Inverse kernel determination problem for a class of pseudo-parabolic integro-differential equations

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Abstract

This study investigates an inverse problem involving the determination of the kernel function in a multidimensional integrodifferential pseudo-parabolic equation of the third order. The study begins with an analysis of the direct problem, where we examine an initial-boundary value problem with homogeneous boundary conditions for a known kernel. Employing the Fourier method, we construct the solution as a series expansion in terms of eigenfunctions of the Laplace operator with Dirichlet boundary conditions. A crucial component of our analysis involves deriving a priori estimates for the series coefficients in terms of the kernel function norm, which play a fundamental role in our subsequent treatment of the inverse problem.
For the inverse problem, we introduce an overdetermination condition specifying the solution value at a fixed spatial point (pointwise measurement). This formulation leads to a Volterra-type integral equation of the second kind. By applying the Banach fixed-point principle within the framework of continuous functions equipped with an exponentially weighted norm, we establish the global existence and uniqueness of solutions to the inverse problem. Our results demonstrate the well-posedness of the problem under
consideration.

About the authors

Durdimurod K. Durdiev

Bukhara Branch of the Institute of Mathematics named after V. I. Romanovskiy at the Academy of Sciences of the Republic of Uzbekistan; Bukhara State University

Email: d.durdiev@mathinst.uz
ORCID iD: 0000-0002-6054-2827
http://www.mathnet.ru/person29112

Dr. Phys. & Math. Sci., Professor; Head of Branch1; Professor, Dept. of Differential Equations2

Uzbekistan, 705018, Bukhara, Muhammad Igbol st., 11; 705018, Bukhara, Muhammad Igbol st., 11

Hilola B. Elmuradova

Bukhara State University

Email: helmuradova@mail.ru
ORCID iD: 0000-0003-4306-2589
https://www.mathnet.ru/person228134

Teacher; PhD Student; Dept. of Differential Equations2

Uzbekistan, 705018, Bukhara, Muhammad Igbol st., 11

Askar A. Rahmonov

Bukhara Branch of the Institute of Mathematics named after V. I. Romanovskiy at the Academy of Sciences of the Republic of Uzbekistan; Bukhara State University

Author for correspondence.
Email: araxmonov@mail.ru
ORCID iD: 0000-0002-7641-9698
https://www.mathnet.ru/person67047

Cand. Phys. & Math. Sci., Associate Professor; Senior Researcher1; Associate Professor, Dept. of Differential Equations2

Uzbekistan, 705018, Bukhara, Muhammad Igbol st., 11; 705018, Bukhara, Muhammad Igbol st., 11

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