On the decomposition method for the approximate solution of nonlinear ordinary differential equations

Abstract

The study of linear and nonlinear differential equations constitute a main topic in the undergraduate curriculum. Several standard methods are normally employed for the solution of such problems. In this paper, the decomposition method is introduced as a powerful alternative scheme for solving ordinary differential equations. The technique is well adapted for obtaining and identifying the closed form solution or for getting an approximate solution comparable to existing numerical schemes. The solution obtained is in the form of a power series with elegantly computable terms. An essential feature of this technique is its rapid convergence and the high degree of accuracy by which it approximates a solution; it is shown that only few iterations are sufficient to obtain accurate approximation to the exact solution. The technique is implemented for well-known nonlinear differential equations that are usually taught at the undergraduate level. The method is illustrated by considering the Abel, Bernoulli and Ricatti differential equations. Most of the symbolic and numerical computations have been performed using the computer algebra system Maple.