Exact Solutions for Systems of Volterra Integral Equations of the First Kind by Homotopy Perturbation Method

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1 Applied Mathematical Sciences, Vol. 2, 28, no. 54, Eact Solutions for Systems of Volterra Integral Equations of the First Kind by Homotopy Perturbation Method J. Biazar 1, M. Eslami and H. Ghazvini Department of Mathematics, Faculty of Sciences Guilan University, P.O. Bo 1914, P.C , Rasht, Iran Abstract In this article, the homotopy perturbation method is proposed to solve systems of Volterra integral equations of the first kind. Theoretical proposes are presented briefly, so that this paper can be read independently. To illustrate the method some eamples are presented. The results reveal that the homotopy perturbation method is very effective and simple and gives the eact solutions. Keywords: Systems of Volterra integral equations; Homotopy perturbation method 1 Introduction Homotopy perturbation method is a powerful device for solving functional equations [1-7]. Until recently the application of the homotopy perturbation method in mathematical problems has been devoted by scientists and engineers [9-16]. Homotopy perturbation method without demanding a small parameter in equations deforms continuously to a simple problem which is easily solved. This method yields a very rapid convergence of the solution series in most cases, usually only a few iterations leading to very accurate solutions. This paper shows that some systems of Volterra integral equations can be easily calculated by the homotopy perturbation method. A system of integral equations of the first kind can be presented as k i (, t)g i (u 1 (t),u 2 (t)...,u n (t))dt = f i (),i=1, 2,...,n. (1) 1 Corresponding author. biazar@guilan.ac.ir

2 2692 J. Biazar, M. Eslami and H. Ghazvini If g i (u 1,...,u n ) are linear, the system (1) can be written as the following simple representation n k ij (, t)u j (t)dt = f i (),i=1, 2,,n. (2) Taking derivatives of order one from both sides of (2) leads to n n k ij (, t) k ij (, )u j ()+ u j (t)dt = f i (). (3) From ith equation, in (3), we write the ith unknown u i, in terms of the other unknown, to derive the following linear system of Volterra integral equations of the second kind u i () = f () n k ii (, ) k ij (, t) k ii (, ) u j()+ n k ij (, t)/ u j (t)dt. (4) k ii (, ) We continue the process of differentiation until u (j) i () = c i,i=1, 2,..., j =, 1, 2,..., and use homotopy perturbation method to solve the functional equations: u (j) i (). 2 Basic idea of homotopy perturbation method To illustrate the basic ideas of the method, we consider nonlinear differential equation with boundary conditions A(u) =f(r), r Ω, (5) B (u, u )=, r Γ, n where A is a general differential operator, B is a boundary operator, f(r) is a known analytic function, and Γ is the boundary of the domain Ω. The operator A can, generally speaking, be divided into two parts L and N, where L is linear, and N is nonlinear operator. Eq. (1), therefore, can be rewritten as follows: L(u)+N(u) f(r) =. By the homotopy technique [1], we construct a homotopy v(r, p) :Ω [, 1] R which satisfies H(v, p) =(1 p)[l(v) L(u )] + p [A(v) f(r)] =, p [, 1], r Ω

3 Volterra integral equations 2693 or H(v, p) =L(v) L(u )+pl(u )+p [N(v) f(r)] =, (6) where p [, 1] is an embedding parameter, u is an initial approimation of Eq. (5), which satisfies the boundary conditions. Obviously, from Eq. (6) we have H(v, ) = L(v) L(u )=, H(v, 1) = A(v) f(r) =, The changing process of p from zero to unity is just that of v(r, p) from u (r) to u(r). In topology, this is called deformation and quantities L(v) L(u ) and A(v) f(r) are called homotopic. Suppose, the solution of Eq. (6) can be written as a power series in p, v = v + pv 1 + p 2 v 2 +. Setting p = 1 results in approimate solution of Eq. (5) u = lim p 1 v = v + v 1 + v Numerical Eamples This section contains three eamples of linear and non-linear systems of integral equations of the first kind [17]. Eample 1. Consider the following system of linear integral equations of the first kind with the eact solutions: f() = 2 and g() = [17] { ((1 2 + t 2 )f(t) (2 t)g(t)) dt = , (( + t2 )f(t) (2 + t)g(t)) dt = Applying the process of deriving system of Volterra integral equations of the second kind from the first kind system leads to: { f() = g()+2 (f(t) g(t)) dt, g() = ( + 2 )f() 1 2 (f(t) g(t)) dt. Since f() = and g() =, we continue the process of differentiation until f (i) () = c and g (j) () = c, therefore, we have { f () = g ()+g ()+4f()+2f (), g () = f () f ()+ 1 g() f() f(). 2

4 2694 J. Biazar, M. Eslami and H. Ghazvini We can readily construct the homotopies which satisfies (1 p)(f () f ()) + p(f () G () G () 4F () 2F ()) =, (1 p)(g () g ()) + p(g () F () F () 1 G()+F ()+F ()) =. 2 Equating the terms with identical powers of p, we have p : f () =F () = 2, g () =G () =, p 1 : F 1 () =, G 1 () =. And by repeating this approach, we obtain F 2 () = F 3 () =... = and G 2 () =G 3 () =... =. Therefore, the approimate solutions of eample 1 can be readily obtained by f = F i = g = i= G i = +++ i= And hence, f() = 2, g() = which are the eact solutions of eample 1. Eample 2. Consider the following non-linear system of Volterra integral equations of the first kind, where the u() = 2 and v() = [17] { (1 2 + t 2 )(u(t)+v 3 (t)) dt = , (5 + t)(u3 (t) v(t)) dt = By differentiation we obtain { u() = v 3 ()+2 (u(t)+v3 (t)) dt , v() =u 3 ()+ 1 5 (u3 (t) v(t)) dt Since u() = and v() =, we continue the process of differentiation until u (i) () = c and v (j) () = c, therefore, we have u () = v()v 2 () 3v 2 ()v () +4u()+4v 3 ()+2u ()+6v 2 ()v (), v () = u 2 ()u ()+ 1 5 (u3 () v()). The homotopies of the differential equations are as follow (1 p)(u () u ()) + p(u () V ()V 2 ()+3V 2 ()V () 4U() 4V 3 () 2uU () 6V 2 ()V ()) =, (1 p)(v () v ()) + p(v () U 2 ()U () 1 5 (U 3 ()+V())) =.

5 Volterra integral equations 2695 Equating the terms with identical powers of p, we have p : u () =U () = 2, v () =V () =, p 1 : U 1 () =, V 1 () =. And by repeating this approach we obtain, V 2 () =V 3 () =... =,U 2 () = U 3 () =... =. Therefore, the approimate solutions of eample 2 can be readily obtained by u = i= U i = 2 +++, v = i= V i = +++. And hence, u() = 2,v() =, which are the eact solutions of eample 2. Eample 3. Consider the following system of integral equations of the first kind [17] { ((1 + t 2 )u 2 (t) + (1 + t) v(t)) dt = , ((2 2 t + 3 ) 3 u(t)+( t + t 2 +1)v 2 (t)) dt = , with the eact solutions u() = 3 and v() = 2. Applying the process of deriving systems of Volterra integral equations of the second kind from the first kind system leads to: { u 2 ()+ v()+ ((t 2)u2 (t)+ v(t)) dt = , 2u 1 3 ()+v 2 ()+ ( 2t +32 )u 1 3 (t) tv 2 (t) dt = Let consider the new functional f() =u 1 3 () and g() =v 1 2 (), then we have { g() = f 6 () ((t 2)f 6 (t)+g(t)) dt , f() = 1 2 g4 () 1 (( 2t )f(t) tg 4 (t)) dt Since f() = and g() =, we continue the process of differentiation until f (i) () = c and g (j) () = c, therefore, we have { f () = g 3 ()g () f()+ 1 2 g4 () 1 2 ( 2t +6)f(t) dt, g () = f 5 ()f ()+f 6 ()+g()+2 f 6 (t) dt. By using the homotopy perturbation method we have (1 p)(f () f ()) + p(f () G 3 ()G () F () 1 2 G4 ()+ 1 ( 2t +6)F (t) dt) =, 2 (1 p)(g () g ()) + p(g () F 5 ()F () F 6 () G() 2 F 6 (t) dt) =.

6 2696 J. Biazar, M. Eslami and H. Ghazvini Equating the terms with identical powers of p, we have p : f () =F () =, g () =G () =, p 1 : F 1 () =, G 1 () =. And by repeating this approach we obtain F 2 () = F 3 () =... = and G 2 () =G 3 () =... =. Therefore, the approimate solutions can be readily obtained by f = i= F i = +++, g = i= G i = +++. And hence, f() =, g() =, and the correspondent u() = 3,v() = 2, which are the eact solutions of eample 3. 4 Conclusion In this paper, we applied an application of the homotopy perturbation method for solving linear and non-linear systems of Volterra integral equations of the first kind. The results reveal that He s homotopy perturbation method is very effective and convenient and gives the eact solutions. References [1] J.H. He, Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering 178 (1999) [2] J.H. He, A coupling method of homotopy technique and perturbation technique for nonlinear problems, International Journal of Non-Linear Mechanics 35 (1) (2) [3] J.H. He, The homotopy perturbation method for nonlinear oscillators with discontinuities, Appl. Math. Comput. 151 (24) [4] J.H. He, Homotopy perturbation method: a new nonlinear analytical Technique, Appl. Math. Comput. 135 (23) [5] J.H. He, Application of homotopy perturbation method to nonlinear wave equation, Chaos, Solitons and Fractals 26 (25) [6] J.H. He, Homotopy perturbation method for bifurcation for nonlinear problems, Int. J. Nonlinear Sci. Numer. Simul 6 (2) (25) [7] J.H. He, Limit cycle and bifurcation of nonlinear problems, Chaos, Solitons and Fractals 26 (3) (25)

7 Volterra integral equations 2697 [8] A.H. Nayfeh, Problems in perturbation. Newyork john wilkey: [9] D.D. Ganji, The application of He s homotopy perturbation method to nonlinear equations arising in heat transfer, Physics Letters A 355 (26) [1] Z.M. Odibat, S.Momani, Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlinear Sci. Numer. Simul. 7(1) (26) [11] N. Bildik, A. Konuralp, The use of Variational Iteration Method, Differential Transform Method and Adomian Decomposition Method for solving different types of nonlinear partial differential equations, Int. J. Nonlinear Sci. Numer. Simul. 7(1) (26) [12] J.H. He, Variational iteration method: a kind of nonlinear analytical technique: some eamples, Int. J. Non-linear Mech.34 (4) (1999) [13] J.H. He, Comparison of homotopy perturbation method and homotopy analysis method, Appl. Math. Comput. 156 (24) [14] S. Abbasbandy, Application of the integral equations: Homotopy perturbation method and Adomian s decomposition method, Appl. Math. Comput. 173 (2-3) (26) [15] J. Biazar, H. Ghazvini, He s homotopy perturbation method for solving systems of Volterra integral equations of the second kind, Chaos, Solitons and Fractals [In press]. [16] J. Biazar, M. Eslami, H. Ghazvini, Homotopy perturbation method for systems of partial differential equations, Int. J. Nonlinear Sci. Numer. Simul. 8 (3) (27) [17] J. Biazar, E. Babolian, R. Islam, Solution of a system of Volterra integral equations of the first kind by Adomian method, Appl. Math. Comput. 139 (23) Received: December 4, 27