Numerical Solution of Initial Value Problems Based on the Double Exponential Transformation

Abstract

The purpose of this paper is to present a method for approximate solution of initial value problems of ordinary differential equation by the double exponential transformation. The original problem is transformed into a Volterra integral equation and it is solved via the indefinite integration formula derived by Muhammad and Mori. A remarkable advantage of the double exponential transformation technique for solving initial value problems in this method is that it is easily implemented and gives a result with high accuracy also for problems with end point singularities and for stiff problems. The high accuracy of the method proposed in this paper is confirmed by numerical examples and an exponential convergence rate $\exp (-cN/\log N)$ is attained in almost all cases.