A Novel Computational Approach for Solving Fully Implicit Singular Systems of Ordinary Differential Equations

Abstract

This paper presents a novel computational approach to solve fully implicit singular nonlinear systems of ordinary differential equations. These systems have a two fold difficulty: being fully implicit and singular at the same time. Such systems cannot be solved in general by software packages such as Maple due to their fully implicit structure. Furthermore, numerical methods like Runge-Kutta cannot be applied. The proposed method here is based on the idea of applying the differential transform method (DTM) directly to these systems while exploiting an important property of Adomian polynomials. This new idea has led to a general and efficient algorithm that can be easily implemented using Maple, Mathematica or Matlab. We stress here that our technique does not require transforming the implicit system in hands to an explicit differential system. Also our technique equips the DTM with a powerful tool to solve other fully implicit differential systems. To illustrate the capability and efficiency of the proposed method, four numerical examples that are not solvable by software packages like Maple are given. Numerical results show that our method has successfully solved these examples by providing the exact solutions in a convergent power series form.