A Boundary Value Problem for a TimeFractional Diffusion Equation in a Non-Cylindrical Shrinking Domain

Abstract

This article deals with the fundamental problems in the mathematical theory of fractional differential equations, specifically focusing on the analytical solvability of boundary value problems in time-dependent domains. The relevance of the study implies the necessity of developing methods for equations with nonlocal operators modeling anomalous diffusion. A one-dimensional diffusion equation containing a RiemannLiouville fractional derivative with respect to time is examined. The characteristic features of the problem, posed in a non-cylindrical domain bounded by a moving linear boundary and a fixed spatial coordinate, are analyzed. The need to handle inhomogeneous boundary data is identified, and the problem is initially reduced to one with homogeneous conditions. On the basis of the study, the author constructs the fundamental solution in a quarter-plane by means of the bilateral Laplace transform and obtains the Green function for the Dirichlet problem. It is shown that the solution can be expressed through an integral representation in terms of a specific boundary density. This density satisfies a Volterra-type integral equation with a weakly singular kernel. Using the contraction mapping principle, it is proved that this equation has a solution. Consequently, the existence of a regular solution to the original boundary value problem is established.