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 Izv. Vyssh. Uchebn. Zaved. Mat., 2020, Number 1, Pages 84–93 (Mi ivm9538)

Integro-differential equations over a closed circuit with Gaussian function in the kernel

A. I. Peschansky

Sevastopol State University, 33 Universitetskaya str., Sevastopol, 299053 Russia

Abstract: Integro-differential equations with kernels including hypergeometric Gaussian function that depends on the arguments ratio are studied over a closed curve in the complex plane. Special cases of the equations considered are the special integro-differential equation with Cauchy kernel, equations with power and logarithmic kernels. By means of the curvilinear convolution operator with the kernel of special kind, the equations with derivatives are reduced to the equations without derivatives. We find out the connection between special cases of the above-mentioned convolution operator and the known integral representations of piecewise analytical functions applied in the study of boundary value problems of the Riemann type. The correct statement of Noetherian property for the investigated class of equations is given. In this case, the operators corresponding to the equations are considered acting from the space of summable functions into the space of fractional integrals of the curvilinear convolution type. Examples of integro-differential equations solvable in a closed form are given.

Keywords: integro-differential equation, operator of the curvilinear convolution, integral representations of a piecewise analytic function, Noetherian property of the equation.

DOI: https://doi.org/10.26907/0021-3446-2020-84-93

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UDC: 517.968
Revised: 17.03.2019
Accepted: 27.03.2019

Citation: A. I. Peschansky, “Integro-differential equations over a closed circuit with Gaussian function in the kernel”, Izv. Vyssh. Uchebn. Zaved. Mat., 2020, no. 1, 84–93

Citation in format AMSBIB
\Bibitem{Pes20} \by A.~I.~Peschansky \paper Integro-differential equations over a closed circuit with Gaussian function in the kernel \jour Izv. Vyssh. Uchebn. Zaved. Mat. \yr 2020 \issue 1 \pages 84--93 \mathnet{http://mi.mathnet.ru/ivm9538} \crossref{https://doi.org/10.26907/0021-3446-2020-84-93}