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Uniqueness and Existence of Solutions to Some Kinds of Singular Convolution Integral Equations with Cauchy Kernel via R-H Problems

Uniqueness and Existence of Solutions to Some Kinds of Singular Convolution Integral Equations... In this article, we study the existence and uniqueness of solutions to some kinds of singular integral equations with Cauchy kernel and convolution kernel. In order to transform our equations into Riemann-Hilbert problems (R-H problems), we establish the relation between Fourier integral transform and Cauchy type integral, and we generalize Sokhotski–Plemelj formula. By means of the classical R-H problems and of regularity theory of Fredholm integral equations, we present the necessary and sufficient conditions of Noether solvability and the explicit solutions for such equations. Especially, we discuss the property and asymptotic behavior of solutions at nodes. This paper will be of great significance for the study of improving and developing complex analysis, integral equations, and R-H problems. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Applicandae Mathematicae Springer Journals

Uniqueness and Existence of Solutions to Some Kinds of Singular Convolution Integral Equations with Cauchy Kernel via R-H Problems

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Publisher
Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to Springer Nature B.V. 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
ISSN
0167-8019
eISSN
1572-9036
DOI
10.1007/s10440-023-00556-8
Publisher site
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Abstract

In this article, we study the existence and uniqueness of solutions to some kinds of singular integral equations with Cauchy kernel and convolution kernel. In order to transform our equations into Riemann-Hilbert problems (R-H problems), we establish the relation between Fourier integral transform and Cauchy type integral, and we generalize Sokhotski–Plemelj formula. By means of the classical R-H problems and of regularity theory of Fredholm integral equations, we present the necessary and sufficient conditions of Noether solvability and the explicit solutions for such equations. Especially, we discuss the property and asymptotic behavior of solutions at nodes. This paper will be of great significance for the study of improving and developing complex analysis, integral equations, and R-H problems.

Journal

Acta Applicandae Mathematicae – Springer Journals

Published: Apr 1, 2023

Keywords: Singular convolution integral equations; Riemann-Hilbert problems; Singular integral operator; Cauchy kernel; Sokhotski–Plemelj formula; 45E10; 45E05; 30E25

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