Solutions of boundary value problems for loaded hyperbolic type equations

Authors

DOI:

https://doi.org/10.31489/2025m2/177-188

Keywords:

differential equations, partial derivatives, loaded equations, boundary value problem, Laplace transform, convolution, wave equations, fractional derivative, integral equations, Cauchy problem

Abstract

This paper investigates a class of second-order partial differential equations describing wave processes with nonlocal effects, including cases involving fractional derivatives. Such equations often arise in the theory of elasticity, aerodynamics, acoustics, and electrodynamics. The presented equations include both integral and differential terms, evaluated either at a fixed point x = x0 or x = α(t). An equation with a fractional derivative of order 0 ≤ β < 1 is considered, making it possible to model memory effects and other nonlocal properties. For each equation, supplemented by initial conditions, either a closed-form analytical solution is obtained or the main steps of its derivation are outlined. The article employs the Laplace transform to solve the resulting integral equation, enabling the solution to be presented in an explicit form.

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Published

2025-06-30

How to Cite

Orumbayeva, N., Kosmakova, M., Tokmagambetova, T., & Manat, A. (2025). Solutions of boundary value problems for loaded hyperbolic type equations. Bulletin of the Karaganda University. Mathematics Series, 118(2), 177–188. https://doi.org/10.31489/2025m2/177-188

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Section

MATHEMATICS