1

120574119894119895119896119895

119894 = 1 2 3 4

(33)

where

120572119894=

119894minus1

sum

119895=1

120572119894119895 120574

119894=

119894

sum

119895=1

120574119894119895 (34)


119899+ Δ119905) minus 119910

119899+1= 119874(Δ119905



119899+Δ119905) and 119896

119894


119910119899+1


Given 119910(119905119899) = 119910


119899+ Δ119905) at 119905

119899is

expanded as

119910 (119905119899+ Δ119905) = 119910

119899+ Δ119905119891119899+

Δ1199052

2

119869119899119891119899+

Δ1199053

6

(1198692

119899119891119899+ 1198691015840

1198991198912

119899)

+

Δ1199054

24

(11986910158401015840

1198991198913

119899+ 21198691015840

1198991198691198991198912

119899+ 21198691198991198691015840

1198991198912

119899+ 1198693

119899119891119899)

+ 119874 (Δ1199055)

(35)

where 119891119899= 119891(119910

119899) 119869119899= (120597119891120597119910)(119910

119899) 1198691015840119899= (12059721198911205971199102)(119910119899) and

11986910158401015840


119894119895= 120572119894119895+120574119894119895

and 120572119894119894= 0


1198961= Δ119905 (1 minus Δ119905120574

11119869119899)minus1

119891119899

= Δ119905119891119899+ Δ119905212057411119869119899119891119899+ Δ11990531205742

111198692

119899119891119899

+ Δ11990541205743

111198693

119899119891119899+ 119874 (Δ119905

5)

(36)


1198962= Δ119905 (1 minus Δ119905120574

22119869119899)minus1

(119891 (119910119899+ 120572211198961) + 120574211198691198991198961) (37)


obtain

1198962= Δ119905119891

119899+ Δ1199052(12057221+ 12057421+ 12057422) 119869119899119891119899

+ Δ1199053[(12057411+ 12057422) (12057221+ 12057421) + 1205742

22] 1198692

119899119891119899+

Δ1199053

2

1205722

211198691015840

1198991198912

119899

+ Δ1199054[(1205741112057422+ 1205742

11+ 1205742

22) (12057221+ 12057421) + 1205743

22] 1198693

119899119891119899

+ Δ1199054120574111205722

211198691015840

1198991198691198991198912

119899+

Δ1199054

2

120574221205722

211198691198991198691015840

1198991198912

119899

+

Δ1199054

6

1205723

2111986910158401015840

1198991198913

119899+ 119874 (Δ119905

5)

(38)


1198963= Δ119905119891

119899+ Δ1199052(12057331+ 12057332+ 12057433) 119869119899119891119899

+ Δ1199053[

[

3

sum

119895=1

((1205723119895+ 1205743119895) sdot

119895

sum

119897=1

(120572119895119897+ 120574119895119897))]

]

1198692

119899119891119899

+

Δ1199053

2

1205722

31198691015840

1198991198912

119899

+ Δ1199054[

[

3

sum

119894=1

((1205723119894+ 1205743119894) sdot

119894

sum

119895=1

((120572119894119895+ 120574119894119895)

sdot

119895

sum

119897=1

(120572119895119897+ 120574119895119897)))]

]

1198693

119899119891119899

+

Δ1199054

2

[1205722

212057332+ 120574331205722

3] 1198691198991198691015840

1198991198912

119899

+ Δ1199054[1205723((12057321+ 12057422) 12057232+ 1205741112057231)] 1198691015840

1198991198691198991198912

119899

+

Δ1199054

6

(12057231+ 12057232)3

11986910158401015840

1198991198913

119899+ 119874 (Δ119905

5)


1198964= Δ119905119891

119899+ Δ1199052(12057341+ 12057342+ 12057342+ 12057444) 119869119899119891119899

+ Δ1199053[

[

4

sum

119895=1

((1205724119895+ 1205744119895) sdot

119895

sum

119897=1

(120572119895119897+ 120574119895119897))]

]

1198692

119899119891119899

+

Δ1199053

2

1205722

41198691015840

1198991198912

119899

+ Δ1199054[

[

4

sum

119894=1

((1205724119894+ 1205744119894) sdot

119894

sum

119895=1

((120572119894119895+ 120574119894119895)

sdot

119895

sum

119897=1

(120572119895119897+ 120574119895119897)))]

]

1198693

119899119891119899

+ Δ1199054[1205724((12057433+ 12057332+ 12057331) 12057243

+ (12057422+ 12057321) 12057242+ 1205741112057241)] 1198691015840

1198991198691198991198912

119899

+

Δ1199054

2

[120574441205722

4+ 120573431205722

3+ 120573421205722

2] 1198691198991198691015840

1198991198912

119899

+

Δ1199054

6

(12057241+ 12057242+ 12057243)3

11986910158401015840

1198991198913

119899+ 119874 (Δ119905

5)

(39)


1 =

4

sum

119894=1

119887119894 (40)

1

2

=

4

sum

119894=1

119887119894sdot (120574119894119894+

119894minus1

sum

119895=1

120573119894119895) (41)

1

6

=

4

sum

119894=1

119887119894sdot (

119894

sum

119895=1

120573119894119895sdot

119895

sum

119897=1

120573119895119897) (42)

1

3

=

4

sum

119894=1

119887119894sdot (

119894minus1

sum

119895=1

120572119894119895)

2

(43)

1

4

=

4

sum

119894=1

119887119894sdot (

119894minus1

sum

119895=1

120572119894119895)

3

(44)

1

24

=

4

sum

119894=1

119887119894sdot (120572119894

119894minus1

sum

119895=1

120572119894119895sdot (120574119895119895+

119895minus1

sum

119897=1

120573119895119897)) (45)

1

12

=

4

sum

119894=1

119887119894sdot (

119894

sum

119895=1

120573119894119895sdot (

119895minus1

sum

119897=1

120572119895119897)

2

) (46)

1

24

=

4

sum

119894=1

119887119894sdot (

119894

sum

119895=1

(120573119894119895sdot

119895

sum

119897=1

(120573119895119897sdot

119897

sum

119896=1

120573119897119896))) (47)


119894119894= 120574


1198991198691015840

119899= 1198691015840

119899119869119899 thus conditions (45)



119894119894= 120574 for





119887412057343120573321205722

2= minus2120574

4+ 41205743minus

5

3

1205742+

120574

6

(48)

119887412057343120573321205722

2= minus

81205744

3

+ 51205743minus 21205742+

7120574

36

(49)

which imply

1205743minus

3

2

1205742+

120574

2

minus

1

24

= 0 (50)



119899+1=

119910119899+sum119904

119895=1





119860 =(

1 1 1 1

12057311205732

1205733

1205734

1205732

1

2

sum

119894=1

1205732119894sdot 120573119894

3

sum

119894=1

1205733119894sdot 120573119894

4

sum

119894=1

1205734119894sdot 120573119894

0 1205722

21205722

31205722

4

) (51)

and 120573119894= sum119894

119895=1120573119894119895


2

31205722



119894119894=




1198871= 04074074074074 119887

2= minus02568608534470 119887

3= 02

1198874= 06494534460396 120572

21= 075 120572

31= 075

12057232= 0 120572

41= 29193596398302 120572

42= 04

12057243= minus25693596398302 120574

21= minus075 120574

31= minus13152686912402

12057432= 075 120574

41= minus28738466294648 120574

42= minus33778743470341

12057443= 45693596398302 120574 = 1068579021301629



1198871+ 1198872+ 1198873+ 1198874= 1 (52)

11988721205732+ 11988731205733+ 11988741205734=

1

2

minus 120574 (53)

1198873120573321205732+ 1198874(120573421205732+ 120573431205733) =

1

6

minus 120574 + 1205742 (54)

11988721205722

21+ 11988731205722

3+ 11988741205722

4=

1

3

(55)

11988721205723

21+ 11988731205723

3+ 11988741205723

4=

1

4

(56)

11988731205723120572321205732+ 11988741205724(120572421205732+ 120572431205733) =

1

8

minus

120574

3

(57)

1198873120573321205722

2+ 1198874(120573421205722

2+ 120573431205722

3) =

1

12

minus

120574

3

(58)

119887412057343120573321205732=

1

24

minus

120574

2

+

3

2

1205742minus 1205743 (59)


119895=1120573119894119895


412057343120573321205722

2=





2= 1205723= 1205724 (55)-(56) imply that 120572

21= 1205722= 34


1= 1127

Letting 12057231= 12057221and 120572




11988741205733=

1 minus 6120574 + 61205742

612057343

12057243=

(1 minus 81205743) 12057343

1 minus 6120574 + 61205742

(60)


12057241=

3

4

minus 12057242minus

(1 minus 81205743) 12057343

1 minus 6120574 + 61205742 (61)


1198874=

16 (minus21205744+ 41205743minus 512057423 + 1205746)

91205734312057332

12057331=

1 minus 6120574 + 61205742

6120573431198874

minus 12057332

(62)


2= 1627 minus

1198873minus 1198874and

12057342=

(4 minus 16120574 minus 27119887312057332)

(271198874)

minus 12057343

12057341=

(12 minus 120574 minus 1198873(12057331+ 12057332))

1198874

minus 12057342minus 12057343

(63)




0 119906119899

1 119906

119899

119872) is the


as

119880119899+1= 119880119899+ 11988711198701+ 11988721198702+ 11988731198703+ 11988741198704 (64)

(119860 minus 120574Δ119905119869119899)119870119894= 119865(119880

119899+

119894minus1

sum

119895=1

120572119894119895119870119895 119905119899+ 120590119894Δ119905)

+ 120583119894Δ1199052 120597119865

120597119905

(119880119899 119905119899) + Δ119905J

119899(

119894minus1

sum

119895=1

120574119894119895119870119895)

119894 = 1 2 3 4

(65)

where 119869119899= (120597119865120597119880)(119880


119894and 120583

119894are





44=



119877 (119911) = (1 + (1 minus 4120574) 119911 + (

1

2

minus 4120574 + 61205742) 1199112

minus(

51205742

4

minus

5120574

8

+

1

16

) 1199114)

times ((1 minus 120574119911)4)

minus1

(66)


stable we need



Apparently


|119877 (119911)| =

100381610038161003816100381610038161003816100381610038161003816

512057424 minus 51205748 + 116

1205744

100381610038161003816100381610038161003816100381610038161003816

= 06304149382 lt 1

(69)





0 1 2 3 4 5 6 7 8 9 10 11 12

0

1

2

3

4

5

6

7

8

x

y


minus7

minus6

minus5

minus4

minus3

minus2

minus1

Stable

Stable

Stable Unstable





(119909 119905) isin (0 2) times (0 1]

119906 (119909 0) = cos (119909) 119909 isin [0 2]


(70)
















1and





1







119906 (119909 0) = cos (119909) 119909 isin [0 2]

119906119909(0 119905) = 0 119906


(71)










119906 (119909 0) = cos (119909) 119909 isin [0 1]


(72)








Δ119905 110 120 140 180















5 Conclusion






Acknowledgment


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1199061015840

119894(119905) =

119863

ℎ2

1205752

119909

1 + 1205752

11990912

119906119894(119905) + 119891 (119906

119894(119905) 119909119894 119905)

1 le 119894 le 119872 minus 1

(9)




implicit ODE system

(1 +

1205752

119909

12

)1199061015840

119894(119905) =

119863

ℎ21205752

119909119906119894(119905) + (1 +

1205752

119909

12

)119891 (119906119894(119905) 119909119894 119905)

1 le 119894 le 119872 minus 1 0 lt 119905 le 119879

(10)


1198601198801015840(119905) = 119865 (119880119883 119905) 0 lt 119905 le 119879 (11)

where 119880(119905) = (1199061(119905) 1199062(119905) 119906



119894 119905)119860 is an (119872minus1)times


0and 119909 = 119909

119872 which can be



1199061015840

0(119905) = 119892

1015840

1(119905) 119906

1015840

119872(119905) = 119892

1015840

2(119905) (12)


119860 =

(

(

(

(


1

12

5

6

1

12




1

12

5

6

1

12


)

)

)

)

(13)


1198650= 1198921015840

1(119905)

119865119894=

119863

ℎ21205752

119909119906119894(119905) + (1 +

1205752

119909

12

)119891 (119906119894(119905) 119909119894 119905)

1 le 119894 le 119872 minus 1

119865119872= 1198921015840

2(119905)

(14)


0(119905) and 119906

119872(119905) in (10) with 119892

1(119905) and 119892

2(119905)





1199060(119905) and 119906



0






1199061(119905) minus 119906

minus1(119905)

2ℎ

= 119906119909(119886 119905) +

ℎ2

6

119906119909119909119909(119886 119905) + 119874 (ℎ

4) (15)


119906119909119909119909=

1

119863

(119906119909119905minus 119891119909minus 119891119906sdot 119906119909) (16)


119906119909119909119909(119886)

=

1

119863

(1198921015840

3(119905) minus 119891

119909(1199060(119905) 119886 119905) minus 119891

119906(1199060(119905) 119886 119905) sdot 119892

3(119905))

(17)


minus1(119905)

119906minus1(119905) = 119906

1(119905) minus 2ℎ119892

3(119905) minus

ℎ3

3119863

(1198921015840

3(119905) minus 119891

119909(1199060(119905) 119886 119905)

minus 119891119906(1199060(119905) 119886 119905) sdot 119892

3(119905))

(18)



is preserved



119906119872+1(119905) = 119906

119872minus1(119905) + 2ℎ119892

4(119905)

+

ℎ3

3119863

(1198921015840

4(119905) minus 119891

119909(119906119872(119905) 119887 119905)

minus 119891119906(119906119872(119905) 119887 119905) sdot 119892

4(119905))

(19)



10

12

+

ℎ3

36119863

(119891119909119906(1199060 119886 119905)

+119891119906119906(1199060 119886 119905) sdot 119892

3(119905) )

1

6

0)

1198601198720sdotsdotsdot119872= (0

1

6

10

12

minus

ℎ3

36119863

(119891119909119906(119906119872 119887 119905)

+119891119906119906(119906119872 119887 119905) sdot 119892

4(119905)) )

(20)


1198650=

119863

ℎ2(21199061minus 21199060minus 2ℎ119892

3)

+

1

12

(10119891 (1199060 119886 119905) + 119891 (119906

1 119886 + ℎ 119905))

minus

ℎ

3

(1198921015840

3minus 119891119909(1199060 119886 119905) minus 119891

119906(1199060 119886 119905) sdot 119892

3) +

ℎ

6

1198921015840

3

+

ℎ3

36119863

(11989210158401015840

3minus 119891119909119905(1199060 119886 119905) minus 119891

119906(1199060 119886 119905) sdot 119892

1015840

3

minus119891119906119905(1199060 119886 119905) sdot 119892

3)

+

1

12

119891(1199061minus 2ℎ119892

3minus

ℎ3

3119863

times (1198921015840

3minus 119891119909(1199060 119886 119905) minus 119891

119906(1199060 119886 119905) sdot 119892

3)

119886 minus ℎ 119905)

119865119872=

2119863

ℎ2(119906119872minus1minus 119906119872+ ℎ1198924)

+

1

12

(10119891 (119906119872 119887 119905) + 119891 (119906

119872minus1 119887 minus ℎ 119905))

minus

ℎ

3

(1198921015840

4minus 119891119909(119906119872 119887 119905) minus 119891

119906(119906119872 119887 119905) sdot 119892

4) minus

ℎ

6

1198921015840

4

minus

ℎ3

36119863

(11989210158401015840

4minus 119891119909119905(119906119872 119887 119905) minus 119891

119906(119906119872 119887 119905) sdot 119892

1015840

4

minus119891119906119905(119906119872 119887 119905) sdot 119892

4)

+

1

12

119891(119906119872minus1+ 2ℎ119892

4+

ℎ3

3119863

times (1198921015840

4minus 119891119909(119906119872 119887 119905) minus 119891

119906(119906119872 119887 119905) sdot 119892

4)

119887 + ℎ 119905)

(21)


00and 119860

119872119872only


ℎ3

36119863

(119891119909119906(1199060 119886 119905) + 119891

119906119906(1199060 119886 119905) sdot 119892

1(119905)) (22)

minus

ℎ3

36119863

(119891119909119906(119906119872 119887 119905) + 119891

119906119906(119906119872 119887 119905) sdot 119892

4(119905)) (23)


0and 119865

119872are

modified as

1198650= 1198650

minus

ℎ3

36119863

[(119891119909119906(1199060 119886 119905) + 119891

119906119906(1199060 119886 119905) sdot 119892

3(119905))] sdot 119906

1015840

0(119905)

(24)

119865119872= 119865119872

+

ℎ3

36119863

[(119891119909119906(119906119872 119887 119905) + 119891

119906119906(119906119872 119887 119905) sdot 119892

4(119905))]

sdot 1199061015840

119872(119905)

(25)

Using (1) we have

1199061015840

0(119905) =

2119863

ℎ2(1199061(119905) minus 119906

0(119905) minus ℎ119892

3(119905))

+ 119891 (1199060(119905) 119886 119905) + 119874 (ℎ)

(26)

1199061015840

119872(119905) =

2119863

ℎ2(119906119872minus1(119905) minus 119906

119872(119905) + ℎ119892

4(119905))

+ 119891 (119906119872(119905) 119887 119905) + 119874 (ℎ)

(27)



1198650= 1198650minus

ℎ3

36119863

[(119891119909119906(1199060 119886 119905) + 119891

119906119906(1199060 119886 119905) sdot 119892

3(119905))]

times (

2119863

ℎ2(1199061minus 1199060minus ℎ1198923(119905)) + 119891 (119906

0 119886 119905))

(28)


119865119872= 119865119872minus

ℎ3

36119863

[(119891119909119906(119906119872 119887 119905) + 119891

119906119906(119906119872 119887 119905) sdot 119892

4(119905))]

times (

2119863

ℎ2(119906119872minus1(119905) minus 119906

119872(119905) + ℎ119892

4(119905))

+119891 (119906119872(119905) 119887 119905) )

(29)




119860 =

(

(

(

(

(

(

(

(

5

6

1

12


1

12

5

6

1

12




1

12

5

6

1

12


1

12

5

6

)

)

)

)

)

)

)

)

(30)







119899+1= 119877(119910








0) = 119910





119899is known as 119910

119899 the 4-stage


as

119910119899+1= 119910119899+ 11988711198961+ 11988721198962+ 11988731198963+ 11988741198964 (31)

119896119894= Δ119905119891(119910

119899+

119894minus1

sum

119895=1

120572119894119895119896119895) + Δ119905119869 (119910

119899)

119894

sum

119895=1

120574119894119895119896119895

119894 = 1 2 3 4

(32)

where 119887119894 120572119894119895 and 120574


119869(119910119899) = 119891119910(119910119899)


119910119899+1= 119910119899+ 11988711198961+ 11988721198962+ 11988731198963+ 11988741198964

119896119894= Δ119905119891(119910

119899+

119894minus1

sum

119895=1

120572119894119895119896119895 119905119899+ 120572119894Δ119905)

+ 120574119894Δ1199052119891119905(119910119899 119905119899) + Δ119905119891

119910(119910119899 119905119899)

119894

sum

119895=1

120574119894119895119896119895

119894 = 1 2 3 4

(33)

where

120572119894=

119894minus1

sum

119895=1

120572119894119895 120574

119894=

119894

sum

119895=1

120574119894119895 (34)


119899+ Δ119905) minus 119910

119899+1= 119874(Δ119905



119899+Δ119905) and 119896

119894


119910119899+1


Given 119910(119905119899) = 119910


119899+ Δ119905) at 119905

119899is

expanded as

119910 (119905119899+ Δ119905) = 119910

119899+ Δ119905119891119899+

Δ1199052

2

119869119899119891119899+

Δ1199053

6

(1198692

119899119891119899+ 1198691015840

1198991198912

119899)

+

Δ1199054

24

(11986910158401015840

1198991198913

119899+ 21198691015840

1198991198691198991198912

119899+ 21198691198991198691015840

1198991198912

119899+ 1198693

119899119891119899)

+ 119874 (Δ1199055)

(35)

where 119891119899= 119891(119910

119899) 119869119899= (120597119891120597119910)(119910

119899) 1198691015840119899= (12059721198911205971199102)(119910119899) and

11986910158401015840


119894119895= 120572119894119895+120574119894119895

and 120572119894119894= 0


1198961= Δ119905 (1 minus Δ119905120574

11119869119899)minus1

119891119899

= Δ119905119891119899+ Δ119905212057411119869119899119891119899+ Δ11990531205742

111198692

119899119891119899

+ Δ11990541205743

111198693

119899119891119899+ 119874 (Δ119905

5)

(36)


1198962= Δ119905 (1 minus Δ119905120574

22119869119899)minus1

(119891 (119910119899+ 120572211198961) + 120574211198691198991198961) (37)


obtain

1198962= Δ119905119891

119899+ Δ1199052(12057221+ 12057421+ 12057422) 119869119899119891119899

+ Δ1199053[(12057411+ 12057422) (12057221+ 12057421) + 1205742

22] 1198692

119899119891119899+

Δ1199053

2

1205722

211198691015840

1198991198912

119899

+ Δ1199054[(1205741112057422+ 1205742

11+ 1205742

22) (12057221+ 12057421) + 1205743

22] 1198693

119899119891119899

+ Δ1199054120574111205722

211198691015840

1198991198691198991198912

119899+

Δ1199054

2

120574221205722

211198691198991198691015840

1198991198912

119899

+

Δ1199054

6

1205723

2111986910158401015840

1198991198913

119899+ 119874 (Δ119905

5)

(38)


1198963= Δ119905119891

119899+ Δ1199052(12057331+ 12057332+ 12057433) 119869119899119891119899

+ Δ1199053[

[

3

sum

119895=1

((1205723119895+ 1205743119895) sdot

119895

sum

119897=1

(120572119895119897+ 120574119895119897))]

]

1198692

119899119891119899

+

Δ1199053

2

1205722

31198691015840

1198991198912

119899

+ Δ1199054[

[

3

sum

119894=1

((1205723119894+ 1205743119894) sdot

119894

sum

119895=1

((120572119894119895+ 120574119894119895)

sdot

119895

sum

119897=1

(120572119895119897+ 120574119895119897)))]

]

1198693

119899119891119899

+

Δ1199054

2

[1205722

212057332+ 120574331205722

3] 1198691198991198691015840

1198991198912

119899

+ Δ1199054[1205723((12057321+ 12057422) 12057232+ 1205741112057231)] 1198691015840

1198991198691198991198912

119899

+

Δ1199054

6

(12057231+ 12057232)3

11986910158401015840

1198991198913

119899+ 119874 (Δ119905

5)


1198964= Δ119905119891

119899+ Δ1199052(12057341+ 12057342+ 12057342+ 12057444) 119869119899119891119899

+ Δ1199053[

[

4

sum

119895=1

((1205724119895+ 1205744119895) sdot

119895

sum

119897=1

(120572119895119897+ 120574119895119897))]

]

1198692

119899119891119899

+

Δ1199053

2

1205722

41198691015840

1198991198912

119899

+ Δ1199054[

[

4

sum

119894=1

((1205724119894+ 1205744119894) sdot

119894

sum

119895=1

((120572119894119895+ 120574119894119895)

sdot

119895

sum

119897=1

(120572119895119897+ 120574119895119897)))]

]

1198693

119899119891119899

+ Δ1199054[1205724((12057433+ 12057332+ 12057331) 12057243

+ (12057422+ 12057321) 12057242+ 1205741112057241)] 1198691015840

1198991198691198991198912

119899

+

Δ1199054

2

[120574441205722

4+ 120573431205722

3+ 120573421205722

2] 1198691198991198691015840

1198991198912

119899

+

Δ1199054

6

(12057241+ 12057242+ 12057243)3

11986910158401015840

1198991198913

119899+ 119874 (Δ119905

5)

(39)


1 =

4

sum

119894=1

119887119894 (40)

1

2

=

4

sum

119894=1

119887119894sdot (120574119894119894+

119894minus1

sum

119895=1

120573119894119895) (41)

1

6

=

4

sum

119894=1

119887119894sdot (

119894

sum

119895=1

120573119894119895sdot

119895

sum

119897=1

120573119895119897) (42)

1

3

=

4

sum

119894=1

119887119894sdot (

119894minus1

sum

119895=1

120572119894119895)

2

(43)

1

4

=

4

sum

119894=1

119887119894sdot (

119894minus1

sum

119895=1

120572119894119895)

3

(44)

1

24

=

4

sum

119894=1

119887119894sdot (120572119894

119894minus1

sum

119895=1

120572119894119895sdot (120574119895119895+

119895minus1

sum

119897=1

120573119895119897)) (45)

1

12

=

4

sum

119894=1

119887119894sdot (

119894

sum

119895=1

120573119894119895sdot (

119895minus1

sum

119897=1

120572119895119897)

2

) (46)

1

24

=

4

sum

119894=1

119887119894sdot (

119894

sum

119895=1

(120573119894119895sdot

119895

sum

119897=1

(120573119895119897sdot

119897

sum

119896=1

120573119897119896))) (47)


119894119894= 120574


1198991198691015840

119899= 1198691015840

119899119869119899 thus conditions (45)



119894119894= 120574 for





119887412057343120573321205722

2= minus2120574

4+ 41205743minus

5

3

1205742+

120574

6

(48)

119887412057343120573321205722

2= minus

81205744

3

+ 51205743minus 21205742+

7120574

36

(49)

which imply

1205743minus

3

2

1205742+

120574

2

minus

1

24

= 0 (50)



119899+1=

119910119899+sum119904

119895=1





119860 =(

1 1 1 1

12057311205732

1205733

1205734

1205732

1

2

sum

119894=1

1205732119894sdot 120573119894

3

sum

119894=1

1205733119894sdot 120573119894

4

sum

119894=1

1205734119894sdot 120573119894

0 1205722

21205722

31205722

4

) (51)

and 120573119894= sum119894

119895=1120573119894119895


2

31205722



119894119894=




1198871= 04074074074074 119887

2= minus02568608534470 119887

3= 02

1198874= 06494534460396 120572

21= 075 120572

31= 075

12057232= 0 120572

41= 29193596398302 120572

42= 04

12057243= minus25693596398302 120574

21= minus075 120574

31= minus13152686912402

12057432= 075 120574

41= minus28738466294648 120574

42= minus33778743470341

12057443= 45693596398302 120574 = 1068579021301629



1198871+ 1198872+ 1198873+ 1198874= 1 (52)

11988721205732+ 11988731205733+ 11988741205734=

1

2

minus 120574 (53)

1198873120573321205732+ 1198874(120573421205732+ 120573431205733) =

1

6

minus 120574 + 1205742 (54)

11988721205722

21+ 11988731205722

3+ 11988741205722

4=

1

3

(55)

11988721205723

21+ 11988731205723

3+ 11988741205723

4=

1

4

(56)

11988731205723120572321205732+ 11988741205724(120572421205732+ 120572431205733) =

1

8

minus

120574

3

(57)

1198873120573321205722

2+ 1198874(120573421205722

2+ 120573431205722

3) =

1

12

minus

120574

3

(58)

119887412057343120573321205732=

1

24

minus

120574

2

+

3

2

1205742minus 1205743 (59)


119895=1120573119894119895


412057343120573321205722

2=





2= 1205723= 1205724 (55)-(56) imply that 120572

21= 1205722= 34


1= 1127

Letting 12057231= 12057221and 120572




11988741205733=

1 minus 6120574 + 61205742

612057343

12057243=

(1 minus 81205743) 12057343

1 minus 6120574 + 61205742

(60)


12057241=

3

4

minus 12057242minus

(1 minus 81205743) 12057343

1 minus 6120574 + 61205742 (61)


1198874=

16 (minus21205744+ 41205743minus 512057423 + 1205746)

91205734312057332

12057331=

1 minus 6120574 + 61205742

6120573431198874

minus 12057332

(62)


2= 1627 minus

1198873minus 1198874and

12057342=

(4 minus 16120574 minus 27119887312057332)

(271198874)

minus 12057343

12057341=

(12 minus 120574 minus 1198873(12057331+ 12057332))

1198874

minus 12057342minus 12057343

(63)




0 119906119899

1 119906

119899

119872) is the


as

119880119899+1= 119880119899+ 11988711198701+ 11988721198702+ 11988731198703+ 11988741198704 (64)

(119860 minus 120574Δ119905119869119899)119870119894= 119865(119880

119899+

119894minus1

sum

119895=1

120572119894119895119870119895 119905119899+ 120590119894Δ119905)

+ 120583119894Δ1199052 120597119865

120597119905

(119880119899 119905119899) + Δ119905J

119899(

119894minus1

sum

119895=1

120574119894119895119870119895)

119894 = 1 2 3 4

(65)

where 119869119899= (120597119865120597119880)(119880


119894and 120583

119894are





44=



119877 (119911) = (1 + (1 minus 4120574) 119911 + (

1

2

minus 4120574 + 61205742) 1199112

minus(

51205742

4

minus

5120574

8

+

1

16

) 1199114)

times ((1 minus 120574119911)4)

minus1

(66)


stable we need



Apparently


|119877 (119911)| =

100381610038161003816100381610038161003816100381610038161003816

512057424 minus 51205748 + 116

1205744

100381610038161003816100381610038161003816100381610038161003816

= 06304149382 lt 1

(69)





0 1 2 3 4 5 6 7 8 9 10 11 12

0

1

2

3

4

5

6

7

8

x

y


minus7

minus6

minus5

minus4

minus3

minus2

minus1

Stable

Stable

Stable Unstable





(119909 119905) isin (0 2) times (0 1]

119906 (119909 0) = cos (119909) 119909 isin [0 2]


(70)
















1and





1







119906 (119909 0) = cos (119909) 119909 isin [0 2]

119906119909(0 119905) = 0 119906


(71)










119906 (119909 0) = cos (119909) 119909 isin [0 1]


(72)








Δ119905 110 120 140 180















5 Conclusion






Acknowledgment


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119906119872+1(119905) = 119906

119872minus1(119905) + 2ℎ119892

4(119905)

+

ℎ3

3119863

(1198921015840

4(119905) minus 119891

119909(119906119872(119905) 119887 119905)

minus 119891119906(119906119872(119905) 119887 119905) sdot 119892

4(119905))

(19)



10

12

+

ℎ3

36119863

(119891119909119906(1199060 119886 119905)

+119891119906119906(1199060 119886 119905) sdot 119892

3(119905) )

1

6

0)

1198601198720sdotsdotsdot119872= (0

1

6

10

12

minus

ℎ3

36119863

(119891119909119906(119906119872 119887 119905)

+119891119906119906(119906119872 119887 119905) sdot 119892

4(119905)) )

(20)


1198650=

119863

ℎ2(21199061minus 21199060minus 2ℎ119892

3)

+

1

12

(10119891 (1199060 119886 119905) + 119891 (119906

1 119886 + ℎ 119905))

minus

ℎ

3

(1198921015840

3minus 119891119909(1199060 119886 119905) minus 119891

119906(1199060 119886 119905) sdot 119892

3) +

ℎ

6

1198921015840

3

+

ℎ3

36119863

(11989210158401015840

3minus 119891119909119905(1199060 119886 119905) minus 119891

119906(1199060 119886 119905) sdot 119892

1015840

3

minus119891119906119905(1199060 119886 119905) sdot 119892

3)

+

1

12

119891(1199061minus 2ℎ119892

3minus

ℎ3

3119863

times (1198921015840

3minus 119891119909(1199060 119886 119905) minus 119891

119906(1199060 119886 119905) sdot 119892

3)

119886 minus ℎ 119905)

119865119872=

2119863

ℎ2(119906119872minus1minus 119906119872+ ℎ1198924)

+

1

12

(10119891 (119906119872 119887 119905) + 119891 (119906

119872minus1 119887 minus ℎ 119905))

minus

ℎ

3

(1198921015840

4minus 119891119909(119906119872 119887 119905) minus 119891

119906(119906119872 119887 119905) sdot 119892

4) minus

ℎ

6

1198921015840

4

minus

ℎ3

36119863

(11989210158401015840

4minus 119891119909119905(119906119872 119887 119905) minus 119891

119906(119906119872 119887 119905) sdot 119892

1015840

4

minus119891119906119905(119906119872 119887 119905) sdot 119892

4)

+

1

12

119891(119906119872minus1+ 2ℎ119892

4+

ℎ3

3119863

times (1198921015840

4minus 119891119909(119906119872 119887 119905) minus 119891

119906(119906119872 119887 119905) sdot 119892

4)

119887 + ℎ 119905)

(21)


00and 119860

119872119872only


ℎ3

36119863

(119891119909119906(1199060 119886 119905) + 119891

119906119906(1199060 119886 119905) sdot 119892

1(119905)) (22)

minus

ℎ3

36119863

(119891119909119906(119906119872 119887 119905) + 119891

119906119906(119906119872 119887 119905) sdot 119892

4(119905)) (23)


0and 119865

119872are

modified as

1198650= 1198650

minus

ℎ3

36119863

[(119891119909119906(1199060 119886 119905) + 119891

119906119906(1199060 119886 119905) sdot 119892

3(119905))] sdot 119906

1015840

0(119905)

(24)

119865119872= 119865119872

+

ℎ3

36119863

[(119891119909119906(119906119872 119887 119905) + 119891

119906119906(119906119872 119887 119905) sdot 119892

4(119905))]

sdot 1199061015840

119872(119905)

(25)

Using (1) we have

1199061015840

0(119905) =

2119863

ℎ2(1199061(119905) minus 119906

0(119905) minus ℎ119892

3(119905))

+ 119891 (1199060(119905) 119886 119905) + 119874 (ℎ)

(26)

1199061015840

119872(119905) =

2119863

ℎ2(119906119872minus1(119905) minus 119906

119872(119905) + ℎ119892

4(119905))

+ 119891 (119906119872(119905) 119887 119905) + 119874 (ℎ)

(27)



1198650= 1198650minus

ℎ3

36119863

[(119891119909119906(1199060 119886 119905) + 119891

119906119906(1199060 119886 119905) sdot 119892

3(119905))]

times (

2119863

ℎ2(1199061minus 1199060minus ℎ1198923(119905)) + 119891 (119906

0 119886 119905))

(28)


119865119872= 119865119872minus

ℎ3

36119863

[(119891119909119906(119906119872 119887 119905) + 119891

119906119906(119906119872 119887 119905) sdot 119892

4(119905))]

times (

2119863

ℎ2(119906119872minus1(119905) minus 119906

119872(119905) + ℎ119892

4(119905))

+119891 (119906119872(119905) 119887 119905) )

(29)




119860 =

(

(

(

(

(

(

(

(

5

6

1

12


1

12

5

6

1

12




1

12

5

6

1

12


1

12

5

6

)

)

)

)

)

)

)

)

(30)







119899+1= 119877(119910








0) = 119910





119899is known as 119910

119899 the 4-stage


as

119910119899+1= 119910119899+ 11988711198961+ 11988721198962+ 11988731198963+ 11988741198964 (31)

119896119894= Δ119905119891(119910

119899+

119894minus1

sum

119895=1

120572119894119895119896119895) + Δ119905119869 (119910

119899)

119894

sum

119895=1

120574119894119895119896119895

119894 = 1 2 3 4

(32)

where 119887119894 120572119894119895 and 120574


119869(119910119899) = 119891119910(119910119899)


119910119899+1= 119910119899+ 11988711198961+ 11988721198962+ 11988731198963+ 11988741198964

119896119894= Δ119905119891(119910

119899+

119894minus1

sum

119895=1

120572119894119895119896119895 119905119899+ 120572119894Δ119905)

+ 120574119894Δ1199052119891119905(119910119899 119905119899) + Δ119905119891

119910(119910119899 119905119899)

119894

sum

119895=1

120574119894119895119896119895

119894 = 1 2 3 4

(33)

where

120572119894=

119894minus1

sum

119895=1

120572119894119895 120574

119894=

119894

sum

119895=1

120574119894119895 (34)


119899+ Δ119905) minus 119910

119899+1= 119874(Δ119905



119899+Δ119905) and 119896

119894


119910119899+1


Given 119910(119905119899) = 119910


119899+ Δ119905) at 119905

119899is

expanded as

119910 (119905119899+ Δ119905) = 119910

119899+ Δ119905119891119899+

Δ1199052

2

119869119899119891119899+

Δ1199053

6

(1198692

119899119891119899+ 1198691015840

1198991198912

119899)

+

Δ1199054

24

(11986910158401015840

1198991198913

119899+ 21198691015840

1198991198691198991198912

119899+ 21198691198991198691015840

1198991198912

119899+ 1198693

119899119891119899)

+ 119874 (Δ1199055)

(35)

where 119891119899= 119891(119910

119899) 119869119899= (120597119891120597119910)(119910

119899) 1198691015840119899= (12059721198911205971199102)(119910119899) and

11986910158401015840


119894119895= 120572119894119895+120574119894119895

and 120572119894119894= 0


1198961= Δ119905 (1 minus Δ119905120574

11119869119899)minus1

119891119899

= Δ119905119891119899+ Δ119905212057411119869119899119891119899+ Δ11990531205742

111198692

119899119891119899

+ Δ11990541205743

111198693

119899119891119899+ 119874 (Δ119905

5)

(36)


1198962= Δ119905 (1 minus Δ119905120574

22119869119899)minus1

(119891 (119910119899+ 120572211198961) + 120574211198691198991198961) (37)


obtain

1198962= Δ119905119891

119899+ Δ1199052(12057221+ 12057421+ 12057422) 119869119899119891119899

+ Δ1199053[(12057411+ 12057422) (12057221+ 12057421) + 1205742

22] 1198692

119899119891119899+

Δ1199053

2

1205722

211198691015840

1198991198912

119899

+ Δ1199054[(1205741112057422+ 1205742

11+ 1205742

22) (12057221+ 12057421) + 1205743

22] 1198693

119899119891119899

+ Δ1199054120574111205722

211198691015840

1198991198691198991198912

119899+

Δ1199054

2

120574221205722

211198691198991198691015840

1198991198912

119899

+

Δ1199054

6

1205723

2111986910158401015840

1198991198913

119899+ 119874 (Δ119905

5)

(38)


1198963= Δ119905119891

119899+ Δ1199052(12057331+ 12057332+ 12057433) 119869119899119891119899

+ Δ1199053[

[

3

sum

119895=1

((1205723119895+ 1205743119895) sdot

119895

sum

119897=1

(120572119895119897+ 120574119895119897))]

]

1198692

119899119891119899

+

Δ1199053

2

1205722

31198691015840

1198991198912

119899

+ Δ1199054[

[

3

sum

119894=1

((1205723119894+ 1205743119894) sdot

119894

sum

119895=1

((120572119894119895+ 120574119894119895)

sdot

119895

sum

119897=1

(120572119895119897+ 120574119895119897)))]

]

1198693

119899119891119899

+

Δ1199054

2

[1205722

212057332+ 120574331205722

3] 1198691198991198691015840

1198991198912

119899

+ Δ1199054[1205723((12057321+ 12057422) 12057232+ 1205741112057231)] 1198691015840

1198991198691198991198912

119899

+

Δ1199054

6

(12057231+ 12057232)3

11986910158401015840

1198991198913

119899+ 119874 (Δ119905

5)


1198964= Δ119905119891

119899+ Δ1199052(12057341+ 12057342+ 12057342+ 12057444) 119869119899119891119899

+ Δ1199053[

[

4

sum

119895=1

((1205724119895+ 1205744119895) sdot

119895

sum

119897=1

(120572119895119897+ 120574119895119897))]

]

1198692

119899119891119899

+

Δ1199053

2

1205722

41198691015840

1198991198912

119899

+ Δ1199054[

[

4

sum

119894=1

((1205724119894+ 1205744119894) sdot

119894

sum

119895=1

((120572119894119895+ 120574119894119895)

sdot

119895

sum

119897=1

(120572119895119897+ 120574119895119897)))]

]

1198693

119899119891119899

+ Δ1199054[1205724((12057433+ 12057332+ 12057331) 12057243

+ (12057422+ 12057321) 12057242+ 1205741112057241)] 1198691015840

1198991198691198991198912

119899

+

Δ1199054

2

[120574441205722

4+ 120573431205722

3+ 120573421205722

2] 1198691198991198691015840

1198991198912

119899

+

Δ1199054

6

(12057241+ 12057242+ 12057243)3

11986910158401015840

1198991198913

119899+ 119874 (Δ119905

5)

(39)


1 =

4

sum

119894=1

119887119894 (40)

1

2

=

4

sum

119894=1

119887119894sdot (120574119894119894+

119894minus1

sum

119895=1

120573119894119895) (41)

1

6

=

4

sum

119894=1

119887119894sdot (

119894

sum

119895=1

120573119894119895sdot

119895

sum

119897=1

120573119895119897) (42)

1

3

=

4

sum

119894=1

119887119894sdot (

119894minus1

sum

119895=1

120572119894119895)

2

(43)

1

4

=

4

sum

119894=1

119887119894sdot (

119894minus1

sum

119895=1

120572119894119895)

3

(44)

1

24

=

4

sum

119894=1

119887119894sdot (120572119894

119894minus1

sum

119895=1

120572119894119895sdot (120574119895119895+

119895minus1

sum

119897=1

120573119895119897)) (45)

1

12

=

4

sum

119894=1

119887119894sdot (

119894

sum

119895=1

120573119894119895sdot (

119895minus1

sum

119897=1

120572119895119897)

2

) (46)

1

24

=

4

sum

119894=1

119887119894sdot (

119894

sum

119895=1

(120573119894119895sdot

119895

sum

119897=1

(120573119895119897sdot

119897

sum

119896=1

120573119897119896))) (47)


119894119894= 120574


1198991198691015840

119899= 1198691015840

119899119869119899 thus conditions (45)



119894119894= 120574 for





119887412057343120573321205722

2= minus2120574

4+ 41205743minus

5

3

1205742+

120574

6

(48)

119887412057343120573321205722

2= minus

81205744

3

+ 51205743minus 21205742+

7120574

36

(49)

which imply

1205743minus

3

2

1205742+

120574

2

minus

1

24

= 0 (50)



119899+1=

119910119899+sum119904

119895=1





119860 =(

1 1 1 1

12057311205732

1205733

1205734

1205732

1

2

sum

119894=1

1205732119894sdot 120573119894

3

sum

119894=1

1205733119894sdot 120573119894

4

sum

119894=1

1205734119894sdot 120573119894

0 1205722

21205722

31205722

4

) (51)

and 120573119894= sum119894

119895=1120573119894119895


2

31205722



119894119894=




1198871= 04074074074074 119887

2= minus02568608534470 119887

3= 02

1198874= 06494534460396 120572

21= 075 120572

31= 075

12057232= 0 120572

41= 29193596398302 120572

42= 04

12057243= minus25693596398302 120574

21= minus075 120574

31= minus13152686912402

12057432= 075 120574

41= minus28738466294648 120574

42= minus33778743470341

12057443= 45693596398302 120574 = 1068579021301629



1198871+ 1198872+ 1198873+ 1198874= 1 (52)

11988721205732+ 11988731205733+ 11988741205734=

1

2

minus 120574 (53)

1198873120573321205732+ 1198874(120573421205732+ 120573431205733) =

1

6

minus 120574 + 1205742 (54)

11988721205722

21+ 11988731205722

3+ 11988741205722

4=

1

3

(55)

11988721205723

21+ 11988731205723

3+ 11988741205723

4=

1

4

(56)

11988731205723120572321205732+ 11988741205724(120572421205732+ 120572431205733) =

1

8

minus

120574

3

(57)

1198873120573321205722

2+ 1198874(120573421205722

2+ 120573431205722

3) =

1

12

minus

120574

3

(58)

119887412057343120573321205732=

1

24

minus

120574

2

+

3

2

1205742minus 1205743 (59)


119895=1120573119894119895


412057343120573321205722

2=





2= 1205723= 1205724 (55)-(56) imply that 120572

21= 1205722= 34


1= 1127

Letting 12057231= 12057221and 120572




11988741205733=

1 minus 6120574 + 61205742

612057343

12057243=

(1 minus 81205743) 12057343

1 minus 6120574 + 61205742

(60)


12057241=

3

4

minus 12057242minus

(1 minus 81205743) 12057343

1 minus 6120574 + 61205742 (61)


1198874=

16 (minus21205744+ 41205743minus 512057423 + 1205746)

91205734312057332

12057331=

1 minus 6120574 + 61205742

6120573431198874

minus 12057332

(62)


2= 1627 minus

1198873minus 1198874and

12057342=

(4 minus 16120574 minus 27119887312057332)

(271198874)

minus 12057343

12057341=

(12 minus 120574 minus 1198873(12057331+ 12057332))

1198874

minus 12057342minus 12057343

(63)




0 119906119899

1 119906

119899

119872) is the


as

119880119899+1= 119880119899+ 11988711198701+ 11988721198702+ 11988731198703+ 11988741198704 (64)

(119860 minus 120574Δ119905119869119899)119870119894= 119865(119880

119899+

119894minus1

sum

119895=1

120572119894119895119870119895 119905119899+ 120590119894Δ119905)

+ 120583119894Δ1199052 120597119865

120597119905

(119880119899 119905119899) + Δ119905J

119899(

119894minus1

sum

119895=1

120574119894119895119870119895)

119894 = 1 2 3 4

(65)

where 119869119899= (120597119865120597119880)(119880


119894and 120583

119894are





44=



119877 (119911) = (1 + (1 minus 4120574) 119911 + (

1

2

minus 4120574 + 61205742) 1199112

minus(

51205742

4

minus

5120574

8

+

1

16

) 1199114)

times ((1 minus 120574119911)4)

minus1

(66)


stable we need



Apparently


|119877 (119911)| =

100381610038161003816100381610038161003816100381610038161003816

512057424 minus 51205748 + 116

1205744

100381610038161003816100381610038161003816100381610038161003816

= 06304149382 lt 1

(69)





0 1 2 3 4 5 6 7 8 9 10 11 12

0

1

2

3

4

5

6

7

8

x

y


minus7

minus6

minus5

minus4

minus3

minus2

minus1

Stable

Stable

Stable Unstable





(119909 119905) isin (0 2) times (0 1]

119906 (119909 0) = cos (119909) 119909 isin [0 2]


(70)
















1and





1







119906 (119909 0) = cos (119909) 119909 isin [0 2]

119906119909(0 119905) = 0 119906


(71)










119906 (119909 0) = cos (119909) 119909 isin [0 1]


(72)








Δ119905 110 120 140 180















5 Conclusion






Acknowledgment


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198650= 1198650minus

ℎ3

36119863

[(119891119909119906(1199060 119886 119905) + 119891

119906119906(1199060 119886 119905) sdot 119892

3(119905))]

times (

2119863

ℎ2(1199061minus 1199060minus ℎ1198923(119905)) + 119891 (119906

0 119886 119905))

(28)


119865119872= 119865119872minus

ℎ3

36119863

[(119891119909119906(119906119872 119887 119905) + 119891

119906119906(119906119872 119887 119905) sdot 119892

4(119905))]

times (

2119863

ℎ2(119906119872minus1(119905) minus 119906

119872(119905) + ℎ119892

4(119905))

+119891 (119906119872(119905) 119887 119905) )

(29)




119860 =

(

(

(

(

(

(

(

(

5

6

1

12


1

12

5

6

1

12




1

12

5

6

1

12


1

12

5

6

)

)

)

)

)

)

)

)

(30)







119899+1= 119877(119910








0) = 119910





119899is known as 119910

119899 the 4-stage


as

119910119899+1= 119910119899+ 11988711198961+ 11988721198962+ 11988731198963+ 11988741198964 (31)

119896119894= Δ119905119891(119910

119899+

119894minus1

sum

119895=1

120572119894119895119896119895) + Δ119905119869 (119910

119899)

119894

sum

119895=1

120574119894119895119896119895

119894 = 1 2 3 4

(32)

where 119887119894 120572119894119895 and 120574


119869(119910119899) = 119891119910(119910119899)


119910119899+1= 119910119899+ 11988711198961+ 11988721198962+ 11988731198963+ 11988741198964

119896119894= Δ119905119891(119910

119899+

119894minus1

sum

119895=1

120572119894119895119896119895 119905119899+ 120572119894Δ119905)

+ 120574119894Δ1199052119891119905(119910119899 119905119899) + Δ119905119891

119910(119910119899 119905119899)

119894

sum

119895=1

120574119894119895119896119895

119894 = 1 2 3 4

(33)

where

120572119894=

119894minus1

sum

119895=1

120572119894119895 120574

119894=

119894

sum

119895=1

120574119894119895 (34)


119899+ Δ119905) minus 119910

119899+1= 119874(Δ119905



119899+Δ119905) and 119896

119894


119910119899+1


Given 119910(119905119899) = 119910


119899+ Δ119905) at 119905

119899is

expanded as

119910 (119905119899+ Δ119905) = 119910

119899+ Δ119905119891119899+

Δ1199052

2

119869119899119891119899+

Δ1199053

6

(1198692

119899119891119899+ 1198691015840

1198991198912

119899)

+

Δ1199054

24

(11986910158401015840

1198991198913

119899+ 21198691015840

1198991198691198991198912

119899+ 21198691198991198691015840

1198991198912

119899+ 1198693

119899119891119899)

+ 119874 (Δ1199055)

(35)

where 119891119899= 119891(119910

119899) 119869119899= (120597119891120597119910)(119910

119899) 1198691015840119899= (12059721198911205971199102)(119910119899) and

11986910158401015840


119894119895= 120572119894119895+120574119894119895

and 120572119894119894= 0


1198961= Δ119905 (1 minus Δ119905120574

11119869119899)minus1

119891119899

= Δ119905119891119899+ Δ119905212057411119869119899119891119899+ Δ11990531205742

111198692

119899119891119899

+ Δ11990541205743

111198693

119899119891119899+ 119874 (Δ119905

5)

(36)


1198962= Δ119905 (1 minus Δ119905120574

22119869119899)minus1

(119891 (119910119899+ 120572211198961) + 120574211198691198991198961) (37)


obtain

1198962= Δ119905119891

119899+ Δ1199052(12057221+ 12057421+ 12057422) 119869119899119891119899

+ Δ1199053[(12057411+ 12057422) (12057221+ 12057421) + 1205742

22] 1198692

119899119891119899+

Δ1199053

2

1205722

211198691015840

1198991198912

119899

+ Δ1199054[(1205741112057422+ 1205742

11+ 1205742

22) (12057221+ 12057421) + 1205743

22] 1198693

119899119891119899

+ Δ1199054120574111205722

211198691015840

1198991198691198991198912

119899+

Δ1199054

2

120574221205722

211198691198991198691015840

1198991198912

119899

+

Δ1199054

6

1205723

2111986910158401015840

1198991198913

119899+ 119874 (Δ119905

5)

(38)


1198963= Δ119905119891

119899+ Δ1199052(12057331+ 12057332+ 12057433) 119869119899119891119899

+ Δ1199053[

[

3

sum

119895=1

((1205723119895+ 1205743119895) sdot

119895

sum

119897=1

(120572119895119897+ 120574119895119897))]

]

1198692

119899119891119899

+

Δ1199053

2

1205722

31198691015840

1198991198912

119899

+ Δ1199054[

[

3

sum

119894=1

((1205723119894+ 1205743119894) sdot

119894

sum

119895=1

((120572119894119895+ 120574119894119895)

sdot

119895

sum

119897=1

(120572119895119897+ 120574119895119897)))]

]

1198693

119899119891119899

+

Δ1199054

2

[1205722

212057332+ 120574331205722

3] 1198691198991198691015840

1198991198912

119899

+ Δ1199054[1205723((12057321+ 12057422) 12057232+ 1205741112057231)] 1198691015840

1198991198691198991198912

119899

+

Δ1199054

6

(12057231+ 12057232)3

11986910158401015840

1198991198913

119899+ 119874 (Δ119905

5)


1198964= Δ119905119891

119899+ Δ1199052(12057341+ 12057342+ 12057342+ 12057444) 119869119899119891119899

+ Δ1199053[

[

4

sum

119895=1

((1205724119895+ 1205744119895) sdot

119895

sum

119897=1

(120572119895119897+ 120574119895119897))]

]

1198692

119899119891119899

+

Δ1199053

2

1205722

41198691015840

1198991198912

119899

+ Δ1199054[

[

4

sum

119894=1

((1205724119894+ 1205744119894) sdot

119894

sum

119895=1

((120572119894119895+ 120574119894119895)

sdot

119895

sum

119897=1

(120572119895119897+ 120574119895119897)))]

]

1198693

119899119891119899

+ Δ1199054[1205724((12057433+ 12057332+ 12057331) 12057243

+ (12057422+ 12057321) 12057242+ 1205741112057241)] 1198691015840

1198991198691198991198912

119899

+

Δ1199054

2

[120574441205722

4+ 120573431205722

3+ 120573421205722

2] 1198691198991198691015840

1198991198912

119899

+

Δ1199054

6

(12057241+ 12057242+ 12057243)3

11986910158401015840

1198991198913

119899+ 119874 (Δ119905

5)

(39)


1 =

4

sum

119894=1

119887119894 (40)

1

2

=

4

sum

119894=1

119887119894sdot (120574119894119894+

119894minus1

sum

119895=1

120573119894119895) (41)

1

6

=

4

sum

119894=1

119887119894sdot (

119894

sum

119895=1

120573119894119895sdot

119895

sum

119897=1

120573119895119897) (42)

1

3

=

4

sum

119894=1

119887119894sdot (

119894minus1

sum

119895=1

120572119894119895)

2

(43)

1

4

=

4

sum

119894=1

119887119894sdot (

119894minus1

sum

119895=1

120572119894119895)

3

(44)

1

24

=

4

sum

119894=1

119887119894sdot (120572119894

119894minus1

sum

119895=1

120572119894119895sdot (120574119895119895+

119895minus1

sum

119897=1

120573119895119897)) (45)

1

12

=

4

sum

119894=1

119887119894sdot (

119894

sum

119895=1

120573119894119895sdot (

119895minus1

sum

119897=1

120572119895119897)

2

) (46)

1

24

=

4

sum

119894=1

119887119894sdot (

119894

sum

119895=1

(120573119894119895sdot

119895

sum

119897=1

(120573119895119897sdot

119897

sum

119896=1

120573119897119896))) (47)


119894119894= 120574


1198991198691015840

119899= 1198691015840

119899119869119899 thus conditions (45)



119894119894= 120574 for





119887412057343120573321205722

2= minus2120574

4+ 41205743minus

5

3

1205742+

120574

6

(48)

119887412057343120573321205722

2= minus

81205744

3

+ 51205743minus 21205742+

7120574

36

(49)

which imply

1205743minus

3

2

1205742+

120574

2

minus

1

24

= 0 (50)



119899+1=

119910119899+sum119904

119895=1





119860 =(

1 1 1 1

12057311205732

1205733

1205734

1205732

1

2

sum

119894=1

1205732119894sdot 120573119894

3

sum

119894=1

1205733119894sdot 120573119894

4

sum

119894=1

1205734119894sdot 120573119894

0 1205722

21205722

31205722

4

) (51)

and 120573119894= sum119894

119895=1120573119894119895


2

31205722



119894119894=




1198871= 04074074074074 119887

2= minus02568608534470 119887

3= 02

1198874= 06494534460396 120572

21= 075 120572

31= 075

12057232= 0 120572

41= 29193596398302 120572

42= 04

12057243= minus25693596398302 120574

21= minus075 120574

31= minus13152686912402

12057432= 075 120574

41= minus28738466294648 120574

42= minus33778743470341

12057443= 45693596398302 120574 = 1068579021301629



1198871+ 1198872+ 1198873+ 1198874= 1 (52)

11988721205732+ 11988731205733+ 11988741205734=

1

2

minus 120574 (53)

1198873120573321205732+ 1198874(120573421205732+ 120573431205733) =

1

6

minus 120574 + 1205742 (54)

11988721205722

21+ 11988731205722

3+ 11988741205722

4=

1

3

(55)

11988721205723

21+ 11988731205723

3+ 11988741205723

4=

1

4

(56)

11988731205723120572321205732+ 11988741205724(120572421205732+ 120572431205733) =

1

8

minus

120574

3

(57)

1198873120573321205722

2+ 1198874(120573421205722

2+ 120573431205722

3) =

1

12

minus

120574

3

(58)

119887412057343120573321205732=

1

24

minus

120574

2

+

3

2

1205742minus 1205743 (59)


119895=1120573119894119895


412057343120573321205722

2=





2= 1205723= 1205724 (55)-(56) imply that 120572

21= 1205722= 34


1= 1127

Letting 12057231= 12057221and 120572




11988741205733=

1 minus 6120574 + 61205742

612057343

12057243=

(1 minus 81205743) 12057343

1 minus 6120574 + 61205742

(60)


12057241=

3

4

minus 12057242minus

(1 minus 81205743) 12057343

1 minus 6120574 + 61205742 (61)


1198874=

16 (minus21205744+ 41205743minus 512057423 + 1205746)

91205734312057332

12057331=

1 minus 6120574 + 61205742

6120573431198874

minus 12057332

(62)


2= 1627 minus

1198873minus 1198874and

12057342=

(4 minus 16120574 minus 27119887312057332)

(271198874)

minus 12057343

12057341=

(12 minus 120574 minus 1198873(12057331+ 12057332))

1198874

minus 12057342minus 12057343

(63)




0 119906119899

1 119906

119899

119872) is the


as

119880119899+1= 119880119899+ 11988711198701+ 11988721198702+ 11988731198703+ 11988741198704 (64)

(119860 minus 120574Δ119905119869119899)119870119894= 119865(119880

119899+

119894minus1

sum

119895=1

120572119894119895119870119895 119905119899+ 120590119894Δ119905)

+ 120583119894Δ1199052 120597119865

120597119905

(119880119899 119905119899) + Δ119905J

119899(

119894minus1

sum

119895=1

120574119894119895119870119895)

119894 = 1 2 3 4

(65)

where 119869119899= (120597119865120597119880)(119880


119894and 120583

119894are





44=



119877 (119911) = (1 + (1 minus 4120574) 119911 + (

1

2

minus 4120574 + 61205742) 1199112

minus(

51205742

4

minus

5120574

8

+

1

16

) 1199114)

times ((1 minus 120574119911)4)

minus1

(66)


stable we need



Apparently


|119877 (119911)| =

100381610038161003816100381610038161003816100381610038161003816

512057424 minus 51205748 + 116

1205744

100381610038161003816100381610038161003816100381610038161003816

= 06304149382 lt 1

(69)





0 1 2 3 4 5 6 7 8 9 10 11 12

0

1

2

3

4

5

6

7

8

x

y


minus7

minus6

minus5

minus4

minus3

minus2

minus1

Stable

Stable

Stable Unstable





(119909 119905) isin (0 2) times (0 1]

119906 (119909 0) = cos (119909) 119909 isin [0 2]


(70)
















1and





1







119906 (119909 0) = cos (119909) 119909 isin [0 2]

119906119909(0 119905) = 0 119906


(71)










119906 (119909 0) = cos (119909) 119909 isin [0 1]


(72)








Δ119905 110 120 140 180















5 Conclusion






Acknowledgment


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119899is known as 119910

119899 the 4-stage


as

119910119899+1= 119910119899+ 11988711198961+ 11988721198962+ 11988731198963+ 11988741198964 (31)

119896119894= Δ119905119891(119910

119899+

119894minus1

sum

119895=1

120572119894119895119896119895) + Δ119905119869 (119910

119899)

119894

sum

119895=1

120574119894119895119896119895

119894 = 1 2 3 4

(32)

where 119887119894 120572119894119895 and 120574


119869(119910119899) = 119891119910(119910119899)


119910119899+1= 119910119899+ 11988711198961+ 11988721198962+ 11988731198963+ 11988741198964

119896119894= Δ119905119891(119910

119899+

119894minus1

sum

119895=1

120572119894119895119896119895 119905119899+ 120572119894Δ119905)

+ 120574119894Δ1199052119891119905(119910119899 119905119899) + Δ119905119891

119910(119910119899 119905119899)

119894

sum

119895=1

120574119894119895119896119895

119894 = 1 2 3 4

(33)

where

120572119894=

119894minus1

sum

119895=1

120572119894119895 120574

119894=

119894

sum

119895=1

120574119894119895 (34)


119899+ Δ119905) minus 119910

119899+1= 119874(Δ119905



119899+Δ119905) and 119896

119894


119910119899+1


Given 119910(119905119899) = 119910


119899+ Δ119905) at 119905

119899is

expanded as

119910 (119905119899+ Δ119905) = 119910

119899+ Δ119905119891119899+

Δ1199052

2

119869119899119891119899+

Δ1199053

6

(1198692

119899119891119899+ 1198691015840

1198991198912

119899)

+

Δ1199054

24

(11986910158401015840

1198991198913

119899+ 21198691015840

1198991198691198991198912

119899+ 21198691198991198691015840

1198991198912

119899+ 1198693

119899119891119899)

+ 119874 (Δ1199055)

(35)

where 119891119899= 119891(119910

119899) 119869119899= (120597119891120597119910)(119910

119899) 1198691015840119899= (12059721198911205971199102)(119910119899) and

11986910158401015840


119894119895= 120572119894119895+120574119894119895

and 120572119894119894= 0


1198961= Δ119905 (1 minus Δ119905120574

11119869119899)minus1

119891119899

= Δ119905119891119899+ Δ119905212057411119869119899119891119899+ Δ11990531205742

111198692

119899119891119899

+ Δ11990541205743

111198693

119899119891119899+ 119874 (Δ119905

5)

(36)


1198962= Δ119905 (1 minus Δ119905120574

22119869119899)minus1

(119891 (119910119899+ 120572211198961) + 120574211198691198991198961) (37)


obtain

1198962= Δ119905119891

119899+ Δ1199052(12057221+ 12057421+ 12057422) 119869119899119891119899

+ Δ1199053[(12057411+ 12057422) (12057221+ 12057421) + 1205742

22] 1198692

119899119891119899+

Δ1199053

2

1205722

211198691015840

1198991198912

119899

+ Δ1199054[(1205741112057422+ 1205742

11+ 1205742

22) (12057221+ 12057421) + 1205743

22] 1198693

119899119891119899

+ Δ1199054120574111205722

211198691015840

1198991198691198991198912

119899+

Δ1199054

2

120574221205722

211198691198991198691015840

1198991198912

119899

+

Δ1199054

6

1205723

2111986910158401015840

1198991198913

119899+ 119874 (Δ119905

5)

(38)


1198963= Δ119905119891

119899+ Δ1199052(12057331+ 12057332+ 12057433) 119869119899119891119899

+ Δ1199053[

[

3

sum

119895=1

((1205723119895+ 1205743119895) sdot

119895

sum

119897=1

(120572119895119897+ 120574119895119897))]

]

1198692

119899119891119899

+

Δ1199053

2

1205722

31198691015840

1198991198912

119899

+ Δ1199054[

[

3

sum

119894=1

((1205723119894+ 1205743119894) sdot

119894

sum

119895=1

((120572119894119895+ 120574119894119895)

sdot

119895

sum

119897=1

(120572119895119897+ 120574119895119897)))]

]

1198693

119899119891119899

+

Δ1199054

2

[1205722

212057332+ 120574331205722

3] 1198691198991198691015840

1198991198912

119899

+ Δ1199054[1205723((12057321+ 12057422) 12057232+ 1205741112057231)] 1198691015840

1198991198691198991198912

119899

+

Δ1199054

6

(12057231+ 12057232)3

11986910158401015840

1198991198913

119899+ 119874 (Δ119905

5)


1198964= Δ119905119891

119899+ Δ1199052(12057341+ 12057342+ 12057342+ 12057444) 119869119899119891119899

+ Δ1199053[

[

4

sum

119895=1

((1205724119895+ 1205744119895) sdot

119895

sum

119897=1

(120572119895119897+ 120574119895119897))]

]

1198692

119899119891119899

+

Δ1199053

2

1205722

41198691015840

1198991198912

119899

+ Δ1199054[

[

4

sum

119894=1

((1205724119894+ 1205744119894) sdot

119894

sum

119895=1

((120572119894119895+ 120574119894119895)

sdot

119895

sum

119897=1

(120572119895119897+ 120574119895119897)))]

]

1198693

119899119891119899

+ Δ1199054[1205724((12057433+ 12057332+ 12057331) 12057243

+ (12057422+ 12057321) 12057242+ 1205741112057241)] 1198691015840

1198991198691198991198912

119899

+

Δ1199054

2

[120574441205722

4+ 120573431205722

3+ 120573421205722

2] 1198691198991198691015840

1198991198912

119899

+

Δ1199054

6

(12057241+ 12057242+ 12057243)3

11986910158401015840

1198991198913

119899+ 119874 (Δ119905

5)

(39)


1 =

4

sum

119894=1

119887119894 (40)

1

2

=

4

sum

119894=1

119887119894sdot (120574119894119894+

119894minus1

sum

119895=1

120573119894119895) (41)

1

6

=

4

sum

119894=1

119887119894sdot (

119894

sum

119895=1

120573119894119895sdot

119895

sum

119897=1

120573119895119897) (42)

1

3

=

4

sum

119894=1

119887119894sdot (

119894minus1

sum

119895=1

120572119894119895)

2

(43)

1

4

=

4

sum

119894=1

119887119894sdot (

119894minus1

sum

119895=1

120572119894119895)

3

(44)

1

24

=

4

sum

119894=1

119887119894sdot (120572119894

119894minus1

sum

119895=1

120572119894119895sdot (120574119895119895+

119895minus1

sum

119897=1

120573119895119897)) (45)

1

12

=

4

sum

119894=1

119887119894sdot (

119894

sum

119895=1

120573119894119895sdot (

119895minus1

sum

119897=1

120572119895119897)

2

) (46)

1

24

=

4

sum

119894=1

119887119894sdot (

119894

sum

119895=1

(120573119894119895sdot

119895

sum

119897=1

(120573119895119897sdot

119897

sum

119896=1

120573119897119896))) (47)


119894119894= 120574


1198991198691015840

119899= 1198691015840

119899119869119899 thus conditions (45)



119894119894= 120574 for





119887412057343120573321205722

2= minus2120574

4+ 41205743minus

5

3

1205742+

120574

6

(48)

119887412057343120573321205722

2= minus

81205744

3

+ 51205743minus 21205742+

7120574

36

(49)

which imply

1205743minus

3

2

1205742+

120574

2

minus

1

24

= 0 (50)



119899+1=

119910119899+sum119904

119895=1





119860 =(

1 1 1 1

12057311205732

1205733

1205734

1205732

1

2

sum

119894=1

1205732119894sdot 120573119894

3

sum

119894=1

1205733119894sdot 120573119894

4

sum

119894=1

1205734119894sdot 120573119894

0 1205722

21205722

31205722

4

) (51)

and 120573119894= sum119894

119895=1120573119894119895


2

31205722



119894119894=




1198871= 04074074074074 119887

2= minus02568608534470 119887

3= 02

1198874= 06494534460396 120572

21= 075 120572

31= 075

12057232= 0 120572

41= 29193596398302 120572

42= 04

12057243= minus25693596398302 120574

21= minus075 120574

31= minus13152686912402

12057432= 075 120574

41= minus28738466294648 120574

42= minus33778743470341

12057443= 45693596398302 120574 = 1068579021301629



1198871+ 1198872+ 1198873+ 1198874= 1 (52)

11988721205732+ 11988731205733+ 11988741205734=

1

2

minus 120574 (53)

1198873120573321205732+ 1198874(120573421205732+ 120573431205733) =

1

6

minus 120574 + 1205742 (54)

11988721205722

21+ 11988731205722

3+ 11988741205722

4=

1

3

(55)

11988721205723

21+ 11988731205723

3+ 11988741205723

4=

1

4

(56)

11988731205723120572321205732+ 11988741205724(120572421205732+ 120572431205733) =

1

8

minus

120574

3

(57)

1198873120573321205722

2+ 1198874(120573421205722

2+ 120573431205722

3) =

1

12

minus

120574

3

(58)

119887412057343120573321205732=

1

24

minus

120574

2

+

3

2

1205742minus 1205743 (59)


119895=1120573119894119895


412057343120573321205722

2=





2= 1205723= 1205724 (55)-(56) imply that 120572

21= 1205722= 34


1= 1127

Letting 12057231= 12057221and 120572




11988741205733=

1 minus 6120574 + 61205742

612057343

12057243=

(1 minus 81205743) 12057343

1 minus 6120574 + 61205742

(60)


12057241=

3

4

minus 12057242minus

(1 minus 81205743) 12057343

1 minus 6120574 + 61205742 (61)


1198874=

16 (minus21205744+ 41205743minus 512057423 + 1205746)

91205734312057332

12057331=

1 minus 6120574 + 61205742

6120573431198874

minus 12057332

(62)


2= 1627 minus

1198873minus 1198874and

12057342=

(4 minus 16120574 minus 27119887312057332)

(271198874)

minus 12057343

12057341=

(12 minus 120574 minus 1198873(12057331+ 12057332))

1198874

minus 12057342minus 12057343

(63)




0 119906119899

1 119906

119899

119872) is the


as

119880119899+1= 119880119899+ 11988711198701+ 11988721198702+ 11988731198703+ 11988741198704 (64)

(119860 minus 120574Δ119905119869119899)119870119894= 119865(119880

119899+

119894minus1

sum

119895=1

120572119894119895119870119895 119905119899+ 120590119894Δ119905)

+ 120583119894Δ1199052 120597119865

120597119905

(119880119899 119905119899) + Δ119905J

119899(

119894minus1

sum

119895=1

120574119894119895119870119895)

119894 = 1 2 3 4

(65)

where 119869119899= (120597119865120597119880)(119880


119894and 120583

119894are





44=



119877 (119911) = (1 + (1 minus 4120574) 119911 + (

1

2

minus 4120574 + 61205742) 1199112

minus(

51205742

4

minus

5120574

8

+

1

16

) 1199114)

times ((1 minus 120574119911)4)

minus1

(66)


stable we need



Apparently


|119877 (119911)| =

100381610038161003816100381610038161003816100381610038161003816

512057424 minus 51205748 + 116

1205744

100381610038161003816100381610038161003816100381610038161003816

= 06304149382 lt 1

(69)





0 1 2 3 4 5 6 7 8 9 10 11 12

0

1

2

3

4

5

6

7

8

x

y


minus7

minus6

minus5

minus4

minus3

minus2

minus1

Stable

Stable

Stable Unstable





(119909 119905) isin (0 2) times (0 1]

119906 (119909 0) = cos (119909) 119909 isin [0 2]


(70)
















1and





1







119906 (119909 0) = cos (119909) 119909 isin [0 2]

119906119909(0 119905) = 0 119906


(71)










119906 (119909 0) = cos (119909) 119909 isin [0 1]


(72)








Δ119905 110 120 140 180















5 Conclusion






Acknowledgment


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1198964= Δ119905119891

119899+ Δ1199052(12057341+ 12057342+ 12057342+ 12057444) 119869119899119891119899

+ Δ1199053[

[

4

sum

119895=1

((1205724119895+ 1205744119895) sdot

119895

sum

119897=1

(120572119895119897+ 120574119895119897))]

]

1198692

119899119891119899

+

Δ1199053

2

1205722

41198691015840

1198991198912

119899

+ Δ1199054[

[

4

sum

119894=1

((1205724119894+ 1205744119894) sdot

119894

sum

119895=1

((120572119894119895+ 120574119894119895)

sdot

119895

sum

119897=1

(120572119895119897+ 120574119895119897)))]

]

1198693

119899119891119899

+ Δ1199054[1205724((12057433+ 12057332+ 12057331) 12057243

+ (12057422+ 12057321) 12057242+ 1205741112057241)] 1198691015840

1198991198691198991198912

119899

+

Δ1199054

2

[120574441205722

4+ 120573431205722

3+ 120573421205722

2] 1198691198991198691015840

1198991198912

119899

+

Δ1199054

6

(12057241+ 12057242+ 12057243)3

11986910158401015840

1198991198913

119899+ 119874 (Δ119905

5)

(39)


1 =

4

sum

119894=1

119887119894 (40)

1

2

=

4

sum

119894=1

119887119894sdot (120574119894119894+

119894minus1

sum

119895=1

120573119894119895) (41)

1

6

=

4

sum

119894=1

119887119894sdot (

119894

sum

119895=1

120573119894119895sdot

119895

sum

119897=1

120573119895119897) (42)

1

3

=

4

sum

119894=1

119887119894sdot (

119894minus1

sum

119895=1

120572119894119895)

2

(43)

1

4

=

4

sum

119894=1

119887119894sdot (

119894minus1

sum

119895=1

120572119894119895)

3

(44)

1

24

=

4

sum

119894=1

119887119894sdot (120572119894

119894minus1

sum

119895=1

120572119894119895sdot (120574119895119895+

119895minus1

sum

119897=1

120573119895119897)) (45)

1

12

=

4

sum

119894=1

119887119894sdot (

119894

sum

119895=1

120573119894119895sdot (

119895minus1

sum

119897=1

120572119895119897)

2

) (46)

1

24

=

4

sum

119894=1

119887119894sdot (

119894

sum

119895=1

(120573119894119895sdot

119895

sum

119897=1

(120573119895119897sdot

119897

sum

119896=1

120573119897119896))) (47)


119894119894= 120574


1198991198691015840

119899= 1198691015840

119899119869119899 thus conditions (45)



119894119894= 120574 for





119887412057343120573321205722

2= minus2120574

4+ 41205743minus

5

3

1205742+

120574

6

(48)

119887412057343120573321205722

2= minus

81205744

3

+ 51205743minus 21205742+

7120574

36

(49)

which imply

1205743minus

3

2

1205742+

120574

2

minus

1

24

= 0 (50)



119899+1=

119910119899+sum119904

119895=1





119860 =(

1 1 1 1

12057311205732

1205733

1205734

1205732

1

2

sum

119894=1

1205732119894sdot 120573119894

3

sum

119894=1

1205733119894sdot 120573119894

4

sum

119894=1

1205734119894sdot 120573119894

0 1205722

21205722

31205722

4

) (51)

and 120573119894= sum119894

119895=1120573119894119895


2

31205722



119894119894=




1198871= 04074074074074 119887

2= minus02568608534470 119887

3= 02

1198874= 06494534460396 120572

21= 075 120572

31= 075

12057232= 0 120572

41= 29193596398302 120572

42= 04

12057243= minus25693596398302 120574

21= minus075 120574

31= minus13152686912402

12057432= 075 120574

41= minus28738466294648 120574

42= minus33778743470341

12057443= 45693596398302 120574 = 1068579021301629



1198871+ 1198872+ 1198873+ 1198874= 1 (52)

11988721205732+ 11988731205733+ 11988741205734=

1

2

minus 120574 (53)

1198873120573321205732+ 1198874(120573421205732+ 120573431205733) =

1

6

minus 120574 + 1205742 (54)

11988721205722

21+ 11988731205722

3+ 11988741205722

4=

1

3

(55)

11988721205723

21+ 11988731205723

3+ 11988741205723

4=

1

4

(56)

11988731205723120572321205732+ 11988741205724(120572421205732+ 120572431205733) =

1

8

minus

120574

3

(57)

1198873120573321205722

2+ 1198874(120573421205722

2+ 120573431205722

3) =

1

12

minus

120574

3

(58)

119887412057343120573321205732=

1

24

minus

120574

2

+

3

2

1205742minus 1205743 (59)


119895=1120573119894119895


412057343120573321205722

2=





2= 1205723= 1205724 (55)-(56) imply that 120572

21= 1205722= 34


1= 1127

Letting 12057231= 12057221and 120572




11988741205733=

1 minus 6120574 + 61205742

612057343

12057243=

(1 minus 81205743) 12057343

1 minus 6120574 + 61205742

(60)


12057241=

3

4

minus 12057242minus

(1 minus 81205743) 12057343

1 minus 6120574 + 61205742 (61)


1198874=

16 (minus21205744+ 41205743minus 512057423 + 1205746)

91205734312057332

12057331=

1 minus 6120574 + 61205742

6120573431198874

minus 12057332

(62)


2= 1627 minus

1198873minus 1198874and

12057342=

(4 minus 16120574 minus 27119887312057332)

(271198874)

minus 12057343

12057341=

(12 minus 120574 minus 1198873(12057331+ 12057332))

1198874

minus 12057342minus 12057343

(63)




0 119906119899

1 119906

119899

119872) is the


as

119880119899+1= 119880119899+ 11988711198701+ 11988721198702+ 11988731198703+ 11988741198704 (64)

(119860 minus 120574Δ119905119869119899)119870119894= 119865(119880

119899+

119894minus1

sum

119895=1

120572119894119895119870119895 119905119899+ 120590119894Δ119905)

+ 120583119894Δ1199052 120597119865

120597119905

(119880119899 119905119899) + Δ119905J

119899(

119894minus1

sum

119895=1

120574119894119895119870119895)

119894 = 1 2 3 4

(65)

where 119869119899= (120597119865120597119880)(119880


119894and 120583

119894are





44=



119877 (119911) = (1 + (1 minus 4120574) 119911 + (

1

2

minus 4120574 + 61205742) 1199112

minus(

51205742

4

minus

5120574

8

+

1

16

) 1199114)

times ((1 minus 120574119911)4)

minus1

(66)


stable we need



Apparently


|119877 (119911)| =

100381610038161003816100381610038161003816100381610038161003816

512057424 minus 51205748 + 116

1205744

100381610038161003816100381610038161003816100381610038161003816

= 06304149382 lt 1

(69)





0 1 2 3 4 5 6 7 8 9 10 11 12

0

1

2

3

4

5

6

7

8

x

y


minus7

minus6

minus5

minus4

minus3

minus2

minus1

Stable

Stable

Stable Unstable





(119909 119905) isin (0 2) times (0 1]

119906 (119909 0) = cos (119909) 119909 isin [0 2]


(70)
















1and





1







119906 (119909 0) = cos (119909) 119909 isin [0 2]

119906119909(0 119905) = 0 119906


(71)










119906 (119909 0) = cos (119909) 119909 isin [0 1]


(72)








Δ119905 110 120 140 180















5 Conclusion






Acknowledgment


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198871= 04074074074074 119887

2= minus02568608534470 119887

3= 02

1198874= 06494534460396 120572

21= 075 120572

31= 075

12057232= 0 120572

41= 29193596398302 120572

42= 04

12057243= minus25693596398302 120574

21= minus075 120574

31= minus13152686912402

12057432= 075 120574

41= minus28738466294648 120574

42= minus33778743470341

12057443= 45693596398302 120574 = 1068579021301629



1198871+ 1198872+ 1198873+ 1198874= 1 (52)

11988721205732+ 11988731205733+ 11988741205734=

1

2

minus 120574 (53)

1198873120573321205732+ 1198874(120573421205732+ 120573431205733) =

1

6

minus 120574 + 1205742 (54)

11988721205722

21+ 11988731205722

3+ 11988741205722

4=

1

3

(55)

11988721205723

21+ 11988731205723

3+ 11988741205723

4=

1

4

(56)

11988731205723120572321205732+ 11988741205724(120572421205732+ 120572431205733) =

1

8

minus

120574

3

(57)

1198873120573321205722

2+ 1198874(120573421205722

2+ 120573431205722

3) =

1

12

minus

120574

3

(58)

119887412057343120573321205732=

1

24

minus

120574

2

+

3

2

1205742minus 1205743 (59)


119895=1120573119894119895


412057343120573321205722

2=





2= 1205723= 1205724 (55)-(56) imply that 120572

21= 1205722= 34


1= 1127

Letting 12057231= 12057221and 120572




11988741205733=

1 minus 6120574 + 61205742

612057343

12057243=

(1 minus 81205743) 12057343

1 minus 6120574 + 61205742

(60)


12057241=

3

4

minus 12057242minus

(1 minus 81205743) 12057343

1 minus 6120574 + 61205742 (61)


1198874=

16 (minus21205744+ 41205743minus 512057423 + 1205746)

91205734312057332

12057331=

1 minus 6120574 + 61205742

6120573431198874

minus 12057332

(62)


2= 1627 minus

1198873minus 1198874and

12057342=

(4 minus 16120574 minus 27119887312057332)

(271198874)

minus 12057343

12057341=

(12 minus 120574 minus 1198873(12057331+ 12057332))

1198874

minus 12057342minus 12057343

(63)




0 119906119899

1 119906

119899

119872) is the


as

119880119899+1= 119880119899+ 11988711198701+ 11988721198702+ 11988731198703+ 11988741198704 (64)

(119860 minus 120574Δ119905119869119899)119870119894= 119865(119880

119899+

119894minus1

sum

119895=1

120572119894119895119870119895 119905119899+ 120590119894Δ119905)

+ 120583119894Δ1199052 120597119865

120597119905

(119880119899 119905119899) + Δ119905J

119899(

119894minus1

sum

119895=1

120574119894119895119870119895)

119894 = 1 2 3 4

(65)

where 119869119899= (120597119865120597119880)(119880


119894and 120583

119894are





44=



119877 (119911) = (1 + (1 minus 4120574) 119911 + (

1

2

minus 4120574 + 61205742) 1199112

minus(

51205742

4

minus

5120574

8

+

1

16

) 1199114)

times ((1 minus 120574119911)4)

minus1

(66)


stable we need



Apparently


|119877 (119911)| =

100381610038161003816100381610038161003816100381610038161003816

512057424 minus 51205748 + 116

1205744

100381610038161003816100381610038161003816100381610038161003816

= 06304149382 lt 1

(69)





0 1 2 3 4 5 6 7 8 9 10 11 12

0

1

2

3

4

5

6

7

8

x

y


minus7

minus6

minus5

minus4

minus3

minus2

minus1

Stable

Stable

Stable Unstable





(119909 119905) isin (0 2) times (0 1]

119906 (119909 0) = cos (119909) 119909 isin [0 2]


(70)
















1and





1







119906 (119909 0) = cos (119909) 119909 isin [0 2]

119906119909(0 119905) = 0 119906


(71)










119906 (119909 0) = cos (119909) 119909 isin [0 1]


(72)








Δ119905 110 120 140 180















5 Conclusion






Acknowledgment


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












44=



119877 (119911) = (1 + (1 minus 4120574) 119911 + (

1

2

minus 4120574 + 61205742) 1199112

minus(

51205742

4

minus

5120574

8

+

1

16

) 1199114)

times ((1 minus 120574119911)4)

minus1

(66)


stable we need



Apparently


|119877 (119911)| =

100381610038161003816100381610038161003816100381610038161003816

512057424 minus 51205748 + 116

1205744

100381610038161003816100381610038161003816100381610038161003816

= 06304149382 lt 1

(69)





0 1 2 3 4 5 6 7 8 9 10 11 12

0

1

2

3

4

5

6

7

8

x

y


minus7

minus6

minus5

minus4

minus3

minus2

minus1

Stable

Stable

Stable Unstable





(119909 119905) isin (0 2) times (0 1]

119906 (119909 0) = cos (119909) 119909 isin [0 2]


(70)
















1and





1







119906 (119909 0) = cos (119909) 119909 isin [0 2]

119906119909(0 119905) = 0 119906


(71)










119906 (119909 0) = cos (119909) 119909 isin [0 1]


(72)








Δ119905 110 120 140 180















5 Conclusion






Acknowledgment


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















1and





1







119906 (119909 0) = cos (119909) 119909 isin [0 2]

119906119909(0 119905) = 0 119906


(71)










119906 (119909 0) = cos (119909) 119909 isin [0 1]


(72)








Δ119905 110 120 140 180















5 Conclusion






Acknowledgment


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Δ119905 110 120 140 180















5 Conclusion






Acknowledgment


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Acknowledgment


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article A Strongly A-Stable Time Integration Method for … · 2019. 7. 31. · erential equation supplemented with Dirichlet or Neumann boundary condition. We successfully