The first boundary-value problem for some mixed equations of thermal conductivity of the second and fourth order

V. N. Khankhasaev, V. M. Plastinina

East Suberia State University of Technology and Management, Ulan-Ude

Abstract: In this paper, we present a computational model for solving the first boundary-value problem for a mixed differential equation in a spatially two-dimensional case using an implicit finite-difference scheme. For a fourth-order equation, we use two different second-order operators that are similar to a one-dimensional mixed thermal conductivity operator in the spatial variable, generalizing the purely hyperbolic case and used in mathematical modeling of the shutdown process for an electric arc. The well-posedness of the boundary-value problem is proved.

Keywords: hyperbolic heat equation, mixed-type equation, locally one-dimensional method, first boundary-value problem, high-order equation

UDC: 517.95; 532.5

MSC: 35M12

DOI: 10.36535/2782-4438-2025-241-83-89