A Convergent Scheme for Solving Initial Value Problems with Polynomial and Exponential Functions

Abstract

This paper presents the development of a convergent numerical scheme for the solution of initial value problems of first order ordinary differential equations. The scheme has been derived via the combination of two functions namely, polynomial and exponential functions. The local truncation error , order of convergence, consistency and stability of the proposed scheme have been analyzed in the present study.  The Taylor’s series expansion has been used to derive the principal term of . The Dahlquist’s test equation is used to investigate the linear stability region. It is observed that the newly proposed scheme is fourth order convergent, consistent and conditionally stable with the region of linear stability. Three IVPs of different nature have been solved numerically to check the applicability of a new proposed scheme. The absolute error has been calculated at each mesh point of the integration interval. The numerical results show that the scheme is computationally effective, adequate and compares favorably with exact solutions. The aid of MATLAB version: 9.2.0.538062 (R2017a) has been used to carry out all numerical calculations.

Keywords: Local truncation error, Absolute error, stability, consistency, convergence.

MSC: 34A12, 45L05, 65L05, 65L20, 65L70