Some Applications Of The Homotopy Perturbation Method To Nonlinear Differential And Integral Equations

The paper reviews Some applications of the homotopy perturbation method to nonlinear differential and integral equations.With the rapid development of nonlinear science,there has appeared ever increasing interest of scientists and engineers in the analytical technique for nonlinear problems.The widely applied techciques are perturbation methods.But like other nonlinear analytical techniques,perturbation method have their own limitations.This so-called small parameter assumption greatly restricts applications of perturbation techniques.1. Almost all perturbation method are based on an assumption that a small parameter must exist in the equation. As is well known,an overwhelming majority of nonlinear problems have no small parameters at all.2. The determination of small parameters seems to be a special art requiring special techniques.An appropriate choice of small parameters lead to ideal results.Howerver,an unsuitable choice of small parameters results in bad effects.Furthermore,the approximate solution solved by the perturbation methods are valid,in most cases,only for the small values of the parameters.It is obvious that all these limitation come from the small parameter assumption.He discovered a new perturbation technique in 1998,which is a coupling method of homotopy technique and perturbation technique. In contrast to the traditional perturbation methods, this technique does not require a small parameter in an equation. In this method,according to the homotopy technique,a homotopy with an imbedding parameter p[]is constructed, and the Imbedding parameter is constructed as a "small parameter", so the method is called the homotopy perturbation method, which can take full advantages of the traditional perturbation methods and homotopy technique. In this method, the approximate solution of equations can be written as a series of the form of the sum of infinite series which convergence its exact solution The result reveal that its fist order of approximation obtain by the proposed method is valid uniformly even for very large parameter and is more accurate than the perturbation solutions. In the decades,the homotopy perturbation method are widely used by solving nonlinear problems and seemed to be a usual method to solve nonlinear problems.This paper have three parts,the first part is introduction, which is introduce the idea of the homotopy perturbation method.The second part reviews some application of homotopy perturbation method.this part have five Section.The first section is to knowing the homotopy perturbation method by two sample examples;The second Section is to introduce the application of the homotopy perturbation method for Laplace transform. It gives a simple and a powerful mathematical toll.In contrast of usual methods which needs integration, the homotopy perturbation method, with constructed homotopy ,requires simple differentitation.the third Section is to use homotopy perturbation method for solving hyperbolic partial differential equations,It points out that the results are exactly the same as the result of Adomian decomposition method.The fourth Section is to use the homotopy perturbation method to drive an analytical solution for the non-linear schrodinger equations.It reveal that the analytical approximation to the solution are reliable and confirms the power and ability of the homotopy perturbation method.The fifth Section is to introduce the application of the homotopy perturbation method to nonlinear equation arising in heat transfer. Comparing with the perturbation method and Numerical, the homotopy perturbation method has a high accuracy.The third part review three application of modified the homotopy perturbation method. The first is applying tayor series in the homotopy perturbation method;the second proposes a scheme to accelerate the rate of convergence of the homotopy perturbation method applied to linear Fredholm internal equations.The fourth reviews that a systematic and regular method is summarized,which can be applies to solve approximate solutions and periods of strongly nonlinear oscillation problem by using homotopy perturbation method combining method of liberalization. We have studied few problems with or without small parameters with the homotopy perturbation technique,the results show that:1. The proposed method does not require small parameters in the equation,so the limitation of the traditional perturbation methods can be eliminated.2. The initial approximation can be freely selected with possible unknown constants.3. The approximations obtained by this method are valid not only for small param-eters,but also for very large parameters.Furthermore their fist-order approximations are of extreme accuracy.4. Although the proposed method can solve nonlinear differential equation,it can be applicable to nonlinear partial differential equations.5. The homotopy perturbation method has flexibility.it can combine some method to modified itself which will solve special problems.