Initial-boundary value problem for a fractional heat equation on an interval.

This article focuses on solving a Dirichlet initial-boundary value problem for a fractional heat equation involving the Riemann–Liouville fractional derivative of order \(1 < \alpha \leq 2\) on a finite interval. Employing the Fokas method, which utilizes Lax pairs and the associated Riemann–Hilbert problem, the authors derive an integral representation of the solution, extending previous work that primarily addressed half-line domains. The paper details the construction of the formal adjoint operator, the derivation of Lax pairs for the fractional heat equation, and the formulation of a global relation leading to an explicit integral formula for the solution on a finite spatial interval. This approach generalizes classical results for the heat equation and contributes to the analysis of fractional differential equations with boundary conditions on bounded domains.