Nelder-Mead Algorithm in Solving Ordinary Differential Equations Whose Solutions Possess Singularities

Abstract

This research considers Initial Value Problems (IVPs) in Ordinary Differential Equations (ODEs) whose solutions possess singularities. Here, we represent the theoretical solution by a rational function as it is more convenient representing a function close to a singularity by a rational function. The process of transforming the IVP to a constrained optimization problem and application of Nelder-Mead algorithm in obtaining approximate solution is presented in this work. Accuracy and efficiency of this scheme is demonstrated on two numerical examples. The proposed approach produced better results compared with existing methods discussed in literature.