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Entropy 2018, 20(12), 983; https://doi.org/10.3390/e20120983

Approximation to Hadamard Derivative via the Finite Part Integral

1
Department of Mathematics, Shanghai University, Shanghai 200444, China
2
Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang 050043, China
*
Author to whom correspondence should be addressed.
Received: 8 October 2018 / Revised: 7 December 2018 / Accepted: 14 December 2018 / Published: 18 December 2018
(This article belongs to the Special Issue The Fractional View of Complexity)
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Abstract

In 1923, Hadamard encountered a class of integrals with strong singularities when using a particular Green’s function to solve the cylindrical wave equation. He ignored the infinite parts of such integrals after integrating by parts. Such an idea is very practical and useful in many physical models, e.g., the crack problems of both planar and three-dimensional elasticities. In this paper, we present the rectangular and trapezoidal formulas to approximate the Hadamard derivative by the idea of the finite part integral. Then, we apply the proposed numerical methods to the differential equation with the Hadamard derivative. Finally, several numerical examples are displayed to show the effectiveness of the basic idea and technique. View Full-Text
Keywords: Hadamard derivative; finite part integral Hadamard derivative; finite part integral
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Yin, C.; Li, C.; Bi, Q. Approximation to Hadamard Derivative via the Finite Part Integral. Entropy 2018, 20, 983.

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