A grid-based method for solving the initial-boundary value problem for a multidimensional third-order hyperbolic partial differential equation

Beshtokov, M. KH.

Indian Journal of Mathematics

A grid-based method for solving the initial-boundary value problem for a multidimensional third-order hyperbolic partial differential equation

Author(s): M. KH. Beshtokov M. KH. Beshtokov Department of Computational Methods,
Institute of applied mathematics and automation,
Kabardino-Balkarian scientific center of RAS,
89A Shortanova Str., 360000 Nalchik,
Russia. beshtokov-murat@yandex.ru

Vol. 67, No. 1 (2025) • pp. 77–107

Published 27-04-2026 Article History Received 13-04-2025 Revised 22-02-2026 Published 27-04-2026

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Abstract

The initial boundary value problem for a multidimensional partial differential equation of the third order of hyperbolic type, which serves as a mathematical model of the movement of moisture and salts in soils, is studied. For an approximate solution of the problem, the original equation is reduced to an integro-differential equation with a small parameter. It is shown that when the small parameter tends to zero, the solution of the resulting modified problem converges to the solution of the original problem. A. A. Samarsky's locally one-dimensional difference scheme is constructed. An a priori estimate is obtained using the method of energy inequalities, from which the uniqueness and stability of the solution of the scheme follow, the convergence of the solution of the locally one-dimensional difference scheme to the solution of the modified differential problem is proved.

Keywords

hyperbolic equation, third-order equation, multidimensional equation, initial-boundary value problem, difference schemes, locally one-dimensional scheme, a priori estimates, stability, and convergence.

2020 Mathematics Subject Classification

35L30, 65M06, 65M12.

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Allahabad Mathematical Society

A grid-based method for solving the initial-boundary value problem for a multidimensional third-order hyperbolic partial differential equation

Abstract

Keywords

2020 Mathematics Subject Classification

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