Solutions of boundary value problems for loaded hyperbolic type equations

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.