Closed-form solutions for several classes of singular integral equations with convolution and Cauchy operator

Abstract

The main goal of this paper is to study the existence of solutions for several new classes of singular convolution integral equations containing the Cauchy operator in the normal type case. To investigate the solutions of such equations, we establish the Noethericity theory of solvability. By means of the properties of complex Fourier transforms, we transform these equations into Riemann boundary value problems with nodes. The analytic solutions and conditions of solvability are obtained via using Riemann–Hilbert approach. Moreover, we also discuss the asymptotic property of solutions near nodes. Thus, this paper generalizes the theories of complex analysis, functional analysis, integral equations and the classical Riemann boundary value problems.

Keywords:

• Singular integral equations of convolution type
• Riemann boundary value problems
• Sokhotski–Plemelj formula
• Cauchy integral operator

AMS Subject Classifications:

• 45E10
• 30E25
• 45E05

Acknowledgements

The authors would like to express their gratitude to the anonymous referees for their invaluable comments and suggestions, which helped to improve the quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported financially by the National Natural Science Foundation of China [grant number 11971015].