Robin's problem for a fourth-order hyperbolic equation
DOI:
https://doi.org/10.17721/1812-5409.2026/1.12Keywords:
differential operator, regular solutions, iterated equations of hyperbolic type, Robin's problemAbstract
The Robin problem, also known as a mixed boundary condition, is often applied to partial differential equations. It defines a relationship between the value of a function and its derivative at the boundary of a domain, enabling the modeling of physical processes that affect both the properties of the surface itself and the flow of matter or energy. Fourth-order hyperbolic equations describe complex processes involving wave propagation and their interactions, enabling the modeling of dispersion effects often present in real-world physical processes. Such equations are used in hydrodynamics to model wave propagation at great depths, where both the velocity and the influence of dispersion on wave behavior are taken into account; in elasticity theory to analyze deformations and wave propagation in elastic materials, where dispersion effects and complex boundary conditions are present; and in seismology.
The aim of this work is to solve the Robin problem for a fourth-order hyperbolic equation with conditions imposing a relationship between certain derivatives of the function at the boundary. For this problem, the obtained solution to the Cauchy problem for a second-order hyperbolic equation is used. The solution to the Cauchy problem is obtained using differential operators that transform arbitrary functions into a regular solution of an m-th order hyperbolic equation. By reducing the problem to a solved Cauchy problem for a second-order hyperbolic equation and matching the boundary conditions of these problems, a system of equations arises whose solution exists and is unique as a solution to the Volterra integral equation and as a solution to the inhomogeneous Bessel equation. The solution to the Robin problem is a combination of the obtained solutions to the corresponding Cauchy problems for second-order equations.
Pages of the article in the issue: 101 - 105
Language of the article: English
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Copyright (c) 2026 Iryna Alexandrovich, Larisa Adzhubey, Andrii Ryzhenko, Viktor Lyashko
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