Approximate Solutions of Nonlinear Partial Differential Equations by Modified q-Homotopy Analysis Method

Abstract

A modified $q$-homotopy analysis method (m$q$-HAM) was proposed for solving $n$th-order nonlinear differential equations. This method improves the convergence of the series solution in the $n$HAM which was proposed in (see Hassan and El-Tawil 2011, 2012). The proposed method provides an approximate solution by rewriting the $n$th-order nonlinear differential equation in the form of $n$ first-order differential equations. The solution of these $n$ differential equations is obtained as a power series solution. This scheme is tested on two nonlinear exactly solvable differential equations. The results demonstrate the reliability and efficiency of the algorithm developed.