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

Abstract

This paper presents a numerical integration method recentlyproposed by means of an interpolating function involving a transcendentalfunction of exponential type for the solution of continuous dynamicalsystems, that is, the initial value problems (IVPs) in ordinary differentialequations (ODEs). The analysis of the local truncation error³Tn (h)´, orderof convergence, consistency and the stability of the proposed methodhave been investigated in the present study. The principal term of Tn (h)for the method has been derived via Taylor’s series expansion. The standardtest problem is taken into account to investigate the linear stabilityregion and the corresponding stability interval of the method. It is observedthat the newly proposed numerical integration method is secondorder convergent, consistent and conditionally stable. In order to test theperformance measure of the proposed method, five IVPs of varying naturehave been illustrated in the context of the maximum absolute global relativeerrors, the absolute relative errors computed at the final mesh point ofthe integration interval under consideration and the `2¡ error norm. Furthermore,the results are compared with two existing second order explicitnumerical methods taken from the relevant literature. The so far obtainedresults have demonstrated that the proposed numerical integration methodagrees with the second order convergence based upon the analysis conducted.Hence the proposed method is considered to be a good approachfor finding the solution of different types of IVPs in ODEs