Numerical Green's Function Method Based on the DE Transformation

Abstract

A method for numerical solution of boundary value problems with ordinary differential equation based on the method of Green's function incorporated with the double exponential transformation is presented. The method proposed does not require solving a system of linear equations and gives an approximate solution of very high accuracy with a small number of function evaluations. The error of the method is $O\left(\exp\left(-C_1N/\log(C_2N)\right)\right)$ where $N$ is a parameter representing the number of function evaluations and $C_1$ and $C_2$ are some positive constants. Numerical examples also prove the high efficiency of the method. An alternative method via an integral equation is presented which can be used when the Green's function corresponding to the given equation is not available.