(untitled) (Scientific journal of the Fergana State University, 2025)
The theory of nonlinear acoustics, hydrodynamics of plasma, fluid filtration problems in porous media, and many other applied problems often lead to boundary value problems for differential equations involving third-order small terms. This suggests that solving boundary value problems for differential equations with third-order small terms is of great importance as one of the relevant directions in the theory of partial differential equations, providing a complete understanding of the analysis and properties of the aforementioned applied problems. As mentioned above, many practically significant problems, including fluid filtration processes in porous media, the movement of free-surface groundwater in multilayered environments, and others, lead to the study of boundary value problems for third-order differential equations. In the article by T.D. Juraev and A. Popelek, the complete classification and canonical form of third-order differential equations are presented. Problems related to third- and higher-order pseudoparabolic equations have been studied in various works, as well as problems for third-order differential equations involving singular Bessel operators. In this paper, the Cauchy problem for a nonhomogeneous third-order hyperbolic model equation is studied.