The asymptotic expansion and extrapolation of trapezoidal rule for integrals with fractional order singularities

Abstract

This paper is aimed at deriving the error asymptotic expansion of trapezoidal rule approximation to integrals for the functions with fractional derivatives or algebraic singularities at some points and from which to design a modified Romberg extrapolation algorithm to effectively compute some singular integrals. Firstly, high-order local fractional derivatives are defined, then a general fractional Taylor's expansion is derived. Secondly, the error asymptotic expansion of trapezoidal rule for these integrals is obtained directly by using the formula of sums of non-integer powers and the general fractional Taylor's expansion. This method is different from the previous work. Thirdly, a modified Romberg extrapolation algorithm is designed to get numerical results efficiently. Finally, numerical examples are presented to verify the correctness of the theoretical analysis and the effectiveness of the extrapolation method.

Keywords:

• local fractional derivative
• fractional Taylor's expansion
• singular integral
• trapezoidal integration formula
• error asymptotic expansion
• modified Romberg method

AMS 2010 Subject Classifications:

• 65D30
• 26A33

Acknowledgements

The authors are very grateful to the anonymous referees for their valuable remarks and comments, which significantly contributed to the quality of the paper. This work was supported by National Natural Science Foundation of China (Grant number 11071123) and Natural Science Youth Foundation of Jiangsu Province [Grant number BK20130860].