Using rational logarithmic basis functions to solve singular differential equations

Abstract

Numerical methods based on polynomial approximation perform poorly when applied to singular initial value problems (IVPs). Hence, we are motivated to derive and implement numerical methods involving non-polynomial basis functions such as logarithmic and rational functions. Specifically, by imbedding a constant parameter into the logarithmic function, we are able to improve any discontinuity issues with the natural logarithm approximant. An efficient method is developed using the Taylor Series expansion to optimize the imbedded parameter. Numerical experiments performed show that this method is more accurate than the improved Euler's method. This method is implemented as a predictor-corrector method.

Keywords

• Singular initial value problem
• rate of convergence
• rational logarithmic basis function