
Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2020, Number 67, Pages 40–54
(Mi vtgu800)




MATHEMATICS
Investigation of some classes of second order partial integrodifferential equations with a powerlogarithmic singularity in the kernel
S. K. Zarifzoda^{}, R. N. Odinaev^{} ^{} Tajik National
University, Dushanbe, Republic of Tajikistan
Abstract:
For a class of secondorder partial integrodifferential equations with a power singularity and logarithmic singularity in the kernel, integral representations of the solution manifold in terms of arbitrary constants are obtained in the class of functions vanishing with a certain asymptotic behavior. Although the kernel of the given equation is not a Fredholm type kernel, the solution of the studied equation in a class of vanishing functions is found in an explicit form. We represent a secondorder integrodifferential equation as a product of two firstorder integrodifferential operators. For these onedimensional integrodifferential operators, in the cases when the roots of the corresponding characteristic equations are real and different, real and equal and complex and conjugate, the inverse operators are found. It is found that the presence of power singularity and logarithmic singularity in the kernel affects the number of arbitrary constants in the general solution. This number, depending on the roots of the corresponding characteristic equations, can reach nine. Also, the cases when the given integrodifferential equation has a unique solution are found. The correctness of the obtained results with the help of the detailed solutions of concrete examples are shown. The method of solving the given problem can be used for solving model and nonmodel integrodifferential equations with a higher order power singularity and logarithmic singularity in the kernel.
Keywords:
integrodifferential equation, power singularity, logarithmic singularity, integral representations, characteristic equation.
DOI:
https://doi.org/10.17223/19988621/67/4
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UDC:
517.968.73
MSC: 45E99, 45K99, 45D99 Received: 24.01.2020
Citation:
S. K. Zarifzoda, R. N. Odinaev, “Investigation of some classes of second order partial integrodifferential equations with a powerlogarithmic singularity in the kernel”, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2020, no. 67, 40–54
Citation in format AMSBIB
\Bibitem{ZarOdi20}
\by S.~K.~Zarifzoda, R.~N.~Odinaev
\paper Investigation of some classes of second order partial integrodifferential equations with a powerlogarithmic singularity in the kernel
\jour Vestn. Tomsk. Gos. Univ. Mat. Mekh.
\yr 2020
\issue 67
\pages 4054
\mathnet{http://mi.mathnet.ru/vtgu800}
\crossref{https://doi.org/10.17223/19988621/67/4}
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