Analytical solutions of moving boundary problems for the time-fractional diffusion equation

TL;DR

This paper derives analytical solutions for moving boundary problems involving the time-fractional diffusion equation, including a fractional analogue of the Stefan problem, using the embedding method and auxiliary functions.

Contribution

It introduces a novel analytical approach to solve moving boundary problems for fractional diffusion equations, extending classical solutions to fractional derivatives.

Findings

Derived explicit solutions for fractional Stefan problems

Established a fractional analogue of the Neumann solution

Extended classical boundary problems to fractional derivatives

Abstract

The time-fractional diffusion equation is considered, where the time derivative is either of Caputo or Riemann-Liouville type. The solution of a general initial-boundary value problem with time-dependent boundary conditions over bounded and unbounded domains is derived using the embedding method. The solution of the initial-boundary value problem, expressed in terms of a two-parameter auxiliary function, is used to obtain analytical solutions of moving boundary problems. In particular, a 'fractional' analogue of the Neumann solution to a classical Stefan problem for melting ice is found.