Chaos, Solitons and Fractals 42 (2009) 2760–2764

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Chaos, Solitons and Fractals

journal homepage: www.elsevier .com/locate /chaos

Homotopy perturbation method for the mixed Volterra–Fredholmintegral equations

Ahmet YildirimDepartment of Mathematics, Science Faculty, Ege University, 35100 Bornova-_Izmir, Turkey

a r t i c l e i n f o

Article history:Accepted 31 March 2009

0960-0779/$ - see front matter � 2009 Elsevier Ltddoi:10.1016/j.chaos.2009.03.147

E-mail address: ahmet.yildirim@ege.edu.tr

a b s t r a c t

This article presents a numerical method for solving nonlinear mixed Volterra–Fredholmintegral equations. The method combined with the noise terms phenomena may providethe exact solution by using two iterations only. Two numerical illustrations are given toshow the pertinent features of the technique. The results reveal that the proposed methodis very effective and simple.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

The solution of the mixed Volterra–Fredholm integral equation has been a subject of considerable interest. This equationarises in the theory of parabolic boundary value problems, the mathematical modeling of the spatio-temporal developmentof an epidemic and various physical and biological problems.

This paper outlines a reliable strategy for solving the nonlinear mixed Volterra–Fredholm integral equation:

uðx; tÞ ¼ f ðx; tÞ þZ t

0

ZX

Fðx; t; n; s;uðn; sÞÞdnds; ðx; tÞ 2 Xx½0; T�; ð1Þ

where u(x,t) is an unknown function, and f(x,t) and F(x,t,n,s,u(n,s)) are analytic functions on D = Xx[0,T], where X is a closedsubset of Rn, n = 1,2,3. The literature which discusses the existence and the uniqueness results are contained in [1–11].However, few numerical methods for solving Eq. (1) are known. For the linear case, the time collocation method was intro-duced by Pachpatta [2] and the projection method by Hacia [3]. Brunner [7] extended the Pachpatta’s [2] results to nonlinearVolterra–Hammerstein integral equations, while Maleknejad and Hadizadeh [9] and Wazwaz [10] used a technique based onthe Adomian decomposition method for solution of Eq. (1). Also Banifatemi et al. [11] applied two-dimensional LegendreWavelets Method to mixed Volterra–Fredholm integral equations.

In this letter, we will use homotopy perturbation method (HPM) to study mixed Volterra–Fredholm integral equations.The HPM introduced by He [12–18]. In this method the solution is considered as the summation of an infinite series

which usually converges rapidly to the exact solutions.Using homotopy technique in topology, a homotopy is constructedwith an embedding parameter p2 [0,1] which is considered as a ‘‘small parameter”. Considerable research works have beenconducted recently in applying this method to a class of linear and non-linear equations [19–35]. We extend the method tosolve mixed Volterra–Fredholm integral equations.

2. Basic ideas of homotopy perturbation method

To illustrate the basic idea of He’s homotopy perturbation method, consider the following general nonlinear differentialequation;

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A. Yildirim / Chaos, Solitons and Fractals 42 (2009) 2760–2764 2761

AðuÞ � f ðrÞ ¼ 0; r 2 X ð2Þ

with boundary conditions;

Bðu; ou=onÞ ¼ 0; r 2 C ð3Þ

where A is a general differential operator, B is a boundary operator, f(r) is a known analytic function, C is the boundary of thedomain X.

The operator A can, generally speaking, be divided in to two parts L and N, where L is linear, and N is nonlinear, thereforeEq. (2) can be written as,

LðuÞ þ NðuÞ � f ðrÞ ¼ 0: ð4Þ

By using homotopy technique, one can construct a homotopy v(r,p):X � [0,1] ? R which satisfies

Hðv ;pÞ ¼ ð1� pÞ LðvÞ � Lðu0Þ½ � þ p½AðvÞ � f ðrÞ� ¼ 0; p 2 ½0;1�; ð5aÞ

or

Hðv ;pÞ ¼ LðvÞ � Lðu0Þ þ pLðu0Þ þ p½NðvÞ � f ðrÞ� ¼ 0; ð5bÞ

where p 2 [0,1] is an embedding parameter, and u0 is the initial approximation of Eq. (2) which satisfies the boundary con-ditions. Clearly, we have

Hðv ;0Þ ¼ LðvÞ � Lðu0Þ ¼ 0; ð6ÞHðv ;1Þ ¼ AðvÞ � f ðrÞ ¼ 0; ð7Þ

the changing process of p from zero to unity is just that of v(r,p) changing from u0(r)to u(r). This is called deformation, andalso, L(v) � L(u0) and A(v) � f(r) are called homotopic in topology. If, the embedding parameter p; (0 6 p 6 1) is considered asa ‘‘small parameter”, applying the classical perturbation technique, we can naturally assume that the solution of Eqs. (6) and(7) can be given as a power series in p, i.e.,

v ¼ v0 þ pv1 þ p2v2 þ . . . ; ð8Þ

and setting p = 1 results in the approximate solution of Eq. (2) as;

u ¼ limp!1

v ¼ v0 þ v1 þ v2 þ . . . ð9Þ

The convergence of series (9) has been proved by He in his paper [16]. It is worth to note that the major advantage of He’shomotopy perturbation method is that the perturbation equation can be freely constructed in many ways (therefore is prob-lem dependent) by homotopy in topology and the initial approximation can also be freely selected. Moreover, the construc-tion of the homotopy for the perturb problem plays very important role for obtaining desired accuracy [20]. This homotopyperturbation method will become a much more interesting method to solving nonlinear problems in science andengineering.

3. The homotopy perturbation method applied to mixed Volterra–Fredholm integral equations

Consider the mixed Volterra–Fredholm integral equation

uðx; tÞ ¼ f ðx; tÞ þZ t

0

ZX

Fðx; t; n; s;uðn; sÞÞdnds; ðx; tÞ 2 Xx½0; T�; ð10Þ

where u(x,t) is an unknown function, and f(x,t) and F(x,t,n,s,u(n,s)) are analytic functions on D = Xx[0,T], where X is a closedsubset of Rn, n = 1,2,3.

We assume that the function f can be divided into the sum of two parts, namely f0 and f1, therefore we set

f ¼ f0 þ f1: ð11Þ

We re-write Eq. (10) as

uðx; tÞ ¼ f0ðx; tÞ þ f1ðx; tÞ þZ t

0

ZX

Fðx; t; n; s;uðn; sÞÞdnds; ðx; tÞ 2 Xx½0; T�; ð12Þ

We may choose a homotopy such that

uðx; tÞ � f0ðx; tÞ � pðf1ðx; tÞ þZ t

0

ZX

Fðx; t; n; s;uðn; sÞÞdndsÞ ¼ 0; ð13Þ

Substituting (8) into (13), and equating the terms with identical powers of p, we have

2762 A. Yildirim / Chaos, Solitons and Fractals 42 (2009) 2760–2764

p0 : u0ðx; tÞ ¼ f0ðx; tÞ; ð14Þ

p1 : u1ðx; tÞ ¼ f1ðx; tÞ þZ t

0

ZX

F x; t; n; s; u0ðn; sÞð Þdnds; ð15Þ

��

pk : ukðx; tÞ ¼Z t

0

ZX

F x; t; n; s;uk�1ðn; tÞð Þdnds; ð16Þ

Relation (14)–(16) will enable us to determine the components un (x,t),n P 0 recurrently, and as a result, the series solutionof u(x,t) is readily obtained.

4. Numerical results

To give a clear overview of the analysis presented above, we have chosen to present two test problems: the first is linearexamined by [7,9] and the second is nonlinear, studied by [8,9].

Example 1. (See [7,9]) We first consider the linear mixed Volterra–Fredholm integral equation

uðx; tÞ ¼ f ðx; tÞ þZ t

0

ZX

Fðx; t; n; sÞuðn; sÞdnds; ðx; tÞ 2 Xx½0; T�; ð17Þ

where

f ðx; tÞ ¼ e�t cosðxÞ þ t cosðxÞ þ 12

t cosðx� 2Þ sinð2Þ� �

; ð18Þ

and

Fðx; t; n; sÞ ¼ � cosðx� nÞe�ðt�sÞ; ð19Þ

with X = [0,2].As stated before, we decompose f(x,t) into two parts as follows:

f0ðx; tÞ ¼ e�t cosðxÞ þ t cosðxÞ½ �; ð20Þ

f1ðx; tÞ ¼12

te�t cosðx� 2Þ sinð2Þ: ð21Þ

The homotopy perturbation technique admits the use of the recursive relation given in (14)–(16) in the form

u0ðx; tÞ ¼ e�t½cosðxÞ þ t cosðxÞ�; ð22Þ

u1ðx; tÞ ¼12

te�t cosðx� 2Þ sinð2Þ �Z t

0

ZX

cosðx� nÞe�ðt�sÞ u0ðx; tÞð Þdnds; ð23Þ

ukþ1ðx; tÞ ¼ �Z t

0

ZX

cosðx� nÞe�ðt�sÞ ukðx; tÞð Þdnds; k P 1: ð24Þ

This gives

u0ðx; tÞ ¼ e�t½cosðxÞ þ t cosðxÞ�; ð25Þ

u1ðx; tÞ ¼ �te�t cosðxÞ þ 12

te�t 12

sinðxþ 4Þ þ sin2ð2Þ sinðxÞ� �

� 12

te�t 12

cosðxÞ sinð4Þ þ 12

sinðxÞ� �

þ 18

t2e�t sinðx� 4Þ � sinðxÞ � 14

cosðxÞ� �

: ð26Þ

It is obvious that self-canceling noise terms te�tcos(x) and � te�tcos(x), identical terms with opposite signs, appear betweenthe components u0(x,t), and justifying that the remaining non-canceled term of u0(x,t) justify the equation, the exact solution

uðx; tÞ ¼ e�t cosðxÞ ð27Þ

is readily obtained.

Example 2. (See [8,9]) We consider the nonlinear mixed Volterra–Fredholm integral equation.

uðx; tÞ ¼ f ðx; tÞ þZ t

0

ZX

Gðx; t; n; sÞ 1� e�uðn;sÞ� �dnds; ðx; tÞ 2 Xx½0; T�; ð28Þ

where

A. Yildirim / Chaos, Solitons and Fractals 42 (2009) 2760–2764 2763

f ðx; tÞ ¼ � ln 1þ xt1þ t2

� �þ xt2

8ð1þ tÞ 1þ t2� � ; ð29Þ

and

Gðx; t; n; sÞ ¼x 1� n2� �

ð1þ tÞ 1þ s2ð Þ ; ð30Þ

with X = [0,1].We first decompose f(x,t) into two components as follows:

f0ðx; tÞ ¼ � ln 1þ xt1þ t2

� �; ð31Þ

f1ðx; tÞ ¼xt2

8ð1þ tÞ 1þ t2� � : ð32Þ

The homotopy perturbation technique admits the use of the recursive relation given in (14)–(16) in the form

u0ðx; tÞ ¼ � ln 1þ xt1þ t2

� �; ð33Þ

u1ðx; tÞ ¼xt2

8ð1þ tÞ 1þ t2� �þ

Z t

0

ZX

Gðx; t; n; sÞ 1� e�u0ðn;sÞ� �

dnds; ð34Þ

��

This gives

u0ðx; tÞ ¼ � ln 1þ xt1þ t2

� �; ð35Þ

u1ðx; tÞ ¼ 0; ð36Þujðx; tÞ ¼ 0; j P 2: ð37Þ

The exact solution

uðx; tÞ ¼ � ln 1þ xt1þ t2

� �; ð38Þ

follows immediately.

5. Conclusion

In this paper, the HPM was employed to solve linear and nonlinear mixed Volterra–Fredholm integral equations. Themethod needs much less computational work compared with traditional methods. It is shown that HPM is a very fastconvergent, precise and cost efficient tool for solving integral equations in the bounded domains.

Acknowledgement

Author thanks to TÜB_ITAK (The Scientific and Technological Research Council Of Turkey) for their financial support.

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