Tenth MSU Conference on Differential Equations and Computational Simulations. Electron. J. Diff. Eqns., Conference 23 (2016), pp. 1-7.

Using rational logarithmic basis functions to solve singular differential equations

John J. Garwood, Samuel N. Jator

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.

Published March 21, 2016.
Math Subject Classifications: 65L05, 65L06.
Key Words: Singular initial value problem; rate of convergence; rational logarithmic basis function.

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John J. Garwood
Department of Mathematics and Statistics
Austin Peay State University
Clarksville, TN 37044, USA
email: jgarwood1@my.apsu.edu
Samuel N. Jator
Department of Mathematics and Statistics
Austin Peay State University
Clarksville, TN 37044, USA
email: jators@apsu.edu

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