Analytical, numerical, and biomedical aspects of boundary value problems for third-order elliptic-type equations with singular coefficients

Mathematical modeling of various real-world processes frequently leads to boundary value problems (BVPs) for third-order partial differential equations of mixed and composite types, which have no classical analogues in mathematical physics. Foundational studies by A.V. Bitsadze and M.S. Salakhitdinov first addressed well-posed boundary value problems for degenerate equations of third-order mixed and mixed-composite types. A key approach in their work involved representing the general solution of a composite-type equation as a sum of functions, which proved essential for operators constructed as products of permuted differential operators. Following these foundational contributions, the study of third-order partial differential equations involving Lavrentiev-Bitsadze, Gellerstedt, heat conduction, and string-type operators has been further advanced by both international and Uzbek mathematicians. Despite these developments, boundary value problems for third-order equations of parabolic-hyperbolic and elliptic-hyperbolic types with singular coefficients remain largely unexplored. In this article, we formulate and investigate boundary value problems for a third-order elliptic equation with a singular coefficient. The existence and uniqueness of classical solutions are rigorously proved. A new extremum principle for third-order equations is developed and applied to establish uniqueness. The existence of a solution is reduced to a singular integral equation of normal type, which is regularized using the classical Carleman-Vekua method, leading to an equivalent Fredholm equation of the second kind. The analytical framework is complemented by a numerical scheme that verifies the theoretical results and illustrates the qualitative behavior of solutions near the degenerate boundary. Furthermore, a numerical illustration is provided to demonstrate the stability and smoothness of the obtained solutions even in the presence of singular coefficients. Finally, the potential biomedical relevance of the model is discussed through its application to steady-state diffusion processes in tumor tissues.

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