Fredholm Integral Equations of the First Kind Solved by Using the Homotopy Perturbation Method

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1 Int. Journl of Mth. Anlysis, Vol. 5, 211, no. 19, Fredholm Integrl Equtions of the First Kind Solved by Using the Homotopy Perturbtion Method Seyyed Mhmood Mirzei Deprtment of Mthemtics, Fculty of Science, Minoodsht Brnch Islmic Azd University, Minoodsht, Irn Abstrct In this reserch, numericl solution for solving the Fredholm integrl equtions is considered. An ppliction of homotopy perturbtion method is pplied to solve the Fredholm integrl equtions. the results revel tht the homotopy perturbtion method is very effective nd simple nd gives the exct solution. For illustrtion nd more explntion of the ide, three exmples re provided. Mthemtics Subject Clssifiction: 45B5, 34D1 Keywords: Homotopy Perturbtion Method, Fredholm integrl eqution 1 Introduction Homotopy perturbtion method introduced by He [3, 4, 5, 6] hs been used by mny mthemticins nd engineers to solve vrious functionl equtions. In this method the solution is considered s the sum of n infinite series which converges rpidly to the ccurte solutions. We re frequently fced with the problem of determining the solution of integrl equtions, one of these integrl equtions is the Fredholm integrl eqution where defined in [7, 8, 9]. In recent yers, lrge mount of literture developed concerning the modified decomposition method introduced by Wzwz by pplying it to lrge size of pplictions in pplied sciences. A new perturbtion method clled homotopy perturbtion method (HPM) ws proposed by He in 1997 nd systemticl description in 2 which is, in fct, coupling of the trditionl perturbtion method nd homotopy in topology [1]. Until recently, the ppliction of the HPM [8] in integrl equtions hs been developed by scientists nd engineers, becuse this method is the most effective nd convenient ones for both wekly nd strongly integrl equtions. In this pper, this method is pplied for the

2 936 S. M. Mirzei Fredholm integrl equtions. The Fredholm integrl equtions re given by φ(x) =f(x)+ K(x, y)φ(y)dy, x b. (1) φ(x) is unknown function tht will be determined, K(x, y) is the kernel of the integrl eqution, f(x) is n nlytic function. 2 Preliminry Notes This section is devoted to reviewing HPM for solving the Fredholm integrl eqution. To illustrte the HPM, we consider (1) s L(u) =u(x) f(x) K(x, y)u(y)dy =, x b. (2) With solution u(x) = φ(x), we define the homotopy H(u, p) by H(u, ) = F (u), H(u, 1) = L(u), where F (u) is functionl opertor with solution, sy, u, which cn be obtined esily. We my choose convex homotopy H(u, p) =(1 p)f (u)+pl(u) =, (3) nd continuously trce n implicitly defined curve from strting point H(u, ) to solution function H(U, 1). The embedding prmeter p monotoniclly increse from zero to unit s the trivil problem L(u) =. The embedding prmeter p [, 1] cn be considered s n expnding prmeter. The HPM uses the homotopy prmeter p s expnding prmeter to obtin u = u + pu 1 + p 2 u 2 +. (4) When p 1,(4) corresponds to (3) becomes the pproximte solution of (2), i.e., U = lim u = u + u 1 + u 2 +. (5) The series (5) is convergent for most cses, nd lso the rte of convergent depends on L(u). Tking F (u) =u(x) f(x), nd substituting (4) in (3) nd equting the terms with identicl power of p, we obtin p : u f(x) = = u = f(x), p 1 : u 1 K(x, y)u (y) dy =,

3 Fredholm integrl equtions 937 u 1 =. nd in generl we hve u (x) =f(x), u n+1 (x) = K(x, y)u (y) dy =, K(x, y)u n (y) dy, n =1, 2, 3,... which is the stndrd Adomin s decomposition method. 3 Numericl Exmple This section contined three exmple of Fredholm integrl equtions. Exmple 3.1 Consider the following Fredholm integrl eqution φ(x) = 5 cos x cos x sin2 (1) +.1 sin y cos xφ(y)dy. (6) Exct solution of this eqution is φ(x) = 5 cos x. We define F (u) =u(x) 5 cos(x), L(u) =u(x) 5 cos(x) cos x sin2 (1).1 sin y cos xu(y)dy, nd substituting F (u) nd L(u) in (3) nd equting the terms whit identicl power of p, we obtin p : u (x) = 5 cos(x), p 1 : u 1 (x) = cos x sin2 (1) +.1 sin y cos xu (y)dy =,

4 938 S. M. Mirzei p k+2 : u k+2 (x) = cos x sin2 (1)+.1 sin y cos xu k+1 (y)dy =, such tht k.with using (5) we hve U(x) = u(x) = lim u = u (x)+u 1 (x)+u 2 (x)+ = 5 cos(x). Exmple 3.2 Consider the following Fredholm integrl eqution φ(x) =x 3 +2x + 1 ( 2 15 x 13 ) 1 15 x2 +.5 y ( x 2 +2x ) φ(y)dy. (7) Exct solution of this eqution is φ(x) =x 3 +2x. We define F (u) =u(x) x 3 2x, ( 15 x 13 ) 1.5 y ( x 2 +2x ) u(y)dy, L(u) =u(x) x 3 2x x2 nd substituting F (u) nd L(u) in (3) nd equting the terms whit identicl power of p, we obtin p : u (x) =x 3 +2x, p 1 : u 1 (x) = 1 ( 2 15 x 13 ) 1 15 x2.5 y ( x 2 +2x ) u (y)dy, p k+2 : u k+2 (x) = 1 ( 2 15 x 13 ) 1 15 x2.5 y ( x 2 +2x ) u k+1 (y)dy =, such tht k.with using (5) we hve U(x) = u(x) = lim u = u (x)+u 1 (x)+u 2 (x)+ = x 3 +2x.

5 Fredholm integrl equtions 939 Exmple 3.3 Consider the following Fredholm integrl eqution φ(x) = x x2 +.1 x.75 yφ(y)dy. (8) Exct solution of this eqution is φ(x) =x 2. We define F (u) =u(x) x 2, L(u) =u(x) x 2 + x x2.1 x.75 yu(y)dy, nd substituting F (u) nd L(u) in (3) nd equting the terms whit identicl power of p, we obtin p : u (x) =x 2, p 1 : u 1 (x) = x x.75 yu (y)dy, p k+2 : u k+2 (x) = x x.75 yu k+1 (y)dy =, such tht k.with using (5) we hve U(x) = u(x) = lim u = u (x)+u 1 (x)+u 2 (x)+ = x 2. Conclusion In this pper, He s homotopy perturbtion method hs been Exctly nd successfully pplied to Finding the solution of Fredholm integrl equtions of the second kind hs been shown. The pproximte solutions obtined by the homotopy perturbtion method re compred with exct solutions. It cn be concluded tht the He s homotopy perturbtion method is Very Strong nd effective nd Applicble technique in finding exct solutions for wide clsses of problems.

6 94 S. M. Mirzei References [1] S. M. Mirzei, Homotopy Perturbtion Method for Solving the Second Kind of Non-Liner Integrl Equtions, Interntionl Mthemticl Forum, 5, 21, no. 23, [2] M.Amirfkhrin, S. M. Mirzei, Modified Neumnn Series for Solving Fredholm Integrl Eqution, Applied Mthemticl Sciences, Vol. 3, 29, no. 13, [3] J. H. Hi, The homotopy perturbtion method for nonliner oscilltors with discontinuities, Applied Mthemtics nd Computtion, 151 (24) [4] J. H. Hi, Appliction of homotopy perturbtion method to wve equtions, Chos, Solitons nd Frctls 26 (25) [5] J. H. Hi, Homotopy perturbtion method for solving boundry vlue problems, Physics Letters A 35 (26) [6] J. H. Hi, Limit cycle nd bifurction of nonliner problems, Chos, Solitons nd Frctls 26 (3) (25) [7] Kythe P.K., Puri P., Computtionl methods for liner integrl eqution,birkhuser, Bosten, 22. [8] Wzwz A.M., A First Course in Integrl Equtions, New Jersey : World Scientific ; [9] Delves L.M., Mohmed J.L., Computtionl methods for integrl equtions, Cmbridge University Press, [1] He. J. H., A coupling method of homotopy technique nd perturbtion techique for non-liner problems, Int. J. Non-Liner Mech. 2;35(1): Received: November, 21