On the Solution existence for collocation discretizations of time-fractional subdiffusion equations

Time-fractional parabolic equations with a Caputo time derivative of order 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×m matrices (where m is the order of the collocation scheme), are verified both analytically, for all m ≥ 1 and all sets of collocation points, and computationally, for all m ≤ 20. The semilinear case is also addressed.