numerical solution for ivp in volterra type linear...

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 490689 4 pageshttpdxdoiorg1011552013490689

Research ArticleNumerical Solution for IVP in Volterra Type LinearIntegrodifferential Equations System

F Ghomanjani1 A KJlJccedilman2 and S Effati1

1 Department of Mathematics Ferdowsi University of Mashhad Mashhad Iran2Department of Mathematics Universiti Putra Malaysia 43400 Serdang Selangor Malaysia

Correspondence should be addressed to A Kılıcman kilicmanyahoocom

Received 23 May 2013 Accepted 9 July 2013

Academic Editor Santanu Saha Ray

Copyright copy 2013 F Ghomanjani et al This 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

Amethod is proposed to determine the numerical solution of system of linear Volterra integrodifferential equations (IDEs) by usingBezier curves The Bezier curves are chosen as piecewise polynomials of degree n and Bezier curves are determined on [119905

0 119905119891] by

119899 + 1 control points The efficiency and applicability of the presented method are illustrated by some numerical examples

1 Introduction

Integrodifferential equations (IDEs) have been found todescribe various kinds of phenomena such as glass formingprocess dropwise condensation nanohydrodynamics andwind ripple in the desert (see [1 2])

There are several numerical and analytical methods forsolving IDEs Some different methods are presented tosolve integral and IDEs in [3 4] Maleknejad et al [5]used rationalized Haar functions method to solve the linearIDEs system Linear IDEs system has been solved by usingGalerkin methods with the hybrid Legendre and block-Pulse functions on interval [0 1) (see [6]) Yusufoglu [7]presented an application of Hersquos homotopy perturbation(HPM) method to solve the IDEs system Hersquos variationaliteration method has been used for solving two systems ofVolterra integrodifferential equations (see [8]) Arikoglu andOzkol [9] presented differential transformmethod (DTM) forintegrodifferential and integral equation systems Hersquos homo-topy perturbation (HPM)methodwas proposed for systemofintegrodifferential equations (see [10]) A numerical methodbased on interpolation of unknown functions at distinctinterpolation points has been introduced for solving linearIDEs system with initial values (see [11]) Recently Biazarintroduced a new modification of homotopy perturbationmethod (NHPM) to obtain the solution of linear IDEs system(see [12]) Taylor expansionmethod has been used for solving

IDEs (see [13 14]) Rashidinia and Tahmasebi developed andmodified Taylor series method (TSM) introduced in [15] tosolve the system of linear Volterra IDEs

In the present work we suggest a technique similar to theone which was used in [16] for solving the system of linearVolterra IDEs in the following form

119899

sum119894=1

120572119898119894

sum119895=0

119901119898119894119895 (119905) 119910

(119895)

119894(119905) +

119899

sum119894=1

int119905

1199050

(119896119898119894 (119905 119909)

120572119898119894

sum119895=0

119910(119895)

119894(119909))119889119909

= 119891119898 (119905) 119898 = 1 2 119899 119905

0le 119905 le 119905

119891

(1)

with the initial conditions

119910(0)

119894(1199050) = 1198881198940 119910

(1)

119894(1199050) = 1198881198941 119910

(120572119898119894minus1)

119894(1199050) = 119888119894(120572119898119894minus1)

(2)

where 119910(119895)

119894(119905) stands for 119895th-order derivative of 119910

119894(119905) 119891119898(119905)

119896119898119894(119905 119909) and 119901

119898119894119895(119905) are known functions (119898 119894 = 1 2

119899 119895 = 0 1 120572119898119894) and 119905

0 119905119891 and 119888

119894119895(119894 = 1 2 119899 119895 =

0 1 120572119898119894

minus 1) are appropriate constantsThe current paper is organized as follows In Section 2

function approximationwill be introducedNumerical exam-ples will be stated in Section 3 Finally Section 4 will give aconclusion briefly

2 Abstract and Applied Analysis

26

24

22

2

18

16

14

12

10 02 04 06 08 1

t

Approximate y1Exact y1

Figure 1 The graph of approximated 1199101(119905) for Example 1

2 Function Approximation

Our strategy is to use Bezier curves to approximate thesolutions 119910

119894(119905) for 1 le 119894 le 119899 which are given below

Define the Bezier polynomials of degree119873 that approximaterespectively the actions of 119910

119894(119905) over the interval [119905

0 119905119891] as

follows

119910119894 (119905) =

119873

sum119903=0

119886119894

119903119861119903119873

(119905 minus 1199050

ℎ) (3)

where ℎ = 119905119891minus 1199050and 119886119903is the control point of Bezier curve

and

119861119903119873

(119905 minus 1199050

ℎ) = (

119873

119903)

1

ℎ119873(119905119891minus 119905)119873minus119903

(119905 minus 1199050)119903 (4)

is the Bernstein polynomial of degree 119873 over the interval[1199050 119905119891] (see [17]) By substituting (3) in (2) 119877

119898(119905) can be

defined for 119905 isin [1199050 119905119891] as

119877119898(119905) =

119899

sum119894=1

120572119898119894

sum119895=0

119901119898119894119895

(119905) 119910(119895)

119894(119905)

+

119899

sum119894=1

int119905

0

(119896119898119894

(119905 119909)

120572119898119894

sum119895=0

119910(119895)

119894(119909))119889119909 minus 119891

119898(119905)

119898 = 1 2 119899

(5)

where (2) is satisfied The convergence was proved in theapproximation with Bezier curves when the degree of theapproximate solution119873 tends to infinity (see [18])

Now the residual function is defined over the interval[1199050 119905119891] as follows

119877 (119905) = int119905119891

1199050

119899

sum119898=1

1003817100381710038171003817119877119898 (119905)10038171003817100381710038172119889119905 (6)

0 02 04 06 08 1t


08

07

06

05

04

03

02

01

0


where sdot is the Euclidean norm Our aim is to solve thefollowing problem over the interval [119905

0 119905119891]

min 119877 (119905)

st 119910(0)

119894(1199050) = 1198881198940

119910(1)

119894(1199050) = 1198881198941 119910

(120572119898119894minus1)

119894(1199050) = 119888119894(120572119898119894

minus1)

(7)

The mathematical programming problem (7) can besolved by many subroutine algorithms and we used Maple12 to solve this optimization problem

Remark 1 In Chapter 1 of [19] it was proved that119873 satisfies

119873 gt119878

1205752120598 (8)

where 119878 = 119910119894(119905) and because of this reason that 119910

119894(119905) is

uniformly continuous on [1199050 119905119891] we have 119904 119905 isin [119905

0 119905119891] that

|119905 minus 119904| lt 120575 and minus(1205982) lt 119910119894(119905) minus 119910

119894(119904) lt 1205982 for more details

see [19]

3 Applications and Numerical Results

Consider the following examples which can be solved byusing the presented method

Example 1 Consider a system of third-order linear VolterraIDEs on the interval [0 1] (see [4])

11991010158401015840

1(119905) + 119905

21199101(119905) minus 119910

10158401015840

2(119905)

+ int119905

0

((119905 minus 119909) 1199101 (119909) + 1199102 (119909)) 119889119909 = 119892

1 (119905)

411990531199101015840

1(119905) + 6119905

21199101 (119905) + 119910

101584010158401015840

2(119905)

+ int119905

0

(1199101(119909) + (119905 + 119909) 119910

2(119909)) 119889119909 = 119892

2(119905)

(9)

Abstract and Applied Analysis 3

Table 1 Computed errors for Example 1

119905 Absolute error for 1199101(119905) Absolute error for 119910

2(119905)

00 0000000 0000000000002 14801 times 10

minus1022475 times 10

minus11

04 0162735585 times 10minus5

312780502 times 10minus7

06 0251133963 times 10minus5

01536077787 times 10minus5

08 18337 times 10minus10

08864238659 times 10minus5




t Absolute error for 1199101(119905) Absolute error for 119910

2(119905)

00 0000000 0000000000002 3840 times 10


minus11

04 05791064832 times 10minus3

0156041748480 times 10minus3

06 017373195072 times 10minus2

0156041748480 times 10minus3

08 069492781056 times 10minus2


10 0000 0000

with the initial conditions1199101(0) = 119910

1015840

1(0) = 1 119910

2(0) = 119910

10158401015840

2(0) =

0 and 1199101015840

2(0) = 1 where

1198921(119905) = (2 + 119905

2) 119890119905minus 119905 minus cos (119905) + sin (119905)

1198922(119905) = 7 sin (119905) minus (1 + 2119905) cos (119905) + 119890

119905(1 + 4119905

2+ 41199053) + 119905 minus 1

(10)

The exact solution of this system is 1199101(119905) = 119890

119905 1199102(119905) =

sin(119905)With 119873 = 5 the approximated solutions for 119910

1(119905) and

1199102(119905) are shown respectively in Figures 1 and 2 and the

computed errors are shown in Table 1 which show the highaccuracy of the proposed method

Example 2 Consider the following system of linear VolterraIDEs equations as follows (see [4])

minus 1199101015840

1minus

1

21199051199101+

3

21199102

=5

2minus 119905 minus

27

21199052+ 1199054+

3

2(minus1 + 2119905

2) minus

1

2(minus3119905 + 4119905

3)

+ intt

minus1

(1199101minus 31199051199102) 119889119909

11990521199101+ 1199101015840

2minus 1199051199102

=2

5+ 3119905 + 3119905

3minus

8

51199055+ 1199052(minus3119905 + 4119905

3)

+ int119905

minus1

((2119905 + 119909) 1199101 + 311990921199102) 119889119909

(11)

under the conditions1199101(0) = 0 and119910

2(0) = minus1 with the exact

solution 1199101(119905) = 4119905

3minus 3119905 119910

2(119905) = 2119905

2minus 1

With 119873 = 5 the computed errors are shown in Table 2which show the high accuracy of the proposed method

4 Conclusions

In this paper Bernsteinrsquos approximation is used to approx-imate the solution of linear Volterra IDEs In this methodwe approximate our unknown function with Bernsteinrsquosapproximation The present results show that Bernsteinrsquosapproximation method for solving linear Volterra IDEs isvery effective and simple and the answers are trusty andtheir accuracy is high and we can execute this method in acomputer simplyThe numerical examples support this claim

Acknowledgment

The authors are very grateful to the referees for their valuablesuggestions and comments that improved the paper

References

[1] T L Bo L Xie and X J Zheng ldquoNumerical approach to windripple in desertrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 8 no 2 pp 223ndash228 2007

[2] L Xu J H He and Y Liu ldquoElectro spun nano-porous sphereswith Chinese drugrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 8 no 2 pp 199ndash202 2007

[3] R Agarwal and D OrsquoRegan Integral and integro-differentialequations theory methods and applications vol 2 The Gordonand breach science publishers Singapore 2000

[4] J Rashidinia and A Tahmasebi ldquoTaylor series method for thesystem of linear Volterra integro-differential equationsrdquo TheJournal of Mathematics and Computer Science vol 4 no 3 pp331ndash343 2012

[5] K Maleknejad F Mirzaee and S Abbasbandy ldquoSolving linearintegro-differential equations systemby using rationalizedHaarfunctions methodrdquo Applied Mathematics and Computation vol155 no 2 pp 317ndash328 2004

[6] K Maleknejad andM Tavassoli Kajani ldquoSolving linear integro-differential equation system by Galerkin methods with hydridfunctionsrdquo Applied Mathematics and Computation vol 159 no3 pp 603ndash612 2004

[7] E Yusufoglu ldquoAn efficient algorithm for solving integro-differ-ential equations systemrdquo Applied Mathematics and Computa-tion vol 192 no 1 pp 51ndash55 2007

[8] J Saberi-Nadjafi and M Tamamgar ldquoThe variational iterationmethod a highly promising method for solving the system ofintegro-differential equationsrdquo Computers amp Mathematics withApplications vol 56 no 2 pp 346ndash351 2008

[9] A Arikoglu and I Ozkol ldquoSolutions of integral and integro-differential equation systems by using differential transformmethodrdquo Computers amp Mathematics with Applications vol 56no 9 pp 2411ndash2417 2008

[10] J BiazarHGhazvini andM Eslami ldquoHersquos homotopy perturba-tion method for systems of integro-differential equationsrdquoChaos Solitons amp Fractals vol 39 no 3 pp 1253ndash1258 2009

[11] E Yusufoglu ldquoNumerical solving initial value problem for Fred-holm type linear integro-differential equation systemrdquo Journalof the Franklin Institute vol 346 no 6 pp 636ndash649 2009


[12] J Biazar and M Eslami ldquoModified HPM for solving systems ofVolterra integral equations of the second kindrdquo Journal of KingSaud University vol 23 no 1 pp 35ndash39 2011

[13] Y Huang and X-F Li ldquoApproximate solution of a class of linearintegro-differential equations by Taylor expansion methodrdquoInternational Journal of Computer Mathematics vol 87 no 6pp 1277ndash1288 2010

[14] A Karamete andM Sezer ldquoA Taylor collocationmethod for thesolution of linear integro-differential equationsrdquo InternationalJournal of Computer Mathematics vol 79 no 9 pp 987ndash10002002

[15] A Tahmasbi and O S Fard ldquoNumerical solution of linear Vol-terra integral equations system of the second kindrdquo AppliedMathematics and Computation vol 201 no 1-2 pp 547ndash5522008

[16] F Ghomanjani and M H Farahi ldquoThe Bezier control pointsmethod for solving delay differential equationrdquo Intelligent Con-trol and Automation vol 3 no 2 pp 188ndash196 2012

[17] J Zheng T W Sederberg and R W Johnson ldquoLeast squaresmethods for solving differential equations using Bezier controlpointsrdquo Applied Numerical Mathematics vol 48 no 2 pp 237ndash252 2004

[18] F Ghomanjani M H Farahi and M Gachpazan ldquoBeziercontrol points method to solve constrained quadratic optimalcontrol of time varying linear systemsrdquo Computational ampApplied Mathematics vol 31 no 3 pp 433ndash456 2012

[19] M I Berenguer A I Garralda-Guillem andM Ruiz Galn ldquoAnapproximation method for solving systems of Volterra integro-differential equationsrdquo Applied Numerical Mathematics vol 67pp 126ndash135 2013

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


26

24

22

2

18

16

14

12

10 02 04 06 08 1

t



2 Function Approximation

Our strategy is to use Bezier curves to approximate thesolutions 119910

119894(119905) for 1 le 119894 le 119899 which are given below

Define the Bezier polynomials of degree119873 that approximaterespectively the actions of 119910

119894(119905) over the interval [119905

0 119905119891] as

follows

119910119894 (119905) =

119873

sum119903=0

119886119894

119903119861119903119873

(119905 minus 1199050

ℎ) (3)

where ℎ = 119905119891minus 1199050and 119886119903is the control point of Bezier curve

and

119861119903119873

(119905 minus 1199050

ℎ) = (

119873

119903)

1

ℎ119873(119905119891minus 119905)119873minus119903

(119905 minus 1199050)119903 (4)

is the Bernstein polynomial of degree 119873 over the interval[1199050 119905119891] (see [17]) By substituting (3) in (2) 119877

119898(119905) can be

defined for 119905 isin [1199050 119905119891] as

119877119898(119905) =

119899

sum119894=1

120572119898119894

sum119895=0

119901119898119894119895

(119905) 119910(119895)

119894(119905)

+

119899

sum119894=1

int119905

0

(119896119898119894

(119905 119909)

120572119898119894

sum119895=0

119910(119895)

119894(119909))119889119909 minus 119891

119898(119905)

119898 = 1 2 119899

(5)

where (2) is satisfied The convergence was proved in theapproximation with Bezier curves when the degree of theapproximate solution119873 tends to infinity (see [18])

Now the residual function is defined over the interval[1199050 119905119891] as follows

119877 (119905) = int119905119891

1199050

119899

sum119898=1

1003817100381710038171003817119877119898 (119905)10038171003817100381710038172119889119905 (6)

0 02 04 06 08 1t


08

07

06

05

04

03

02

01

0


where sdot is the Euclidean norm Our aim is to solve thefollowing problem over the interval [119905

0 119905119891]

min 119877 (119905)

st 119910(0)

119894(1199050) = 1198881198940

119910(1)

119894(1199050) = 1198881198941 119910

(120572119898119894minus1)

119894(1199050) = 119888119894(120572119898119894

minus1)

(7)

The mathematical programming problem (7) can besolved by many subroutine algorithms and we used Maple12 to solve this optimization problem

Remark 1 In Chapter 1 of [19] it was proved that119873 satisfies

119873 gt119878

1205752120598 (8)

where 119878 = 119910119894(119905) and because of this reason that 119910

119894(119905) is

uniformly continuous on [1199050 119905119891] we have 119904 119905 isin [119905

0 119905119891] that

|119905 minus 119904| lt 120575 and minus(1205982) lt 119910119894(119905) minus 119910

119894(119904) lt 1205982 for more details

see [19]

3 Applications and Numerical Results

Consider the following examples which can be solved byusing the presented method

Example 1 Consider a system of third-order linear VolterraIDEs on the interval [0 1] (see [4])

11991010158401015840

1(119905) + 119905

21199101(119905) minus 119910

10158401015840

2(119905)

+ int119905

0

((119905 minus 119909) 1199101 (119909) + 1199102 (119909)) 119889119909 = 119892

1 (119905)

411990531199101015840

1(119905) + 6119905

21199101 (119905) + 119910

101584010158401015840

2(119905)

+ int119905

0

(1199101(119909) + (119905 + 119909) 119910

2(119909)) 119889119909 = 119892

2(119905)

(9)




2(119905)

00 0000000 0000000000002 14801 times 10


minus11

04 0162735585 times 10minus5


06 0251133963 times 10minus5

01536077787 times 10minus5


08864238659 times 10minus5





2(119905)

00 0000000 0000000000002 3840 times 10


minus11

04 05791064832 times 10minus3

0156041748480 times 10minus3

06 017373195072 times 10minus2

0156041748480 times 10minus3

08 069492781056 times 10minus2


10 0000 0000


1015840

1(0) = 1 119910

2(0) = 119910

10158401015840

2(0) =

0 and 1199101015840

2(0) = 1 where

1198921(119905) = (2 + 119905


1198922(119905) = 7 sin (119905) minus (1 + 2119905) cos (119905) + 119890

119905(1 + 4119905

2+ 41199053) + 119905 minus 1

(10)


119905 1199102(119905) =


1(119905) and




minus 1199101015840

1minus

1

21199051199101+

3

21199102

=5

2minus 119905 minus

27

21199052+ 1199054+

3

2(minus1 + 2119905

2) minus

1

2(minus3119905 + 4119905

3)

+ intt

minus1

(1199101minus 31199051199102) 119889119909

11990521199101+ 1199101015840

2minus 1199051199102

=2

5+ 3119905 + 3119905

3minus

8

51199055+ 1199052(minus3119905 + 4119905

3)

+ int119905

minus1

((2119905 + 119909) 1199101 + 311990921199102) 119889119909

(11)



solution 1199101(119905) = 4119905

3minus 3119905 119910

2(119905) = 2119905

2minus 1


4 Conclusions


Acknowledgment


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












2(119905)

00 0000000 0000000000002 14801 times 10


minus11

04 0162735585 times 10minus5


06 0251133963 times 10minus5

01536077787 times 10minus5


08864238659 times 10minus5





2(119905)

00 0000000 0000000000002 3840 times 10


minus11

04 05791064832 times 10minus3

0156041748480 times 10minus3

06 017373195072 times 10minus2

0156041748480 times 10minus3

08 069492781056 times 10minus2


10 0000 0000


1015840

1(0) = 1 119910

2(0) = 119910

10158401015840

2(0) =

0 and 1199101015840

2(0) = 1 where

1198921(119905) = (2 + 119905


1198922(119905) = 7 sin (119905) minus (1 + 2119905) cos (119905) + 119890

119905(1 + 4119905

2+ 41199053) + 119905 minus 1

(10)


119905 1199102(119905) =


1(119905) and




minus 1199101015840

1minus

1

21199051199101+

3

21199102

=5

2minus 119905 minus

27

21199052+ 1199054+

3

2(minus1 + 2119905

2) minus

1

2(minus3119905 + 4119905

3)

+ intt

minus1

(1199101minus 31199051199102) 119889119909

11990521199101+ 1199101015840

2minus 1199051199102

=2

5+ 3119905 + 3119905

3minus

8

51199055+ 1199052(minus3119905 + 4119905

3)

+ int119905

minus1

((2119905 + 119909) 1199101 + 311990921199102) 119889119909

(11)



solution 1199101(119905) = 4119905

3minus 3119905 119910

2(119905) = 2119905

2minus 1


4 Conclusions


Acknowledgment


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









numerical solution for ivp in volterra type linear...

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