Modified Homopoty Perturbation Method To Solve Nonlinear Integral Equation And Its Convergence Analysis

Nonlinear equations are the approximate description of the natural displine, m-any problems in mathematics, natural science and engineering can be described bynonlinear integral equations. But, only few portions of nonlinear integral equationshave analytic solutions and the majorities do not have any. Therefore, it’s importantto solve nonlinear integral equations and obtain the approximate solutions, and thishas practical significance.In recent decades, scholars have proposed many numerical methods to solvenonlinear integral equations, such as Adomain decomposition method, projectionmethod, variational iteration method, Taylor expansion method, Legendre waveletsmethod, homotopy perturbation method and so on.Homotpy perturbation method was first proposed by J.H.He in1998, thismethod combines the traditional perturbation method with homotopy technique todeform a difficult problem into a simple solving problem. Usually, a rapidlyconvergent series solution can be obtained in most cases by using this method andseveral terms of the series solution can be used for approximating to the exactsolution with high degree of accuracy, this method has been successfully applied tomany fields. But the investigations for homotopy perturbation method areinadequate:(1) Since it is difficult to verify whether an operator is compressive ornot, so there is no literature that can give strict proof of the convergence so far.Only some scholars pointed out that the convergence of the HPM can be proved bycompressed image principle.(2) HPM is divergent for some strong nonlinearproblems. Thus, the purpose of this paper is to modify the HPM, make it stillconvergent when solving strong nonlinear problems and give strict proof.The main contents of this paper are as follows:(1) We introduce HPM and make modification to solve two-dimensionalFredholm integral equations of the second kind;(2) HPM is divergent when solving strong nonlinear integral equations, inorder to overcome this shortcoming, we propose modified HPM based on intervaldivision and use M-discriminant method in C space to give the strict proof of theoperator. Meanwhile, the error estimate is obtained;(3) We combine direct method with HPM to solve another kind of nonlinearVolterra-Fredholm integral equations; Numerical results demonstrate that HPM is a fast simple numerical methodamong the various methods of solving nonlinear integral equations.