Taylor expansion method for integrals with algebraic-logarithmic singularities

Abstract

In this paper, a method based on Taylor's polynomials is presented to approximate a singular integral with the integrand of the form f(t) k(t, c), where f(t) is a given smooth function and k(t, c) is a kernel possing algebraic-logarithmic singularity at the point c∈[a, b], (−1≤a, b, c≤1). The function f(t) is approximated by Taylor polynomials. Therefore, the integrand can be expressed in terms of algebraic-logarithmic functions. To get higher accuracy, the integration domain is divided into M subintervals and the methods are used on subintervals. The main advantage of the proposed method is its simplicity in use and its high accuracy. Numerical examples illustrated the pertinent features of the method.

Keywords:

• indefinite integral
• algebraic-logarithmic singularity
• Taylor polynomials

2000 AMS Subject Classification :

• 65D30

Acknowledgements

The author is grateful to the referees for their comments and suggestions that have helped in improving the manuscript. This work is supported by Islamic Azad University–Tabriz Branch.