S.S. Kabdrakhova
symbat2909.sks@gmail.com
Institute of Mathematics and Mathematical Modeling,
Al-Farabi Kazakh National University,
Almaty, Kazakhstan

Boundary value problems for third-order partial differential equations play a significant role in mechanics, nonlinear acoustics, and magnetohydrodynamics. These equations govern phenomena such as longitudinal vibrations in composite rods made of elastic and viscoelastic segments. The well-posedness of these problems and the methodologies for their analysis are important areas of study. There exists a substantial body of work focused on the existence of solutions for boundary value problems related to third-order hyperbolic equations. However, questions surrounding well-posedness and the development of approximate solutions for non-local problems remain relevant.

This study investigates a semi-periodic boundary value problem for a third-order equation with two independent variables. A specific transformation is applied to the unknown function and its time derivative, reducing the equation to a system of two hyperbolic equations with mixed derivatives. To approximate solutions for this boundary value problem, a modified broken Euler method is utilized. Conditions for estimating the convergence of this method to the solution of the problem are established.