SOLVING THE CAUCHY PROBLEM FOR THE NONLINEAR DIFFUSION EQUATION USING HE'S HOMOTOPY PERTURBATION METHOD

Abstract

This thesis applies He's Homotopy Perturbation Method (HPM) to obtain an approximate solution of the Cauchy problem posed for a nonlinear diffusion equation. The essence of the method lies in constructing an auxiliary equation with an embedded homotopy parameter, and subsequently seeking the solution in the form of a power series with respect to that parameter. The nonlinear term of the equation is decomposed into a sequence of recursively computed approximations, so that a highly accurate approximate analytical solution is built up from just the first few terms. The paper presents the algorithm of the method step by step, substantiates the principle of choosing the initial approximation for the Cauchy problem, and discusses the convergence properties of the resulting series. The fact that the method requires no small parameter, needs no linearization, and involves a simple computational procedure makes it a convenient tool for investigating strongly nonlinear partial differential equations.