Research Article The Exponential Cubic B-Spline …downloads.hindawi.com/archive/2015/367056.pdfe exponential cubic B-spline algorithm is presented to nd the numerical solutions of

Research ArticleThe Exponential Cubic B-Spline Algorithm forKorteweg-de Vries Equation

Ozlem Ersoy and Idris Dag

Department of Mathematics-Computer Faculty of Science and Art Eskisehir Osmangazi University 26480 Eskisehir Turkey

Correspondence should be addressed to Ozlem Ersoy ozersoyoguedutr

Received 16 September 2014 Revised 5 January 2015 Accepted 18 January 2015

Academic Editor Yinnian He

Copyright copy 2015 O Ersoy and I DagThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The exponential cubic B-spline algorithm is presented to find the numerical solutions of the Korteweg-de Vries (KdV) equationThe problem is reduced to a system of algebraic equations which is solved by using a variant of Thomas algorithm Numericalexperiments are carried out to demonstrate the efficiency of the suggested algorithm

1 Introduction

The splines consist of piecewise functions defined on thedistributed knots on problem domain and have certaincontinuity inside problem subdomain and at the knotsUntil now some types of splines have been developed andespecially polynomial splines The exponential splines aredefined as more general splines by McCartin [1ndash3] Thebasis of the exponential splines known as the exponential B-splines is also given in the studies of McCartin Existenceof the free parameter in the exponential B-splines yields thedifferent shapes of the splines functions He has also showeda reliable algorithm by using the exponential spline functionsto solve the hyperbolic conservation laws McCartin andJameson [4]HoweverMcCartin stated that application of theexponential splineexponential B-spline functions has beenneglected in the numerical analysis So use of the exponentialspline in the numerical methods for finding solutions of thedifferential equations is not common and few papers existin the literature McCartin has shown that the exponentialsplines admit a basis known as the exponential B-splinesThese B-splines have been started using to form approximatesfunctions recently which are adapted to set up the numer-ical methods to find solutions of the differential equationsrecently An application of the simple exponential splines isconsidered for setting up the collocation method to solvethe numerical solution of singular perturbation problem [5]

Cardinal exponential B-splines are applied in solving sin-gularly perturbed boundary problems [6] A variant of B-spline exponential collocation method was also built up forcomputing numerical solutions of the singularly perturbedboundary value problem [7] Very recently the exponentialB-spline collocation method has been applied to obtain thesolutions of the one-dimensional linear convection-diffusionequation [8]

Types of spline functions are utilized to form approximatesolutions for Korteweg-de Vries equation (KdVE) The stan-dard Galerkin formulation using the smooth splines on uni-formmesh is set up for 1-periodic solutions of KdVE by Bakerand his coauthors [9] The Galerkin finite element methodtogether with the cubic B-splines is used to solve the KdVEin the paper [10] The quadratic B-spline Galerkin methodis built up to find solutions of the KdVE [11] A collocationsolution of the Korteweg-de Vries equation using septic B-splines is proposed by Soliman [12] A variant of the Galerkinfinite element method is designed for solving the KdVE byAksan and Ozdes [13] The collocation method using quinticB-splines is developed to solve the KdVE [14] A numericalmethod is developed for the KdVE by using splitting finitedifference technique and quintic B-spline functions [15] Thespline finite element method using quadratic polynomialspline for the numerical solution of the KdVE is given by GMicula andMMicula [16] A cubic B-spline Taylor-Galerkinmethod is developed to find numerical solution of the KdVE

Hindawi Publishing CorporationAdvances in Numerical AnalysisVolume 2015 Article ID 367056 8 pageshttpdxdoiorg1011552015367056

2 Advances in Numerical Analysis

Table 1 Values of 119861119894(119909) and its first and second derivatives at the knot points

119909 119909119894minus2

119909119894minus1

119909119894

119909119894+1

119909119894+2

119861119894

0119904 minus 119901ℎ

2 (119901ℎ119888 minus 119904)1

119904 minus 119901ℎ

2 (119901ℎ119888 minus 119904)0

1198611015840

1198940

119901 (1 minus 119888)

2 (119901ℎ119888 minus 119904)0

119901 (119888 minus 1)

2 (119901ℎ119888 minus 119904)0

11986110158401015840

1198940

1199012119904

2 (119901ℎ119888 minus 119904)minus1199012119904

119901ℎ119888 minus 119904

1199012119904

2 (119901ℎ119888 minus 119904)0

by Canıvar et al in [17] A study based on cubic B-spline finiteelement method for the solution of the KdVE is suggested byKapoor et al [18] A Bubnov-Galerkin finite element methodwith quintic B-spline functions taken as element shape andweight functions is presented for the solution of the KdVE[19]The paper deals with the numerical solution of the KdVEusing quartic B-splines Galerkin method as both shape andweight functions over the finite intervals [20] A blendedspline quasi-interpolation scheme is employed to solve theone-dimensional nonlinear KdVE [21] A multilevel quarticspline quasi-interpolation scheme is fulfilled to exhibit a largenumber of physical phenomena for KdVE [22]

The aim of the present paper is to develop an approximatesolution of KdVE by collocation method In Section 2 theexponential B-spline collocation algorithm is defined for theKdVE In Section 3 the three numerical experiments areconstructed to demonstrate the efficiency of the proposedmethod and the results are documented in tables and graphsare depicted

We will solve the KdVE

119880119905+ 120576119880119880

119909+ 120583119880119909119909119909= 0 119886 le 119909 le 119887 (1)

where 120576 120583 are positive parameters and the subscripts 119909 and119905 denote differentiation The boundary conditions will bechosen as

119880 (119886 119905) = 119880 (119887 119905) = 0

119880119909 (119886 119905) = 119880119909 (119887 119905) = 0

(2)

KdVE is prototypical example of exactly solvable math-ematical model of waves on shallow water surface It arisesfor evolution interaction of waves and generation in physicsDue to the term 119880

119905 (1) is called the evolution equation the

nonlinear term causes the steepness of the wave and thedispersive term defines the spreading of the wave It is knownthat the effect of the steepness and spreading results in solitonsolutions for the KdVE

2 Exponential B-Spline Collocation Method

The region [119886 119887] is partitioned into equal subintervals bypoints 119909

119894 119894 = 0 119873 On these points together with

additional points 119909119894 119894 = minus3 minus2 minus1119873+1119873+2119873+3 outside

the domain the exponential B-splines 119861119894(119909) can be defined

as

119861119894 (119909)

=

1198872((119909119894minus2minus 119909) minus

1

119901(sinh (119901 (119909

119894minus2minus 119909)))) [119909

119894minus2 119909119894minus1]

1198861+ 1198871(119909119894minus 119909) + 119888

1exp (119901 (119909

119894minus 119909))

+1198891exp (minus119901 (119909

119894minus 119909)) [119909

119894minus1 119909119894]

1198861+ 1198871(119909 minus 119909

119894) + 1198881exp (119901 (119909 minus 119909

119894))

+1198891exp (minus119901 (119909 minus 119909

119894)) [119909

119894 119909119894+1]

1198872((119909 minus 119909

119894+2) minus

1

119901(sinh (119901 (119909 minus 119909

119894+2)))) [119909

119894+1 119909119894+2]

0 otherwise(3)

where

1198861=

119901ℎ119888

119901ℎ119888 minus 119904 119887

1=119901

2[119888 (119888 minus 1) + 119904

2

(119901ℎ119888 minus 119904) (1 minus 119888)]

1198872=

119901

2 (119901ℎ119888 minus 119904)

1198881=1

4[exp (minus119901ℎ) (1 minus 119888) + 119904 (exp (minus119901ℎ) minus 1)

(119901ℎ119888 minus 119904) (1 minus 119888)]

1198891=1

4[exp (119901ℎ) (119888 minus 1) + 119904 (exp (119901ℎ) minus 1)

(119901ℎ119888 minus 119904) (1 minus 119888)]

119904 = sinh (119901ℎ) 119888 = cosh (119901ℎ)

119901 is a free parameter ℎ =119887 minus 119886

119873

(4)

119861minus1(119909) 119861

0(119909) 119861

119873+1(119909) forms a basis for the expo-

nential spline space on the interval [119886 119887] On the fourconsecutive subintervals an exponential B-spline 119861

119894(119909) is

defined and it is second-order continuously differentiablefunctions

In Table 1 the values of 119861119894(119909) 1198611015840

119894(119909) and 11986110158401015840

119894(119909) at the

points 119909119894 which can be obtained from (3) are listed where

1015840 denotes differentiation with respect to space variable 119909

Advances in Numerical Analysis 3

The global approximation119880119873(119909 119905) to the solution119880(119909 119905)

will be searched in terms of the unknown parameters 120575119894

and exponential B-spline function defined on the problemdomain

119880119873 (119909 119905) =

119873+1

sum

119894=minus1

120575119894119861119894 (119909) (5)

Substitution of the points 119909119894in (5) in its first and its second

derivatives respectively yields the numerical solution interms of parameters

119880119894= 119880 (119909

119894 119905) = 119898

1120575119894minus1+ 120575119894+ 1198981120575119894+1

1198801015840

119894= 1198801015840(119909119894 119905) = 119898

2120575119894minus1minus 1198982120575119894+1

11988010158401015840

119894= 11988010158401015840(119909119894 119905) = 119898

3120575119894minus1minus 21198983120575119894+ 1198983120575119894+1

(6)

where 1198981= (119904 minus 119901ℎ)2(119901ℎ119888 minus 119904) 119898

2= 119901(1 minus 119888)2(119901ℎ119888 minus 119904)

and1198983= 11990121199042(119901ℎ119888 minus 119904)

Over the subregion [119909119894 119909119894+1] the local approximation is

given by

119880119890

119873(119909 119905) = 120575119894minus1119861119894minus1 (119909) + 120575119894119861119894 (119909)

+ 120575119894+1119861119894+1 (119909) + 120575119894+2119861119894+2 (119909)

(7)

where 120575119894minus1

120575119894 and 120575

119894+1act as subregion parameters and 119861

119894minus1

119861119894 and 119861

119894+1are known as the subregion shape parameters

To be able to apply the collocation method formed withthe exponential B-splines KdV equation is space-splitted as

119880119905+ 120576119880119880

119909+ 120583119881119909119909= 0

119880119909minus 119881 = 0

(8)

This system includes the second-order derivatives so thatsmooth approximation can be done with the exponential B-splines To integrate system (8) in time discretize 119880

119905by the

usual finite difference scheme and 119880 119880119909 and 119881

119909119909by Crank-

Nicolson method and we get

119880119899+1minus 119880119899

Δ119905+ 120576(119880119880119909)119899+1+ (119880119880

119909)119899

2+ 120583119881119899+1

119909119909+ 119881119899

119909119909

2= 0

119880119899+1

119909+ 119880119899

119909

2minus119881119899+1+ 119881119899

2= 0

(9)

where 119880119899+1 = 119880(119909 (119899 + 1)Δ119905) represent the solution at the(119899 + 1)th time level Here 119905119899+1 = 119905119899 + Δ119905 and Δ119905 is the timestep superscripts denote 119899th time level 119905119899 = 119899Δ119905

One linearizes terms (119880119880119909)119899+1 and (119880119880119909)119899 in (9) as

(119880119880119909)119899+1= 119880119899+1119880119899

119909+ 119880119899119880119899+1

119909minus 119880119899119880119899

119909

(119880119880119909)119899= 119880119899119880119899

119909

(10)

to obtain the time-integrated KdVE

119880119899+1minus 119880119899+120576Δ119905

2(119880119899+1119880119899

119909+ 119880119899119880119899+1

119909)

minus120583Δ119905

2(119881119899+1

119909119909+ 119881119899

119909119909) = 0

119880119899+1

119909+ 119880119899

119909

2minus119881119899+1+ 119881119899

2= 0

(11)

We approximate 119880119899 and 119881119899 in terms of the element parame-ters and exponential B-splines separately as

119880119873 (119909 119905) =

119873+1

sum

119894=minus1

120575119894119861119894 (119909) 119881

119873 (119909 119905) =

119873+1

sum

119894=minus1

120601119894119861119894 (119909) (12)

Putting the approximate solution (12) and its derivativesinto (11) and evaluating the resulting equations at the points119909119894 119894 = 0 119873 yield the following system of equations

1198981120575119899+1

119898minus1minus120583Δ119905

2120574120601119899+1

119898minus1+ 120575119899+1

119898

+ 120583Δ119905120574120601119899+1

119898+ 1198981120575119899+1

119898+1minus120583Δ119905

2120574120601119899+1

119898+1

= 1198981(1 minus

120576Δ119905

2119871) 120575119899

119898minus1minusΔ119905

2(120583120574 + 120576119898

1119870)120601119899

119898minus1

+ (1 minus120576Δ119905

2119871) 120575119899


2(minus2120583120574 + 120576119870) 120601

119899

119898minus1

+ 1198981(1 minus

120576Δ119905

2119871) 120575119899


2(120583120574 + 120576119898

1119870)120601119899

119898minus1

120573120575119899+1

119898minus1minus 1198981120601119899+1

119898minus1minus 120601119899+1

119898minus 120573120575119899+1

119898+1minus 1198981120601119899+1

119898+1

= minus120573120575119899

119898minus1+ 1198981120601119899

119898minus1+ 120601119899

119898+ 120573120575119899

119898+1+ 1198981120601119899

119898+1

119898 = 0 119873 119899 = 0 1

(13)

where

119870 = 1198981120575119894minus1+ 120575119894+ 1198981120575119894+1

119871 = 1198981120601119894minus1+ 120601119894+ 1198981120601119894+1

120573 =119901 (1 minus 119888)

2 (119901ℎ119888 minus 119904) 120574 =

1199012119904

2 (119901ℎ119888 minus 119904)

(14)

The system consists of 2119873 + 2 linear equation in 2119873 + 6unknown parameters d119899+1 = (120575119899+1

minus1 120601119899+1

minus1 120575119899+1

0 120601119899+1

0 120575

119899+1

119873+1

120601119899+1

119873+1) A unique solution of the system can be obtained by

imposing the boundary conditions 119880(119886 119905) = 0 119880(119887 119905) =0 119881(119886 119905) = 0 119881(119887 119905) = 0 to have the following the equations

1198981120575minus1+ 1205750+ 11989811205751= 0

1198981120601minus1+ 1206010+ 11989811206011= 0

1198981120575119873minus1

+ 120575119873+ 1198981120575119873+1

= 0

1198981120601119873minus1

+ 120601119873+ 1198981120601119873+1

= 0

(15)


Elimination of the parameters 120575minus1 120601minus1 120575119873+1

120601119873+1

using (15) from system (13) gives a solvable system of 2119873 +2 linear equation including 2119873 + 2 unknown parametersPlacing solution parameters in (12) when computed fromthe system via a variant of the Thomas algorithm gives theapproximate solution over the subregion [119886 119887] We needthe initial parameter vectors 119889

1= (120575minus1 1205750 120575

119873 120575119873+1)

1198892= (120601minus1 1206010 120601

119873 120601119873+1) to start the iteration process for

system of (13) To do that the following requirements help todetermine the initial parameters

(119880119873)119909(119886 0) = 0 = 1198982120575

0

minus1+ 11989821205750

1

(119880119873)119909(119909119894 0) = 119898

11205750

119894minus1+ 1205750

119894+ 11989811205750

119894+1= 119880 (119909

119894 0)

119894 = 0 119873

(119880119873)119909(119887 0) = 0 = 1198982120575

0

119873minus1+ 11989821205750

119873+1

(119881119873)119909(119886 0) = 0 = 1198982120601

0

minus1+ 11989821206010

1

(119881119873)119909(119909119894 0) = 119898

11206010

119894minus1+ 1206010

119894+ 11989811206010

119894+1= 119881 (119909

119894 0)

119894 = 0 119873

(119881119873)119909(119886 0) = 1198982120601

0

119873minus1+ 11989821206010

119873+1

(16)

3 Numerical Tests

Since the conservation laws remain constant at all timefirst three numerical conservations are calculated using therectangular rule for integrals

1198621= int

infin

minusinfin

119880119889119909 asymp int

119887

119886

119880119889119909 asympℎ

2

119873minus1

sum

119894=0

(119880119894+ 119880119894+1)

1198622= int

infin

minusinfin

(1198802) 119889119909 asymp int

119887

119886

(1198802) 119889119909 asymp

ℎ

2

119873minus1

sum

119894=0

[(1198802)119894+ (1198802)119894+1]

1198623= int

infin

minusinfin

(1198803minus3120583

1205761198802

119909)119889119909 asymp int

119887

119886

(1198803minus3120583

1205761198802

119909)119889119909

asympℎ

2

119873minus1

sum

119894=0

[(1198803)119894minus3120583

120576(1198802

119909)119894+ (1198803)119894+1minus3120583

120576(1198802

119909)119894+1]

(17)

The error norm 119871infin

119871infin=1003816100381610038161003816119880 minus 119880119873

1003816100381610038161003816infin= max119895

100381610038161003816100381610038161003816119880119895minus (119880119873)119899

119895

100381610038161003816100381610038161003816(18)

is calculated to show the error between analytical andnumerical solutions

(a)The soliton solution of the KdVE is

119880 (119909 119905) = 3119888 sec ℎ2 (119860119909 minus 119861119905 + 119863) (19)

where 119860 = (12)radic120576119888120583 and 119861 = 120576119888119860 This solutionrepresents propagation of single soliton having velocity 120576119888and amplitude 3119888

1

08

05 15 21

06

04

02

00

u

x

t = 0 t = 2 t = 25 t = 3t = 1 t = 15t = 05

Figure 1 A single soliton at some times

05 15 210

x

0016

0014

0012

001

0008

0006

0004

0002

0

|Error|

Figure 2 Error distribution

The analytical solution (19) is used as the initial conditionwhen 119905 = 0 The Dirichlet boundary conditions 119880(0 119905) =119881(0 119905) = 0 and119880(2 119905) = 119881(2 119905) = 0 are adapted to the systemto control numerical solutions at the boundaries Parameters120576 = 1 120583 = 484 times 10

minus4 119888 = 03 119863 = minus6 space stepℎ = 001 and time step Δ119905 = 0005 on the interval [0 2] fromtime 119905 = 0 to 119905 = 3 are chosen At time 119905 = 3 numericalmagnitude of the single soliton is calculated as 09 so thatthe numerical amplitude is obtained to be almost the same asthe analytical amplitude Figure 1 illustrates the amplitudes atsome times The distribution of the absolute values of errorscan be observed in Figure 2119871infin

error norms and invariants are presented at theselected times in Table 2 as seen from the table that 119871

2error

norm is found small enough and conservation invariants areexcellent throughout the simulation The method gives goodresults when the free parameter 119901 = 164 times 10

minus5 is usedInvariants 119862

2and 119862

3remain constant during the run and 119862

1


Table 2 Error norms and invariants for single soliton

Time 119871infintimes 103

119901 times 105

119871infintimes 103

119901 = 11198621

1198622

1198623

Method

00 0 0 0144597 0086759 0046849

Pre

05 0521 (119901 = 137) 051 0144602 0086759 0046849

10 0589 (119901 = 164) 075 0144593 0086759 0046849

15 0533 (119901 = 164) 103 0144592 0086759 0046849

20 0595 (119901 = 164) 126 0144591 0086759 0046849

25 0657 (119901 = 164) 150 0144597 0086759 0046849

30 0740 (119901 = 164) 161 0144597 0086759 0046849

30 008 0014460 008676 004685 [23]30 104 0014460 008675 004685 [24]30 004 0144597 0086761 00468524 [17]30 014 0144601 0086760 0046850 [25]

Table 3 Invariant for Maxwellian

120576 = 10 120583 = 004

119905 1198621

1198622

1198623

0 177245 125331 08729225 177243 125332 08741250 177248 125333 08743775 177170 125332 087442100 177203 125333 087442125 177367 125330 087442

remains the constant up to the third decimal digits seen inTable 2

(b) Wave generation is performed by using theMaxwellian initial condition

119880 (119909 0) = exp (minus1199092) (20)

and boundary conditions

119880 (minus15 119905) = 119880 (15 119905) = 0 119905 gt 0 (21)

ℎ = 01 and Δ119905 = 001 and 120576 = 10 are taken Wehave verified the case in which 120583

119888is some critical parameter

and according to the parameters 120583 ≪ 120583119888 initial condition

breaks up into a number of solitons and for values 120583 ≫

120583119888 soliton turns into exhibiting the rapidly oscillating wave

packets When 120583 asymp 120583119888together with parameters 120576 = 10

120583 = 004 ℎ = 01 and Δ119905 = 001 the solution takes theform of the leading soliton and an oscillating tail This caseis shown in Figure 3

For 120583 = 004 we observe a solitary wave plus anoscillating tail (Figure 3) The actual velocity of the wave 119881

119899

has been measured and also computed from the measuredamplitude using the formula119881

119886= 1198861205763 We find that119881

119899= 04

and 119881119886= (11978 times 1)3 = 039926 so the solitary waves are

indeed solitons In Table 3 invariant for Maxwellian 120576 = 10and 120583 = 004


120576 = 10 120583 = 001

119905 1198621

1198622

1198623

0 177245 125331 09857225 177244 125362 09927750 177250 125374 09955875 177243 125376 099576100 177242 125377 099582125 177222 125378 099582


120576 = 10 120583 = 0001

119905 1198621

1198622

1198623

0 177245 125331 10195625 177245 125356 10246150 177245 125365 10264475 177245 125365 102654100 177244 125365 102656125 177244 125365 102656

When 120583 = 001 we find three solitonsWe havemeasuredthe velocity of the largest solitary wave as 119881

119899= 052 and

calculated the expected velocity from the observed amplitude154468 as 119881

119886= (15535 times 1)3 = 051783 In Figure 4

Maxwellian initial condition is depicted for 119905 = 125 120576 = 10120583 = 001 ℎ = 01 and Δ119905 = 001 The invariants are given inTables 4 and 5 for 120576 = 10 120583 = 001 and 120576 = 10 120583 = 0001respectively

For 120583 = 0001 we observed nine solitons moving to theright in Figure 5 The measured velocity of leading soliton is119881119899= 062 and the corresponding velocities calculated from

their measured amplitudes are 119881119886= (17949 times 1)3 = 060

The agreement is good The initial perturbation breaks upinto a number of solitons in the course of time depending onthe value of 120583 chosen So if we decrease the value of 120583 thenthe number of solitons amplitude and the velocity increase


0 5 10 15

x

minus15 minus10 minus5

14

12

1

08

06

04

02

0

u

minus02

Figure 3 Soliton at 119905 = 125

0 5 10 15

x


14

16

12

1

08

06

04

02

0

u

minus02

Figure 4 Generated waves at 119905 = 125 120576 = 10 120583 = 001 ℎ = 01and Δ119905 = 001

(c) As a final test example initial condition

119880 (119909 0) =1

2[1 minus tanh(|119909| minus 25

5)] (22)

together with boundary conditions

119880 (minus50 119905) = 119880 (150 119905) = 0 119905 gt 0 (23)

cause the production of a train of solitons depending on thevalue of 120583 for the KdVE Computation is done on region[minus50 150] up to time 119905 = 800 with parameters 120576 = 02 120583 =01 Δ119905 = 005 and ℎ = 04 Visual representation of thesolution in Figures 6(a)ndash6(f) is drawn that 10 solitons havebeen broken up from the given initial condition

0 5 10 15

x


2

15

1

05

0

u

Figure 5 119905 = 125Maxwellian initial condition 120576 = 10 120583 = 0001ℎ = 0025 and Δ119905 = 0005

Table 6 Conservation laws 120576 = 02 120583 = 01

119905 1198621

1198622

1198623

0 500001 450004 406207100 499999 450011 404055200 500009 450032 405267300 499996 450044 404286400 499989 450048 426020500 499984 450049 406564600 499997 450051 406786700 500009 450051 404270800 500011 450049 404629

The first three conservation laws are recorded at sometimes in Table 6 These are favorably constant The observedvelocity of the leading soliton having the amplitude 196342 is119881119899= 0128which was in close agreement with that calculated

from its observed amplitude of119881119886= (19379times02)3 = 0129

4 Conclusion

The numerical solution of the KdVE is obtained by thecollocation method using the exponential basis functionsPerformance of the present method is shown by calculating119871infinmdashthe error norm and conservation laws The present

method gives accurate results and simulations such as thepropagation of soliton and generation of waves which aresubstantiated fairly Using the exponential cubic B-splinesalternative numerical methods can be set up for findingnumerical solutions of the differential equations with highaccuracy when an appropriate free parameter is chosen


2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 0

(a)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 100

(b)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 200

(c)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 400

(d)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 600

(e)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 800

(f)

Figure 6 (a) The solution graph for 119905 = 0 (b) The solution graph for 119905 = 100 (c) The solution graph for 119905 = 200 (d) The solution graph for119905 = 400 (e) The solution graph for 119905 = 600 (f) The solution graph for 119905 = 800


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] B J McCartin ldquoTheory computation and application of expo-nential splinesrdquo Tech Rep DOEER03077-171 1981

[2] B J McCartin ldquoComputation of exponential splinesrdquo SIAMJournal on Scientific and Statistical Computing vol 11 no 2 pp242ndash262 1990

[3] B J McCartin ldquoTheory of exponential splinesrdquo Journal ofApproximation Theory vol 66 no 1 pp 1ndash23 1991

[4] B J McCartin and A Jameson ldquoNumerical solution of non-linear hyperbolic conservation laws using exponential splinesrdquoComputational Mechanics vol 6 no 2 pp 77ndash91 1990

[5] M Sakai and R A Usmani ldquoA class of simple exponential 119861-splines and their application to numerical solution to singularperturbation problemsrdquo Numerische Mathematik vol 55 no 5pp 493ndash500 1989

[6] D Radunovic ldquoMultiresolution exponential B-splines and sin-gularly perturbed boundary problemrdquo Numerical Algorithmsvol 47 no 2 pp 191ndash210 2008

[7] S C Rao and M Kumar ldquoExponential B-spline collocationmethod for self-adjoint singularly perturbed boundary valueproblemsrdquo Applied Numerical Mathematics vol 58 no 10 pp1572ndash1581 2008

[8] R Mohammadi ldquoExponential B-spline solution of convection-diffusion equationsrdquoAppliedMathematics vol 4 no 6 pp 933ndash944 2013

[9] G A Baker V A Dougalis and O A Karakashian ldquoConver-gence of Galerkin approximations for the Korteweg-de Vriesequationrdquo Mathematics of Computation vol 40 no 162 pp419ndash433 1983

[10] G A Gardner and L R T Gardner ldquoA finite element solutionfor the Korteweg de vries equation using cubic B-spline shapefunctionsrdquo in Proceedings of the International Conference onModelling and Simulation vol 1 AMSE Conferance PressTassin-la-Demi-Lune France 1988

[11] L R Gardner G A Gardner and A H Ali ldquoSimulationsof solitons using quadratic spline finite elementsrdquo ComputerMethods in Applied Mechanics and Engineering vol 92 no 2pp 231ndash243 1991

[12] A A Soliman ldquoCollocation solution of the Korteweg-de Vriesequation using septic splinesrdquo International Journal of ComputerMathematics vol 81 no 3 pp 325ndash331 2004

[13] E N Aksan and A Ozdes ldquoNumerical solution of Korteweg-de Vries equation by Galerkin B-spline finite element methodrdquoAppliedMathematics and Computation vol 175 no 2 pp 1256ndash1265 2006

[14] S I Zaki ldquoA quintic B-spline finite element scheme for theKdVB equationrdquo Computational and Applied Mathematics vol190 pp 532ndash547 2006

[15] P C Jain R Shankar and D Bhardwaj ldquoNumerical solutionof the Korteweg-de Vries (KDV) equationrdquo Chaos Solitons andFractals vol 8 no 6 pp 943ndash951 1997

[16] G Micula and M Micula ldquoOn the numerical approach ofKorteweg-de Vries-Burger equations by spline finite elementand collocation methodsrdquo Seminar on Fixed Point Theory Cluj-Napoca vol 3 pp 261ndash270 2002

[17] A Canıvar M Sari and I Dag ldquoA Taylor-Galerkin finiteelement method for the KdV equation using cubic B-splinesrdquoPhysica B Condensed Matter vol 405 no 16 pp 3376ndash33832010

[18] S Kapoor S Rawat and S Dhawan ldquoNumerical investigationof separated solitary waves solution for KdV equation throughfinite element techniquerdquo International Journal of ComputerApplications vol 40 no 14 pp 27ndash33 2012

[19] NKAmein andMA Ramadan ldquoA small time solutions for theKdV equation using Bubnov-Galerkin finite element methodrdquoJournal of the Egyptian Mathematical Society vol 19 no 3 pp118ndash125 2011

[20] B Saka and I Dag ldquoQuartic B-spline GALerkin approach to thenumerical solution of theKdVB equationrdquoAppliedMathematicsand Computation vol 215 no 2 pp 746ndash758 2009

[21] R Yu R Wang and C Zhu ldquoA numerical method for solvingKdV equation with blended b-spline quasi-interpolationrdquo Jour-nal of Information and Computational Science vol 10 no 16 pp5093ndash5101 2013

[22] R-G Yu R-H Wang and C-G Zhu ldquoA numerical methodfor solving KdV equation with multilevel B-spline quasi-interpolationrdquo Applicable Analysis vol 92 no 8 pp 1682ndash16902013

[23] S I Zaki ldquoA quintic B-spline finite elements scheme for theKdVB equationrdquo Computer Methods in Applied Mechanics andEngineering vol 188 no 1 pp 121ndash134 2000

[24] B Saka ldquoCosine expansion-based differential quadraturemethod for numerical solution of the KdV equationrdquo ChaosSolitons and Fractals vol 40 no 5 pp 2181ndash2190 2009

[25] Dag and Y Dereli ldquoNumerical solutions of KdV equation usingradial basis functionsrdquoAppliedMathematical Modelling vol 32no 4 pp 535ndash546 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

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


Table 1 Values of 119861119894(119909) and its first and second derivatives at the knot points

119909 119909119894minus2

119909119894minus1

119909119894

119909119894+1

119909119894+2

119861119894

0119904 minus 119901ℎ

2 (119901ℎ119888 minus 119904)1

119904 minus 119901ℎ

2 (119901ℎ119888 minus 119904)0

1198611015840

1198940

119901 (1 minus 119888)

2 (119901ℎ119888 minus 119904)0

119901 (119888 minus 1)

2 (119901ℎ119888 minus 119904)0

11986110158401015840

1198940

1199012119904

2 (119901ℎ119888 minus 119904)minus1199012119904

119901ℎ119888 minus 119904

1199012119904

2 (119901ℎ119888 minus 119904)0

by Canıvar et al in [17] A study based on cubic B-spline finiteelement method for the solution of the KdVE is suggested byKapoor et al [18] A Bubnov-Galerkin finite element methodwith quintic B-spline functions taken as element shape andweight functions is presented for the solution of the KdVE[19]The paper deals with the numerical solution of the KdVEusing quartic B-splines Galerkin method as both shape andweight functions over the finite intervals [20] A blendedspline quasi-interpolation scheme is employed to solve theone-dimensional nonlinear KdVE [21] A multilevel quarticspline quasi-interpolation scheme is fulfilled to exhibit a largenumber of physical phenomena for KdVE [22]

The aim of the present paper is to develop an approximatesolution of KdVE by collocation method In Section 2 theexponential B-spline collocation algorithm is defined for theKdVE In Section 3 the three numerical experiments areconstructed to demonstrate the efficiency of the proposedmethod and the results are documented in tables and graphsare depicted

We will solve the KdVE

119880119905+ 120576119880119880

119909+ 120583119880119909119909119909= 0 119886 le 119909 le 119887 (1)

where 120576 120583 are positive parameters and the subscripts 119909 and119905 denote differentiation The boundary conditions will bechosen as

119880 (119886 119905) = 119880 (119887 119905) = 0

119880119909 (119886 119905) = 119880119909 (119887 119905) = 0

(2)

KdVE is prototypical example of exactly solvable math-ematical model of waves on shallow water surface It arisesfor evolution interaction of waves and generation in physicsDue to the term 119880

119905 (1) is called the evolution equation the

nonlinear term causes the steepness of the wave and thedispersive term defines the spreading of the wave It is knownthat the effect of the steepness and spreading results in solitonsolutions for the KdVE

2 Exponential B-Spline Collocation Method

The region [119886 119887] is partitioned into equal subintervals bypoints 119909

119894 119894 = 0 119873 On these points together with

additional points 119909119894 119894 = minus3 minus2 minus1119873+1119873+2119873+3 outside

the domain the exponential B-splines 119861119894(119909) can be defined

as

119861119894 (119909)

=

1198872((119909119894minus2minus 119909) minus

1

119901(sinh (119901 (119909

119894minus2minus 119909)))) [119909

119894minus2 119909119894minus1]

1198861+ 1198871(119909119894minus 119909) + 119888

1exp (119901 (119909

119894minus 119909))

+1198891exp (minus119901 (119909

119894minus 119909)) [119909

119894minus1 119909119894]

1198861+ 1198871(119909 minus 119909

119894) + 1198881exp (119901 (119909 minus 119909

119894))

+1198891exp (minus119901 (119909 minus 119909

119894)) [119909

119894 119909119894+1]

1198872((119909 minus 119909

119894+2) minus

1

119901(sinh (119901 (119909 minus 119909

119894+2)))) [119909

119894+1 119909119894+2]

0 otherwise(3)

where

1198861=

119901ℎ119888

119901ℎ119888 minus 119904 119887

1=119901

2[119888 (119888 minus 1) + 119904

2

(119901ℎ119888 minus 119904) (1 minus 119888)]

1198872=

119901

2 (119901ℎ119888 minus 119904)

1198881=1

4[exp (minus119901ℎ) (1 minus 119888) + 119904 (exp (minus119901ℎ) minus 1)

(119901ℎ119888 minus 119904) (1 minus 119888)]

1198891=1

4[exp (119901ℎ) (119888 minus 1) + 119904 (exp (119901ℎ) minus 1)

(119901ℎ119888 minus 119904) (1 minus 119888)]

119904 = sinh (119901ℎ) 119888 = cosh (119901ℎ)

119901 is a free parameter ℎ =119887 minus 119886

119873

(4)

119861minus1(119909) 119861

0(119909) 119861

119873+1(119909) forms a basis for the expo-

nential spline space on the interval [119886 119887] On the fourconsecutive subintervals an exponential B-spline 119861

119894(119909) is

defined and it is second-order continuously differentiablefunctions

In Table 1 the values of 119861119894(119909) 1198611015840

119894(119909) and 11986110158401015840

119894(119909) at the

points 119909119894 which can be obtained from (3) are listed where

1015840 denotes differentiation with respect to space variable 119909





119880119873 (119909 119905) =

119873+1

sum

119894=minus1

120575119894119861119894 (119909) (5)



119880119894= 119880 (119909

119894 119905) = 119898

1120575119894minus1+ 120575119894+ 1198981120575119894+1

1198801015840

119894= 1198801015840(119909119894 119905) = 119898

2120575119894minus1minus 1198982120575119894+1

11988010158401015840

119894= 11988010158401015840(119909119894 119905) = 119898

3120575119894minus1minus 21198983120575119894+ 1198983120575119894+1

(6)


2= 119901(1 minus 119888)2(119901ℎ119888 minus 119904)

and1198983= 11990121199042(119901ℎ119888 minus 119904)


given by

119880119890

119873(119909 119905) = 120575119894minus1119861119894minus1 (119909) + 120575119894119861119894 (119909)

+ 120575119894+1119861119894+1 (119909) + 120575119894+2119861119894+2 (119909)

(7)


120575119894 and 120575


119894minus1

119861119894 and 119861



119880119905+ 120576119880119880

119909+ 120583119881119909119909= 0

119880119909minus 119881 = 0

(8)


119905by the


119909119909by Crank-


119880119899+1minus 119880119899

Δ119905+ 120576(119880119880119909)119899+1+ (119880119880

119909)119899

2+ 120583119881119899+1

119909119909+ 119881119899

119909119909

2= 0

119880119899+1

119909+ 119880119899

119909

2minus119881119899+1+ 119881119899

2= 0

(9)



(119880119880119909)119899+1= 119880119899+1119880119899

119909+ 119880119899119880119899+1

119909minus 119880119899119880119899

119909

(119880119880119909)119899= 119880119899119880119899

119909

(10)


119880119899+1minus 119880119899+120576Δ119905

2(119880119899+1119880119899

119909+ 119880119899119880119899+1

119909)

minus120583Δ119905

2(119881119899+1

119909119909+ 119881119899

119909119909) = 0

119880119899+1

119909+ 119880119899

119909

2minus119881119899+1+ 119881119899

2= 0

(11)


119880119873 (119909 119905) =

119873+1

sum

119894=minus1

120575119894119861119894 (119909) 119881

119873 (119909 119905) =

119873+1

sum

119894=minus1

120601119894119861119894 (119909) (12)


1198981120575119899+1

119898minus1minus120583Δ119905

2120574120601119899+1

119898minus1+ 120575119899+1

119898

+ 120583Δ119905120574120601119899+1

119898+ 1198981120575119899+1

119898+1minus120583Δ119905

2120574120601119899+1

119898+1

= 1198981(1 minus

120576Δ119905

2119871) 120575119899


2(120583120574 + 120576119898

1119870)120601119899

119898minus1

+ (1 minus120576Δ119905

2119871) 120575119899


2(minus2120583120574 + 120576119870) 120601

119899

119898minus1

+ 1198981(1 minus

120576Δ119905

2119871) 120575119899


2(120583120574 + 120576119898

1119870)120601119899

119898minus1

120573120575119899+1

119898minus1minus 1198981120601119899+1

119898minus1minus 120601119899+1

119898minus 120573120575119899+1

119898+1minus 1198981120601119899+1

119898+1

= minus120573120575119899

119898minus1+ 1198981120601119899

119898minus1+ 120601119899

119898+ 120573120575119899

119898+1+ 1198981120601119899

119898+1

119898 = 0 119873 119899 = 0 1

(13)

where

119870 = 1198981120575119894minus1+ 120575119894+ 1198981120575119894+1

119871 = 1198981120601119894minus1+ 120601119894+ 1198981120601119894+1

120573 =119901 (1 minus 119888)

2 (119901ℎ119888 minus 119904) 120574 =

1199012119904

2 (119901ℎ119888 minus 119904)

(14)


minus1 120601119899+1

minus1 120575119899+1

0 120601119899+1

0 120575

119899+1

119873+1

120601119899+1



1198981120575minus1+ 1205750+ 11989811205751= 0

1198981120601minus1+ 1206010+ 11989811206011= 0

1198981120575119873minus1

+ 120575119873+ 1198981120575119873+1

= 0

1198981120601119873minus1

+ 120601119873+ 1198981120601119873+1

= 0

(15)



120601119873+1


1= (120575minus1 1205750 120575

119873 120575119873+1)

1198892= (120601minus1 1206010 120601



(119880119873)119909(119886 0) = 0 = 1198982120575

0

minus1+ 11989821205750

1

(119880119873)119909(119909119894 0) = 119898

11205750

119894minus1+ 1205750

119894+ 11989811205750

119894+1= 119880 (119909

119894 0)

119894 = 0 119873

(119880119873)119909(119887 0) = 0 = 1198982120575

0

119873minus1+ 11989821205750

119873+1

(119881119873)119909(119886 0) = 0 = 1198982120601

0

minus1+ 11989821206010

1

(119881119873)119909(119909119894 0) = 119898

11206010

119894minus1+ 1206010

119894+ 11989811206010

119894+1= 119881 (119909

119894 0)

119894 = 0 119873

(119881119873)119909(119886 0) = 1198982120601

0

119873minus1+ 11989821206010

119873+1

(16)

3 Numerical Tests


1198621= int

infin

minusinfin

119880119889119909 asymp int

119887

119886

119880119889119909 asympℎ

2

119873minus1

sum

119894=0

(119880119894+ 119880119894+1)

1198622= int

infin

minusinfin

(1198802) 119889119909 asymp int

119887

119886

(1198802) 119889119909 asymp

ℎ

2

119873minus1

sum

119894=0

[(1198802)119894+ (1198802)119894+1]

1198623= int

infin

minusinfin

(1198803minus3120583

1205761198802

119909)119889119909 asymp int

119887

119886

(1198803minus3120583

1205761198802

119909)119889119909

asympℎ

2

119873minus1

sum

119894=0

[(1198803)119894minus3120583

120576(1198802

119909)119894+ (1198803)119894+1minus3120583

120576(1198802

119909)119894+1]

(17)


119871infin=1003816100381610038161003816119880 minus 119880119873

1003816100381610038161003816infin= max119895

100381610038161003816100381610038161003816119880119895minus (119880119873)119899

119895

100381610038161003816100381610038161003816(18)



119880 (119909 119905) = 3119888 sec ℎ2 (119860119909 minus 119861119905 + 119863) (19)


1

08

05 15 21

06

04

02

00

u

x

t = 0 t = 2 t = 25 t = 3t = 1 t = 15t = 05


05 15 210

x

0016

0014

0012

001

0008

0006

0004

0002

0

|Error|





2error



2and 119862


1




119901 times 105


119901 = 11198621

1198622

1198623

Method

00 0 0 0144597 0086759 0046849

Pre

05 0521 (119901 = 137) 051 0144602 0086759 0046849

10 0589 (119901 = 164) 075 0144593 0086759 0046849

15 0533 (119901 = 164) 103 0144592 0086759 0046849

20 0595 (119901 = 164) 126 0144591 0086759 0046849

25 0657 (119901 = 164) 150 0144597 0086759 0046849

30 0740 (119901 = 164) 161 0144597 0086759 0046849

30 008 0014460 008676 004685 [23]30 104 0014460 008675 004685 [24]30 004 0144597 0086761 00468524 [17]30 014 0144601 0086760 0046850 [25]


120576 = 10 120583 = 004

119905 1198621

1198622

1198623

0 177245 125331 08729225 177243 125332 08741250 177248 125333 08743775 177170 125332 087442100 177203 125333 087442125 177367 125330 087442



119880 (119909 0) = exp (minus1199092) (20)


119880 (minus15 119905) = 119880 (15 119905) = 0 119905 gt 0 (21)









119899


119886= 1198861205763 We find that119881

119899= 04




120576 = 10 120583 = 001

119905 1198621

1198622

1198623

0 177245 125331 09857225 177244 125362 09927750 177250 125374 09955875 177243 125376 099576100 177242 125377 099582125 177222 125378 099582


120576 = 10 120583 = 0001

119905 1198621

1198622

1198623

0 177245 125331 10195625 177245 125356 10246150 177245 125365 10264475 177245 125365 102654100 177244 125365 102656125 177244 125365 102656


119899= 052 and


119886= (15535 times 1)3 = 051783 In Figure 4






0 5 10 15

x


14

12

1

08

06

04

02

0

u

minus02


0 5 10 15

x


14

16

12

1

08

06

04

02

0

u

minus02



119880 (119909 0) =1


5)] (22)


119880 (minus50 119905) = 119880 (150 119905) = 0 119905 gt 0 (23)


0 5 10 15

x


2

15

1

05

0

u



119905 1198621

1198622

1198623

0 500001 450004 406207100 499999 450011 404055200 500009 450032 405267300 499996 450044 404286400 499989 450048 426020500 499984 450049 406564600 499997 450051 406786700 500009 450051 404270800 500011 450049 404629



4 Conclusion




2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 0

(a)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 100

(b)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 200

(c)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 400

(d)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 600

(e)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 800

(f)





References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119880119873 (119909 119905) =

119873+1

sum

119894=minus1

120575119894119861119894 (119909) (5)



119880119894= 119880 (119909

119894 119905) = 119898

1120575119894minus1+ 120575119894+ 1198981120575119894+1

1198801015840

119894= 1198801015840(119909119894 119905) = 119898

2120575119894minus1minus 1198982120575119894+1

11988010158401015840

119894= 11988010158401015840(119909119894 119905) = 119898

3120575119894minus1minus 21198983120575119894+ 1198983120575119894+1

(6)


2= 119901(1 minus 119888)2(119901ℎ119888 minus 119904)

and1198983= 11990121199042(119901ℎ119888 minus 119904)


given by

119880119890

119873(119909 119905) = 120575119894minus1119861119894minus1 (119909) + 120575119894119861119894 (119909)

+ 120575119894+1119861119894+1 (119909) + 120575119894+2119861119894+2 (119909)

(7)


120575119894 and 120575


119894minus1

119861119894 and 119861



119880119905+ 120576119880119880

119909+ 120583119881119909119909= 0

119880119909minus 119881 = 0

(8)


119905by the


119909119909by Crank-


119880119899+1minus 119880119899

Δ119905+ 120576(119880119880119909)119899+1+ (119880119880

119909)119899

2+ 120583119881119899+1

119909119909+ 119881119899

119909119909

2= 0

119880119899+1

119909+ 119880119899

119909

2minus119881119899+1+ 119881119899

2= 0

(9)



(119880119880119909)119899+1= 119880119899+1119880119899

119909+ 119880119899119880119899+1

119909minus 119880119899119880119899

119909

(119880119880119909)119899= 119880119899119880119899

119909

(10)


119880119899+1minus 119880119899+120576Δ119905

2(119880119899+1119880119899

119909+ 119880119899119880119899+1

119909)

minus120583Δ119905

2(119881119899+1

119909119909+ 119881119899

119909119909) = 0

119880119899+1

119909+ 119880119899

119909

2minus119881119899+1+ 119881119899

2= 0

(11)


119880119873 (119909 119905) =

119873+1

sum

119894=minus1

120575119894119861119894 (119909) 119881

119873 (119909 119905) =

119873+1

sum

119894=minus1

120601119894119861119894 (119909) (12)


1198981120575119899+1

119898minus1minus120583Δ119905

2120574120601119899+1

119898minus1+ 120575119899+1

119898

+ 120583Δ119905120574120601119899+1

119898+ 1198981120575119899+1

119898+1minus120583Δ119905

2120574120601119899+1

119898+1

= 1198981(1 minus

120576Δ119905

2119871) 120575119899


2(120583120574 + 120576119898

1119870)120601119899

119898minus1

+ (1 minus120576Δ119905

2119871) 120575119899


2(minus2120583120574 + 120576119870) 120601

119899

119898minus1

+ 1198981(1 minus

120576Δ119905

2119871) 120575119899


2(120583120574 + 120576119898

1119870)120601119899

119898minus1

120573120575119899+1

119898minus1minus 1198981120601119899+1

119898minus1minus 120601119899+1

119898minus 120573120575119899+1

119898+1minus 1198981120601119899+1

119898+1

= minus120573120575119899

119898minus1+ 1198981120601119899

119898minus1+ 120601119899

119898+ 120573120575119899

119898+1+ 1198981120601119899

119898+1

119898 = 0 119873 119899 = 0 1

(13)

where

119870 = 1198981120575119894minus1+ 120575119894+ 1198981120575119894+1

119871 = 1198981120601119894minus1+ 120601119894+ 1198981120601119894+1

120573 =119901 (1 minus 119888)

2 (119901ℎ119888 minus 119904) 120574 =

1199012119904

2 (119901ℎ119888 minus 119904)

(14)


minus1 120601119899+1

minus1 120575119899+1

0 120601119899+1

0 120575

119899+1

119873+1

120601119899+1



1198981120575minus1+ 1205750+ 11989811205751= 0

1198981120601minus1+ 1206010+ 11989811206011= 0

1198981120575119873minus1

+ 120575119873+ 1198981120575119873+1

= 0

1198981120601119873minus1

+ 120601119873+ 1198981120601119873+1

= 0

(15)



120601119873+1


1= (120575minus1 1205750 120575

119873 120575119873+1)

1198892= (120601minus1 1206010 120601



(119880119873)119909(119886 0) = 0 = 1198982120575

0

minus1+ 11989821205750

1

(119880119873)119909(119909119894 0) = 119898

11205750

119894minus1+ 1205750

119894+ 11989811205750

119894+1= 119880 (119909

119894 0)

119894 = 0 119873

(119880119873)119909(119887 0) = 0 = 1198982120575

0

119873minus1+ 11989821205750

119873+1

(119881119873)119909(119886 0) = 0 = 1198982120601

0

minus1+ 11989821206010

1

(119881119873)119909(119909119894 0) = 119898

11206010

119894minus1+ 1206010

119894+ 11989811206010

119894+1= 119881 (119909

119894 0)

119894 = 0 119873

(119881119873)119909(119886 0) = 1198982120601

0

119873minus1+ 11989821206010

119873+1

(16)

3 Numerical Tests


1198621= int

infin

minusinfin

119880119889119909 asymp int

119887

119886

119880119889119909 asympℎ

2

119873minus1

sum

119894=0

(119880119894+ 119880119894+1)

1198622= int

infin

minusinfin

(1198802) 119889119909 asymp int

119887

119886

(1198802) 119889119909 asymp

ℎ

2

119873minus1

sum

119894=0

[(1198802)119894+ (1198802)119894+1]

1198623= int

infin

minusinfin

(1198803minus3120583

1205761198802

119909)119889119909 asymp int

119887

119886

(1198803minus3120583

1205761198802

119909)119889119909

asympℎ

2

119873minus1

sum

119894=0

[(1198803)119894minus3120583

120576(1198802

119909)119894+ (1198803)119894+1minus3120583

120576(1198802

119909)119894+1]

(17)


119871infin=1003816100381610038161003816119880 minus 119880119873

1003816100381610038161003816infin= max119895

100381610038161003816100381610038161003816119880119895minus (119880119873)119899

119895

100381610038161003816100381610038161003816(18)



119880 (119909 119905) = 3119888 sec ℎ2 (119860119909 minus 119861119905 + 119863) (19)


1

08

05 15 21

06

04

02

00

u

x

t = 0 t = 2 t = 25 t = 3t = 1 t = 15t = 05


05 15 210

x

0016

0014

0012

001

0008

0006

0004

0002

0

|Error|





2error



2and 119862


1




119901 times 105


119901 = 11198621

1198622

1198623

Method

00 0 0 0144597 0086759 0046849

Pre

05 0521 (119901 = 137) 051 0144602 0086759 0046849

10 0589 (119901 = 164) 075 0144593 0086759 0046849

15 0533 (119901 = 164) 103 0144592 0086759 0046849

20 0595 (119901 = 164) 126 0144591 0086759 0046849

25 0657 (119901 = 164) 150 0144597 0086759 0046849

30 0740 (119901 = 164) 161 0144597 0086759 0046849

30 008 0014460 008676 004685 [23]30 104 0014460 008675 004685 [24]30 004 0144597 0086761 00468524 [17]30 014 0144601 0086760 0046850 [25]


120576 = 10 120583 = 004

119905 1198621

1198622

1198623

0 177245 125331 08729225 177243 125332 08741250 177248 125333 08743775 177170 125332 087442100 177203 125333 087442125 177367 125330 087442



119880 (119909 0) = exp (minus1199092) (20)


119880 (minus15 119905) = 119880 (15 119905) = 0 119905 gt 0 (21)









119899


119886= 1198861205763 We find that119881

119899= 04




120576 = 10 120583 = 001

119905 1198621

1198622

1198623

0 177245 125331 09857225 177244 125362 09927750 177250 125374 09955875 177243 125376 099576100 177242 125377 099582125 177222 125378 099582


120576 = 10 120583 = 0001

119905 1198621

1198622

1198623

0 177245 125331 10195625 177245 125356 10246150 177245 125365 10264475 177245 125365 102654100 177244 125365 102656125 177244 125365 102656


119899= 052 and


119886= (15535 times 1)3 = 051783 In Figure 4






0 5 10 15

x


14

12

1

08

06

04

02

0

u

minus02


0 5 10 15

x


14

16

12

1

08

06

04

02

0

u

minus02



119880 (119909 0) =1


5)] (22)


119880 (minus50 119905) = 119880 (150 119905) = 0 119905 gt 0 (23)


0 5 10 15

x


2

15

1

05

0

u



119905 1198621

1198622

1198623

0 500001 450004 406207100 499999 450011 404055200 500009 450032 405267300 499996 450044 404286400 499989 450048 426020500 499984 450049 406564600 499997 450051 406786700 500009 450051 404270800 500011 450049 404629



4 Conclusion




2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 0

(a)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 100

(b)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 200

(c)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 400

(d)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 600

(e)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 800

(f)





References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120601119873+1


1= (120575minus1 1205750 120575

119873 120575119873+1)

1198892= (120601minus1 1206010 120601



(119880119873)119909(119886 0) = 0 = 1198982120575

0

minus1+ 11989821205750

1

(119880119873)119909(119909119894 0) = 119898

11205750

119894minus1+ 1205750

119894+ 11989811205750

119894+1= 119880 (119909

119894 0)

119894 = 0 119873

(119880119873)119909(119887 0) = 0 = 1198982120575

0

119873minus1+ 11989821205750

119873+1

(119881119873)119909(119886 0) = 0 = 1198982120601

0

minus1+ 11989821206010

1

(119881119873)119909(119909119894 0) = 119898

11206010

119894minus1+ 1206010

119894+ 11989811206010

119894+1= 119881 (119909

119894 0)

119894 = 0 119873

(119881119873)119909(119886 0) = 1198982120601

0

119873minus1+ 11989821206010

119873+1

(16)

3 Numerical Tests


1198621= int

infin

minusinfin

119880119889119909 asymp int

119887

119886

119880119889119909 asympℎ

2

119873minus1

sum

119894=0

(119880119894+ 119880119894+1)

1198622= int

infin

minusinfin

(1198802) 119889119909 asymp int

119887

119886

(1198802) 119889119909 asymp

ℎ

2

119873minus1

sum

119894=0

[(1198802)119894+ (1198802)119894+1]

1198623= int

infin

minusinfin

(1198803minus3120583

1205761198802

119909)119889119909 asymp int

119887

119886

(1198803minus3120583

1205761198802

119909)119889119909

asympℎ

2

119873minus1

sum

119894=0

[(1198803)119894minus3120583

120576(1198802

119909)119894+ (1198803)119894+1minus3120583

120576(1198802

119909)119894+1]

(17)


119871infin=1003816100381610038161003816119880 minus 119880119873

1003816100381610038161003816infin= max119895

100381610038161003816100381610038161003816119880119895minus (119880119873)119899

119895

100381610038161003816100381610038161003816(18)



119880 (119909 119905) = 3119888 sec ℎ2 (119860119909 minus 119861119905 + 119863) (19)


1

08

05 15 21

06

04

02

00

u

x

t = 0 t = 2 t = 25 t = 3t = 1 t = 15t = 05


05 15 210

x

0016

0014

0012

001

0008

0006

0004

0002

0

|Error|





2error



2and 119862


1




119901 times 105


119901 = 11198621

1198622

1198623

Method

00 0 0 0144597 0086759 0046849

Pre

05 0521 (119901 = 137) 051 0144602 0086759 0046849

10 0589 (119901 = 164) 075 0144593 0086759 0046849

15 0533 (119901 = 164) 103 0144592 0086759 0046849

20 0595 (119901 = 164) 126 0144591 0086759 0046849

25 0657 (119901 = 164) 150 0144597 0086759 0046849

30 0740 (119901 = 164) 161 0144597 0086759 0046849

30 008 0014460 008676 004685 [23]30 104 0014460 008675 004685 [24]30 004 0144597 0086761 00468524 [17]30 014 0144601 0086760 0046850 [25]


120576 = 10 120583 = 004

119905 1198621

1198622

1198623

0 177245 125331 08729225 177243 125332 08741250 177248 125333 08743775 177170 125332 087442100 177203 125333 087442125 177367 125330 087442



119880 (119909 0) = exp (minus1199092) (20)


119880 (minus15 119905) = 119880 (15 119905) = 0 119905 gt 0 (21)









119899


119886= 1198861205763 We find that119881

119899= 04




120576 = 10 120583 = 001

119905 1198621

1198622

1198623

0 177245 125331 09857225 177244 125362 09927750 177250 125374 09955875 177243 125376 099576100 177242 125377 099582125 177222 125378 099582


120576 = 10 120583 = 0001

119905 1198621

1198622

1198623

0 177245 125331 10195625 177245 125356 10246150 177245 125365 10264475 177245 125365 102654100 177244 125365 102656125 177244 125365 102656


119899= 052 and


119886= (15535 times 1)3 = 051783 In Figure 4






0 5 10 15

x


14

12

1

08

06

04

02

0

u

minus02


0 5 10 15

x


14

16

12

1

08

06

04

02

0

u

minus02



119880 (119909 0) =1


5)] (22)


119880 (minus50 119905) = 119880 (150 119905) = 0 119905 gt 0 (23)


0 5 10 15

x


2

15

1

05

0

u



119905 1198621

1198622

1198623

0 500001 450004 406207100 499999 450011 404055200 500009 450032 405267300 499996 450044 404286400 499989 450048 426020500 499984 450049 406564600 499997 450051 406786700 500009 450051 404270800 500011 450049 404629



4 Conclusion




2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 0

(a)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 100

(b)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 200

(c)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 400

(d)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 600

(e)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 800

(f)





References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119901 times 105


119901 = 11198621

1198622

1198623

Method

00 0 0 0144597 0086759 0046849

Pre

05 0521 (119901 = 137) 051 0144602 0086759 0046849

10 0589 (119901 = 164) 075 0144593 0086759 0046849

15 0533 (119901 = 164) 103 0144592 0086759 0046849

20 0595 (119901 = 164) 126 0144591 0086759 0046849

25 0657 (119901 = 164) 150 0144597 0086759 0046849

30 0740 (119901 = 164) 161 0144597 0086759 0046849

30 008 0014460 008676 004685 [23]30 104 0014460 008675 004685 [24]30 004 0144597 0086761 00468524 [17]30 014 0144601 0086760 0046850 [25]


120576 = 10 120583 = 004

119905 1198621

1198622

1198623

0 177245 125331 08729225 177243 125332 08741250 177248 125333 08743775 177170 125332 087442100 177203 125333 087442125 177367 125330 087442



119880 (119909 0) = exp (minus1199092) (20)


119880 (minus15 119905) = 119880 (15 119905) = 0 119905 gt 0 (21)









119899


119886= 1198861205763 We find that119881

119899= 04




120576 = 10 120583 = 001

119905 1198621

1198622

1198623

0 177245 125331 09857225 177244 125362 09927750 177250 125374 09955875 177243 125376 099576100 177242 125377 099582125 177222 125378 099582


120576 = 10 120583 = 0001

119905 1198621

1198622

1198623

0 177245 125331 10195625 177245 125356 10246150 177245 125365 10264475 177245 125365 102654100 177244 125365 102656125 177244 125365 102656


119899= 052 and


119886= (15535 times 1)3 = 051783 In Figure 4






0 5 10 15

x


14

12

1

08

06

04

02

0

u

minus02


0 5 10 15

x


14

16

12

1

08

06

04

02

0

u

minus02



119880 (119909 0) =1


5)] (22)


119880 (minus50 119905) = 119880 (150 119905) = 0 119905 gt 0 (23)


0 5 10 15

x


2

15

1

05

0

u



119905 1198621

1198622

1198623

0 500001 450004 406207100 499999 450011 404055200 500009 450032 405267300 499996 450044 404286400 499989 450048 426020500 499984 450049 406564600 499997 450051 406786700 500009 450051 404270800 500011 450049 404629



4 Conclusion




2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 0

(a)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 100

(b)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 200

(c)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 400

(d)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 600

(e)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 800

(f)





References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15

x


14

12

1

08

06

04

02

0

u

minus02


0 5 10 15

x


14

16

12

1

08

06

04

02

0

u

minus02



119880 (119909 0) =1


5)] (22)


119880 (minus50 119905) = 119880 (150 119905) = 0 119905 gt 0 (23)


0 5 10 15

x


2

15

1

05

0

u



119905 1198621

1198622

1198623

0 500001 450004 406207100 499999 450011 404055200 500009 450032 405267300 499996 450044 404286400 499989 450048 426020500 499984 450049 406564600 499997 450051 406786700 500009 450051 404270800 500011 450049 404629



4 Conclusion




2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 0

(a)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 100

(b)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 200

(c)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 400

(d)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 600

(e)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 800

(f)





References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 0

(a)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 100

(b)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 200

(c)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 400

(d)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 600

(e)

2

15

1

05

0

u

0 50 100 150

x

minus50minus05

t = 800

(f)





References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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Research Article The Exponential Cubic B-Spline …downloads.hindawi.com/archive/2015/367056.pdfe exponential cubic B-spline algorithm is presented to nd the numerical solutions of