Convergent Numerical Method Using Transcendental Function of Exponential Type to Solve Continuous Dynamical Systems

Abstract

This paper presents a numerical integration method recently proposed by means of an interpolating function involving a transcendental function of exponential type for the solution of continuous dynamical systems, that is, the initial value problems (IVPs) in ordinary differential equations (ODEs). The analysis of the local truncation error ³ Tn (h) ´ , order of convergence, consistency and the stability of the proposed method have been investigated in the present study. The principal term of Tn (h) for the method has been derived via Taylor’s series expansion. The standard test problem is taken into account to investigate the linear stability region and the corresponding stability interval of the method. It is observed that the newly proposed numerical integration method is second order convergent, consistent and conditionally stable. In order to test the performance measure of the proposed method, five IVPs of varying nature have been illustrated in the context of the maximum absolute global relative errors, the absolute relative errors computed at the final mesh point of the integration interval under consideration and the ` 2− error norm. Furthermore, the results are compared with two existing second order explicit numerical methods taken from the relevant literature. The so far obtained results have demonstrated that the proposed numerical integration method agrees with the second order convergence based upon the analysis conducted. Hence the proposed method is considered to be a good approach for finding the solution of different types of IVPs in ODEs.