On the Asymptotic Behavior of Eigenvalues and Eigenfunctions of an Integral Convolution Operator with a Logarithmic Kernel on a Finite Interval

Автор: Polosin A.
Журнал: Differential Equations
Том: 58
Номер: 9
Год издания: 2022
Издательство: Pleiades Publishing, Inc.
Местоположение издательства: New York, USA
Первая страница: 1242
Последняя страница: 1257
DOI: 10.1134/S0012266122090099
Аннотация: It is well known that one-dimensional integral equations of the convolution typeconsidered on a finite interval cannot generally be solved by quadratures in contrast to similarequations considered on the entire line or on the half-line. For this reason, when studyingtheir spectrum, some asymptotic method must be used. The paper considers a convolutiontypeintegral operator with a logarithmic kernel defined on a finite interval. Using the Fouriertransform, the problem is successively reduced to a conjugation problem and to a singularintegral equation on the half-line in which the integral operator is not a contraction. It is shownthat the leading part of the integral equation thus obtained can be inverted explicitly. Thecases of even and odd eigenfunctions are considered, in each of which the asymptotics of theeigenvalues and eigenfunctions of the original operator is found.
Добавил в систему: Полосин Алексей Андреевич

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