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

Although perturbation techniques are widely applied to analyze nonlinear problems in science and engineering, they are however so strongly dependent on small parameters appeared in equations under consideration that they are restricted only to weakly nonlinear problems. For strongly nonlinear problems which don’t contain any small parameters, perturbation techniques are invalid. So, it seems necessary and worthwhile developing at new kind of analytic technique independent of small parameters. Liao proposed a new analytic technique in his Ph.D. dissertation [1], namely the Homotopy Analysis Method (HAM). Based on homotopy of topology, the validity of the HAM is independent of whether or not there exist small parameters in considered equations. Therefore, the HAM can overcome the foregoing restrictions and limitations of perturbation techniques so that it provides us with a powerful tool to analyze strongly nonlinear problems. [2] In [2] some basic ideas about the HAM was described. In [3] some developments of the HAM was presented. Also some lemmas and theorems was proved. In [4] a reliable approach for convergence of the HAM was discussed. In [10]– [14] [46]–[48] [51]–[53] the HAM was applied on some equations. Also, some modifications and improvements was discussed by authors (e.g. see [15]–[17]). In [19, 20] the homotopy perturbation technique was presented. In [21]–[41] [49]–[50][54]–[55] the homotopy perturbation technique was applied on different equations by some authors and with some modifications (e.g. linear and nonlinear forth-order boundary value problems, functional integral equations, nonlinear problems, system of nonlinear Fredholm integral equations, forth-order integro-differential equations, eighth-order boundary value problems, nonlinear oscillators, partial differential equations, quadratic Riccati differential equation, Volterra integral equations, two-dimensional Fredholm integral equations, Stokes equations and nonlinear ill-posed operator equations). Also the homotopy perturbation method and the HAM was compared by some authors (e.g. [5]–[9]). We now review [39] to show how HPM applied to the following integral equations. Consider the following integral equation: