A Modified Homotopy Perturbation Method for Solving Linear and Nonlinear Integral Equations. 1 Introduction

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1 ISSN (print), (online) Interntionl Journl of Nonliner Science Vol.13(212) No.3,pp A Modified Homotopy Perturbtion Method for Solving Liner nd Nonliner Integrl Equtions N. Aghzdeh, S. Mohmmdi Deprtment of Applied Mthemtics, Azrbijn University of Trbit Mollem, Tbriz , Irn (Received 23 June 211, ccepted 15 Jnury 212) Abstrct: In this pper, we present modifiction to so-clled homotopy perturbtion method for solving liner nd non-liner integrl equtions. This method gives n pproximte nlytic solution to the equtions (usully the exct solution of the equtions). Some numericl exmples presented to show the ccurcy nd efficiency of the method. Keywords: homotopy perturbtion method; integrl eqution; Fredholm; Volterr 1 Introduction Although perturbtion techniques re widely pplied to nlyze nonliner problems in science nd engineering, they re however so strongly dependent on smll prmeters ppered in equtions under considertion tht they re restricted only to wekly nonliner problems. For strongly nonliner problems which don t contin ny smll prmeters, perturbtion techniques re invlid. So, it seems necessry nd worthwhile developing t new kind of nlytic technique independent of smll prmeters. Lio proposed new nlytic technique in his Ph.D. disserttion [1], nmely the Homotopy Anlysis Method (HAM). Bsed on homotopy of topology, the vlidity of the HAM is independent of whether or not there exist smll prmeters in considered equtions. Therefore, the HAM cn overcome the foregoing restrictions nd limittions of perturbtion techniques so tht it provides us with powerful tool to nlyze strongly nonliner problems. [2] In [2] some bsic ides bout the HAM ws described. In [3] some developments of the HAM ws presented. Also some lemms nd theorems ws proved. In [4] relible pproch for convergence of the HAM ws discussed. In [1] [14] [46] [48] [51] [53] the HAM ws pplied on some equtions. Also, some modifictions nd improvements ws discussed by uthors (e.g. see [15] [17]). In [19, 2] the homotopy perturbtion technique ws presented. In [21] [41] [49] [5][54] [55] the homotopy perturbtion technique ws pplied on different equtions by some uthors nd with some modifictions (e.g. liner nd nonliner forth-order boundry vlue problems, functionl integrl equtions, nonliner problems, system of nonliner Fredholm integrl equtions, forth-order integro-differentil equtions, eighth-order boundry vlue problems, nonliner oscilltors, prtil differentil equtions, qudrtic Riccti differentil eqution, Volterr integrl equtions, two-dimensionl Fredholm integrl equtions, Stokes equtions nd nonliner ill-posed opertor equtions). Also the homotopy perturbtion method nd the HAM ws compred by some uthors (e.g. [5] [9]). We now review [39] to show how HPM pplied to the following integrl equtions. Consider the following integrl eqution: Let γ(x) = f(x) + L(u) = u(x) f(x) k(x, t)γ(t)dt, c x d. (1) k(x, t)u(t)dt =, (2) Corresponding uthor. E-mil ddress: ghzdeh@zruniv.edu Copyright c World Acdemic Press, World Acdemic Union IJNS /61

2 N. Aghzdeh, S. Mohmmdi: A Modified Homotopy Perturbtion Method for Solving Liner nd Nonliner Integrl Equtions 39 with solution u(x) = γ(x), we cn define homotopy H(u, p) by H(u, ) = F (u), H(u, 1) = L(u), (3) where F (u) is functionl opertor with solution u.we choose convex homotopy H(u, p) = (1 p)f (u) + pl(u) =, (4) nd continuously trce n implicitly defined curve from strting point H(u ) to solution H(γ, 1). In fct HPM uses the homotopy prmeter p s n expnding prmeter [43, 44] to obtin u = u + pu 1 + p 2 u 2 +, (5) when p 1, (5) corresponds to (4) nd gives n pproximtion to the solution of (2) s: γ = lim p 1 u = u + u 1 + u 2 +, (6) The series (6) converges in most cses, nd the rte of convergence depends on L(u). Tking F (u) = u(x) f(x), nd substituting (5) in (4) nd equting the terms with identicl power of p, we obtin nd in generl we hve 2 The new modified HPM p : u f(x) = u = f(x), p 1 : u 1. u 1 = k(x, t)u (t)dt =, k(x, t)u (t)dt, u (x) = f(x), (7) u n+1 (x) = k(x, t)u n (t)dt, n = 1, 2,.... (8) In [39] modifiction pplied to HPM for liner integrl equtions with degenerte kernels, which used number m. In this pper, m is function (or number), nd our modifiction cn be pplied to liner or non-liner integrl equtions, nd there is no limittion on kernel type. 2.1 Appliction to liner second kind Fredholm integrl eqution In this section, we pply the modified perturbtion method to (1). To this end, we define new convex homotopy perturbtion s H(u, p, m) = (1 p)f (u) + pl(u) + p(1 p)mk f =, (9) where F (u) = u(x) f(x), L(u) = u(x) f(x) k(x, t)u(t)dt = nd K f = k(x, t)f(t)dt, hence we cn write [ ] or (1 p)(u f) + p u f p u f k(x, t)u(t)dt + p(1 p)mk f =, (1) k(x, t)u(t)dt + p(1 p)mk f =, (11) IJNS homepge:

3 31 Interntionl Journl of Nonliner Science, Vol.13(212), No.3, pp Substituting (5) into (11) nd equting the coefficients of like terms with the identicl powers of p, we obtin p : p 1 : p 2 : p 3 :. u f(x) = u = f(x), u 1 k(x, t)u (t)dt + mk f =, u 1 = (1 m)k f, u 2 k(x, t)u 1 (t)dt mk f =, u 2 = (1 m)k K f + mk f, u 3 u 3 = p n+1 : u n+1 = k(x, t)u 2 (t)dt =, k(x, t)u 2 (t)dt, k(x, t)u n (t)dt, n = 2, 3,..., now we find m such tht u 2 =. Since if u 2 = then u 3 = u 4 = =, nd the exct solution will be obtined s u(x) = u (x) + u 1 (x), hence for ll vlues of x we should hve (1 m)k K f + mk f =, or K K f K K f K f Note tht the method cn be pplied s for Volterr integrl equtions, in sme mnner. 2.2 Appliction to non-liner Fredholm integrl equtions Consider the following non-liner Fredholm integrl eqution u(x) = f(x) + k(x, t)t (u(t))dt, x b (12) where the function k is given nd T is given nonliner opertor, nd u the solution to be determined. We ssume tht (12) hs the unique solution. We define convex homotopy perturbtion s where hence, we cn write or H(u, p, m) = (1 p)f (u) + pl(u) + p(1 p)mk T (f) =, (13) F (u) = u(x) f(x) nd L(u) = u(x) f(x) (1 p)(u f) + p Substituting (5) into (16) results into [ u f p u + pu 1 + p 2 u 2 + f(x) p u f k(x, t)u(t)dt ] k(x, t)t (u(t))dt =, (14) + p(1 p)mk T (f) =, (15) k(x, t)u(t)dt + p(1 p)mk T (f) =, (16) k(x, t)t (u + pu 1 + p 2 u 2 + )dt + p(1 p)mk T (f) = (17) IJNS emil for contribution: editor@nonlinerscience.org.uk

4 N. Aghzdeh, S. Mohmmdi: A Modified Homotopy Perturbtion Method for Solving Liner nd Nonliner Integrl Equtions 311 In (17) we cn write T (u + pu 1 + p 2 u 2 + ) s follows T (u + pu 1 + p 2 u 2 + ) = A + pa 1 + p 2 A 2 +, (18) where A k re Adomin polynomils which depend upon u, u 1, u 2,..., u k. [39, 45] By differentiting both sides of (18) we cn write d k dp k T (u + pu 1 + p 2 u 2 + ) p= = dk dp k (A + pa 1 + p 2 A 2 + ) p=. (19) From (19) we hve d k A k = A k (u, u 1, u 2,..., u k ) = 1 k! dp k T (u + pu 1 + p 2 u 2 + ) p=, k =, 1,... (2) By substituting (18) into (17) we hve u + pu 1 + p 2 u 2 + f(x) p Equting the terms with identicl powers of p, we hve p : p 1 : p 2 : p 3 :. k(x, t)(a + pa 1 + p 2 A 2 + )dt + p(1 p)mk T (f) =. (21) u f(x) = u = f(x), u 1 k(x, t)a (t)dt + mk T (f) =, u 1 = (1 m)k T (f), u 2 k(x, t)a 1 (t)dt mk T (f) =, u 2 = K ((1 m)k T (f)t (f)) + mk T (f), u 3 u 3 = p n+1 : u n+1 = k(x, t)a 2 (t)dt =, k(x, t)a 2 (t)dt, k(x, t)a n (t)dt, n = 3, 4..., now we find m such tht u 2 =. Since if u 2 = then u 3 = u 4 = =, nd the exct solution will be obtined s u(x) = u (x) + u 1 (x), hence for ll vlues of x we should hve or (1 m)k (K T (f)t (f)) + mk T (f) =, K (K T (f)t (f)) K (K T (f)t (f)) K T (f) Note tht the method cn be pplied s for Volterr integrl equtions, in sme mnner. 3 Numericl Exmples Exmple 1 Consider the eqution u(x) = (1 2π) cos x + sin x + with exct solution u(x) = sin x + cos x. Using the method, we hve u = f(x), π u 1 = (1 m)k f, where 4 cos x cos tu(t)dt, (22) K K f K K f K f IJNS homepge:

5 312 Interntionl Journl of Nonliner Science, Vol.13(212), No.3, pp nd K f = u(x) = u + u 1. k(x, t)f(t)dt In this exmple K f = 2(1 2π)π cos x nd K K f = 4(1 2π)π 2 cos x hence 2π 1 + 2π, u = (1 2π) cos x + sin x, u 1 = 2π cos x so we hve u(x) = cos x + sin x, which is the exct solution. Exmple 2 Consider the following Fredholm integrl eqution u(x) = e 2x e 2x+ 5 3 t u(t)dt, with exct solution u(x) = e 2x. In this exmple K f = ( 1 + e 1 3 )e x nd K K f = ( 1 + e 1 3 ) 2 e x hence 1 1 e 1 3 so we hve u(x) = e 2x, which is the exct solution. u = e 2x+ 1 3, Exmple 3 Consider the following Volterr integrl eqution u 1 = ( 1 + e 1 3 )e 2x u(x) = 12x + x 2 2x 3 x6 11 sin x x (x t) 3 u(t)dt, with exct solution u(x) = x 2 + sin x. In this exmple ) K f = 2 (66x 11x 3 + 3x5 5 + x6 6 x7 7 x1 66 sin x 252 nd hence K K f = 1584x 264x x5 5 11x x x1 126 x x sin x x( x x x x 8 22x x 1 + x 13 ) sin x x( x x x x x 8 44x x 1 + x 13 ) sin x u = 12x + x 2 2x 3 x6 11 sin x, 3 u 1 = 143(x( x x 4 42x x 6 + x 9 ) sin x) 2 9(x( x x x x x 8 44x x 1 + x 13 ) sin x) u(x) = 12x + x 2 2x 3 x6 11 sin x (x( x x 4 42x x 6 + x 9 ) sin x) 2 9(x( x x x x x 8 44x x 1 + x 13 ) sin x) IJNS emil for contribution: editor@nonlinerscience.org.uk

6 N. Aghzdeh, S. Mohmmdi: A Modified Homotopy Perturbtion Method for Solving Liner nd Nonliner Integrl Equtions 313 Exmple 4 Consider the following Volterr integrl eqution u(x) = x x sinh(x t)u(t)dt, with exct solution u(x) = x x3 6. Here, K f = x sinh x nd K K f = x sinh x 2x cosh x 2 hence 2x + x cosh x 3 sinh x x cosh x sinh x u = x, 2(x sinh x)2 u 1 = x cosh x sinh x u(x) = 2x2 + x 2 cosh x + 3x sinh x 2 sinh x 2 x cosh x sinh x Exmple 5 Consider the following Volterr integrl eqution u(x) = 1 x 16 (7 cos x + 9 cos 3x + 4x sin x) (x t) cos(x t)u(t)dt, with exct solution u(x) ( = 1 3 (2 cos 3x + 1). Here, K f = ( x 2 ) cos x cos 3x 4x(15 + 8x 2 ) sin x ) nd hence K K f = 1 ( 5( x x 4 ) cos x cos 3x + 4x( x x 4 ) sin x ) ( x x 4 ) cos x cos 3x + 4x( x x 4 ) sin x 5( x x 4 ) cos x cos 3x + 4x( x x 4 ) sin x u = 1 (7 cos x + 9 cos 3x + 4x sin x), 16 5(3( x 2 ) cos x 135 cos 3x + 4x(15 + 8x 2 ) sin x) 2 u 1 = 48(5( x x 4 ) cos x cos 3x + 4x( x x 4 ) sin x) u(x) = 1 ( 96(7 cos x + 9 cos 3x + 4x sin x) 1536 Exmple 6 Consider the following nonliner Volterr integrl eqution 16 ( 3( x 2 ) cos x 135 cos 3x + 4x(15 + 8x 2 ) sin x ) 2 5( x x 4 ) cos x cos 3x + 4x( x x 4 ) sin x u(x) = 1 + sin 2 x 3 with exct solution u(x) = cos x. Using the method, we hve nd u = f(x), x u 1 = (1 m)k T (f), where K T (f) = u(x) = u + u 1. sin(x t)u 2 (t)dt, K (K T (f)t (f)) K (K T (f)t (f)) K T (f) k(x, t)t (f(t))dt In this problem K T (f) = 1 4 ( cos x 6 cos 2x + cos 4x) nd K (K T (f)t (f)) = 1 ( cos x cos 2x 3( cos 3x 154 cos 4x + cos 6x + 31x sin x)) 56 ) IJNS homepge:

7 314 Interntionl Journl of Nonliner Science, Vol.13(212), No.3, pp Tble 1: The vlues of bsolute error for exmples 3 6. x exmple 3 exmple 4 exmple 5 exmple e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-4 1.4e e e e e e-2 hence cos x 7315 cos 2x + 3( cos 3x 154 cos 4x + cos 6x + 31x sin x) cos x cos 2x cos 3x 392 cos 4x + 3 cos 6x + 93x sin x, u = 1 + sin 2 x, u 1 = u(x) = 1 + sin x 2 + 7( cos x 6 cos 2x + cos 4x) 2 4( cos x cos 2x cos 3x 392 cos 4x + 3 cos 6x + 93x sin x), 7( cos x 6 cos 2x + cos 4x) 2 4( cos x cos 2x cos 3x 392 cos 4x + 3 cos 6x + 93x sin x). References [1] S.J. Lio, The homotopy nlysis method nd its pplictions in mechnics, Ph.D. thesis, Shnghi Jiotong University [2] S.J. Lio, Homotopy nlysis method: new nlytic method for nonliner problems, Applied Mthemtics nd Mechnics, 19(1)(1998): [3] S.J. Lio, Notes on the homotopy nlysis method: Some definitions nd theorems, Commun Nonliner Sci Numer Simult, 14(29): [4] Z.M. Odibt, A study on the convergence of homotopy nlysis method, Applied Mthemtics nd Computtion, 217(21): [5] S. Ling nd D. Jeffrey, Comprison of homotopy nlysis method nd homotopy perturbtion method through n evolution eqution, Commun Nonliner Sci Numer Simult, 14(29): [6] M.S.H. Chowdhury, I. Hshim nd O. Abdulziz, Comprison of homotopy nlysis method nd homotopy perturbtion method for purely nonliner fin-type problems, Commun Nonliner Sci nd Numer Simult, 14(29): [7] M. Sjid nd T. Hyt, Comprison of HAM nd HPM methods in nonliner het conduction nd convection equtions, Nonliner Anlysis: Rel World Applictions, 9(28): [8] S. Lio, Comprison between the homotopy nlysis method nd homotopy perturbtion method, Applied Mthemtics nd Computtion, 169(25): [9] S. Ling nd D.J. Jeffrey,Comprison of homotopy nlysis method nd homotopy perturbtion method through n evolution eqution, Commun Nonliner Sci Numer Simult, 14(29): [1] Z. Niu nd C. Wng, A one-step optiml homotopy nlysis method for nonliner differentil equtions, Commun Nonliner Sci Numer Simult, 15(21): [11] S.J. Lio, An optiml homotopy nlysis pproch for strongly nonliner differentil equtions, Commun Nonliner Sci Numer Simult, 15(21): [12] H. Jfri nd M.A. Firoozjee, Multistge homotopy nlysis method for solving nonliner integrl equtions, Applictions nd Applied Mthemtics: An Interntionl Journl, Specil Issue (1)(21): [13] Y.H. Qin, W. Zhng, B.W. Lin nd S.K. Li, Anlyticl pproximte periodic solutions for two-of-freedom coupled vn der pol-duffing oscilltors by extended homotopy nlysis method, Act Mech, DOI 1.17/s IJNS emil for contribution: editor@nonlinerscience.org.uk

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