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Approximation to Hadamard Derivative via the Finite Part Integral

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,...

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Autores principales: Yin, Chuntao, Li, Changpin, Bi, Qinsheng
Formato: Online Artículo Texto
Lenguaje:English
Publicado: MDPI 2018
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7512583/
https://www.ncbi.nlm.nih.gov/pubmed/33266706
http://dx.doi.org/10.3390/e20120983
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author Yin, Chuntao
Li, Changpin
Bi, Qinsheng
author_facet Yin, Chuntao
Li, Changpin
Bi, Qinsheng
author_sort Yin, Chuntao
collection PubMed
description 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.
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spelling pubmed-75125832020-11-09 Approximation to Hadamard Derivative via the Finite Part Integral Yin, Chuntao Li, Changpin Bi, Qinsheng Entropy (Basel) Article 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. MDPI 2018-12-18 /pmc/articles/PMC7512583/ /pubmed/33266706 http://dx.doi.org/10.3390/e20120983 Text en © 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
spellingShingle Article
Yin, Chuntao
Li, Changpin
Bi, Qinsheng
Approximation to Hadamard Derivative via the Finite Part Integral
title Approximation to Hadamard Derivative via the Finite Part Integral
title_full Approximation to Hadamard Derivative via the Finite Part Integral
title_fullStr Approximation to Hadamard Derivative via the Finite Part Integral
title_full_unstemmed Approximation to Hadamard Derivative via the Finite Part Integral
title_short Approximation to Hadamard Derivative via the Finite Part Integral
title_sort approximation to hadamard derivative via the finite part integral
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7512583/
https://www.ncbi.nlm.nih.gov/pubmed/33266706
http://dx.doi.org/10.3390/e20120983
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