Stability and convergence analysis of the quadratic spline collocation method for time-dependent fractional diffusion equations

摘要

A quadratic spline collocation method combined with the Crank–Nicolson time discretization is proposed for time-dependent two-sided fractional diffusion equations. By carefully analyzing the mathematical properties of the coefficient matrix, the new scheme is proved to be unconditionally stable in the sense of discrete L2-norm for α ∈ [α*, 2), where α is the order of the space-fractional derivative of the fractional diffusion equation, and α* ≈ 1.2576 (see Lemma 3.1). Furthermore, the fractional-order spline interpolation error over the collocation points is studied, and subsequently we show that the spline collocation solution of the fractional diffusion equation converges to the exact one with order O(h3−α+τ2) under the discrete L2-norm, where τ and h are the temporal and spatial step sizes, respectively. Finally, numerical experiments are given to verify the theoretical results.