Application Of Homotopy Perturbation Method To Solve Nonlinear Equation

Non-linear phenomena appear in various fields of modern science and technology, whose mathematical model is usually described by non-linear equations, and thus solving non-linear equations has important theoretical and practical significance.Soliton theory is one of specialized disciplines of nonlinear science, which has been established a number of efficient numerical methods for solving nonlinear equations system in the past 30 years. Such as, homotopy perturbation method (HPM) [1], Adomian decomposition method [2] and so on. However, the development of the new methods or the improvement of the original methods for solving nonlinear equations is an important research area of soliton theory. In this paper, we consider two questions by using homotopy perturbation method:the approximate analytic solution of an initial value problem and a mixed problem of sine-Gordon equation with the absolute errors of approximate analytic solutions and the exact solutions are given ;the approximate analytic solutions of an ainitial value problem of fully non-linear approximation sine-Gordon equation are obtained by using homotopy perturbation method(HPM) and the modified Adomian decomposition method. The absolute error of approximate analytic solution and the exact solution which obtained from two methods are also presented.We found that the homotopy perturbation method(HPM) has higher accuracy than the modified Adomian decomposition method.