Implicit Three-Step Hybrid Block Method for Solving First Order Initial Value Problems in Ordinary Differential Equations

Abstract

Recently, the development of numerical method for approximating solutions of initial value problems (IVPs) in ordinary differential equations (ODEs) has attracted considerable attention and many researchers have shown interest in constructing efficient methods with good stability properties for the numerical integration of ODEs. This research focuses on the derivation of new implicit three step block hybrid method for the solution of first order IVPs in ODEs. The new method is derived based on multistep collocation using Chebyshev polynomials as bases functions at some selected points to get a continuous linear multistep method. The continuous methods are evaluated at some off-grid points to generate the discrete schemes for step number k=3 which conveniently constitutes the block method. Basic properties of the developed method is examined and the method is found to be zero stable, consistent, convergent and of uniform order 8. The efficiency of the method is tested on some numerical examples in the literature. On comparison, the method developed performed favorably when compared with the existing methods. As
such the method is recommended for the solution of general first order initial value problems in ordinary differential equations.