On the Solution Existence for Collocation Discretizations of Time-Fractional Subdiffusion Equations

Abstract

Time-fractional parabolic equations with a Caputo time derivative of order \(\alpha \in (0,1)\) are discretized in time using continuous collocation methods. For such discretizations, we give sufficient conditions for existence and uniqueness of their solutions. Two approaches are explored: the Lax–Milgram Theorem and the eigenfunction expansion. The resulting sufficient conditions, which involve certain \(m\times m\) matrices (where m is the order of the collocation scheme), are verified both analytically, for all \(m\ge 1\) and all sets of collocation points, and computationally, for all \( m\le 20\). The semilinear case is also addressed.