The asymptotic expansion and extrapolation of trapezoidal rule for integrals with fractional order singularities. (English) Zbl 1317.65066
Summary: 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 methodReferences:
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