2023/10/04 by Xiao Ye, Ye, Xiao, Jun Liu +7
Mathematics · #65M06 #65M12 #Differential Equations and Numerical Methods #FOS: Mathematics #Fractional Differential Equations Solutions #G.1.8 #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.2310.02775
openalex publication_date 2023/10/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
In this paper, we consider the variable-order time fractional mobile/immobile diffusion (TF-MID) equation in two-dimensional spatial domain, where the fractional order α(t) satisfies 0<α*≤ α(t)≤ α*<1. We combine the quadratic spline collocation (QSC) method and the L1+ formula to propose a QSC-L1+ scheme. It can be proved that, the QSC-L1+ scheme is unconditionally stable and convergent with O(τ^min\3-α^*-α(0),2\ + Δx2+Δy2), where τ, Δx and Δy are the temporal and spatial step sizes, respectively. With some proper assumptions on α(t), the QSC-L1+ scheme has second temporal convergence order even on the uniform mesh, without any restrictions on the solution of the equation. We further construct a novel alternating direction implicit (ADI) framework to develop an ADI-QSC-L1+ scheme, which has the same unconditionally stability and convergence orders. In addition, a fast implementation for the ADI-QSC-L1+ scheme based on the exponential-sum-approximation (ESA) technique is proposed. Moreover, we also introduce the optimal QSC method to improve the spatial convergence to fourth-order. Numerical experiments are attached to support the theoretical analysis, and to demonstrate the effectiveness of the proposed schemes.