A Comparison of Explicit Semi-Analytical Numerical Integration Methods for Solving Stiff ODE Systems

Abstract

Abstract:In this study, a comparison among three semi-analytical numerical integration algorithms for solving stiff ODE systems is presented. The algorithms are based on Differential Transform Method (DTM) which are Multiple-Step DTM (MsDTM), Enhanced MsDTM (EMsDTM) and MsDTM with Padé approximants (MsDTM-P). These methods can be classified as explicit one step semi-analytical numerical integration methods. The error and stability analysis of each method is presented. New important relationships among the methods are introduced. To demonstrate our results, a comparison of the accuracy, stability and computational efficiency of the methods is presented through solving some linear and nonlinear problems arising in applied science and engineering.

A New Generalized Taylor-Like Explicit Method for Stiff Ordinary Differential Equations

“…Moreover, the Taylor-like method [11] is an arbitrary high order A-stable method that avoids extremely small stepsizes during the integration procedure. To avoid the analytical computation of the successive derivatives involved in the Taylor-like methods, numerical differentiation [21,22], automatic differentiation [23], differential transformation [24][25][26] and Infinity Computer with a new numeral system [27][28][29][30][31][32] can be used. In fact, Taylor-like explicit methods [5,7,[9][10][11] have computational drawbacks with zero-component derivative or zero-vector norm in their component or vector forms, respectively.…”