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.
Highlights
During the last several decades, many efforts have been made in the study of fractional calculus and entropy to investigate the dynamical behavior [1,2,3,4,5,6]
Beyond the complexity appearing in complex systems, the fractionality emerging in fractional dynamical systems has gradually attracted interest
We show how to get the approximate value of the Hadamard derivative via the finite part integral
Summary
During the last several decades, many efforts have been made in the study of fractional calculus and entropy to investigate the dynamical behavior [1,2,3,4,5,6]. The authors in [25] devoted their work to the fractional-order entropy analysis of earthquake data series. The Hadamard derivative is a nonlocal fractional derivative with a singular logarithmic kernel with memory; it is suitable to describe complex systems For these reasons, the study of the Hadamard derivative is necessary and useful for the entropy analysis. Diethelm gave an implicit algorithm for the approximate solution of the fractional differential equation with the Riemann–Liouville derivative in the sense of the finite part integral in [27].
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