Existence and uniqueness of the solution of a mixed problem for a parabolic equation under nonconventional boundary conditions

TL;DR

This paper proves the existence and uniqueness of solutions for a specific parabolic PDE with complex boundary conditions, providing explicit solutions using advanced integral methods.

Contribution

It introduces new conditions ensuring unique solvability and derives explicit solutions for a parabolic problem with non-local, non-self-adjoint boundary conditions.

Findings

Established conditions for unique solvability

Derived explicit analytical solutions

Applied residue and contour integral methods

Abstract

In this study, we investigate a mixed problem linked to a second-order parabolic equation, characterized by temporal dependencies and variable~coefficients, and constrained by non-local, non-self-adjoint boundary conditions. By defining precise conditions on the input data, we establish the unique solvability of the problem through a synthesis of the residue and contour integral methods. Moreover, our research yields an explicit analytical solution, facilitating the direct resolution of the stated problem.