Solutions of boundary value problems for loaded hyperbolic type equations
Orumbayeva N.T. Kosmakova M.T. Tokmagambetova T.D. Manat A.M.
2025E.A. Buketov Karaganda University Publish house
Bulletin of the Karaganda University. Mathematics Series
2025#118Issue 2177 - 188 pp.
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.
boundary value problem , Cauchy problem , convolution , differential equations , fractional derivative , integral equations , Laplace transform , loaded equations , partial derivatives , wave equations
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Karaganda Buketov University, Institute of Applied Mathematics, 28 Universitetskaya Street, Karaganda, 100028, Kazakhstan
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