Initial Value Problems Exactly Solvable by Nonlinear Numerical Schemes

Abstract

In this paper, couple of explicit nonlinear single-step numerical schemes have been investigated for producing exact solutions to IVPs (initial value problems) in ODEs (ordinary differential equations). The first term of the local truncation errors also called the leading term of the schemes under consideration is equated to zero to achieve the objectives because a necessary condition on a numerical scheme to produce exact results is that the leading term of its LTE (local truncation error) must vanish. In this regard, a technique to determine some general forms of ODEs exactly solved by a given non-linear numerical scheme is developed. Schemes under investigation are Fatunla’s First Order Nonlinear Scheme, Higinio’s First Order Nonlinear Scheme for autonomous ODEs, Higinio’s Second Order Nonlinear Scheme, and Higinio's Third Order Nonlinear Scheme. These numerical schemes have been examined to obtain the general form of ODEs exactly solvable by such schemes.