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

Abstract

AbstractTime-fractional parabolic equations with a Caputo time derivative of order $$\alpha \in (0,1)$$ α ∈ ( 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 $$m\times m$$ m × m matrices (where m is the order of the collocation scheme), are verified both analytically, for all $$m\ge 1$$ m ≥ 1 and all sets of collocation points, and computationally, for all $$ m\le 20$$ m ≤ 20 . The semilinear case is also addressed.