2025/05/10 by Peng Ding, Zhiping Mao, Ding, Peng +1
Engineering · Mathematics · #Advanced Control Systems Design #FOS: Mathematics #Fractional Differential Equations Solutions #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.2505.06565
openalex publication_date 2025/05/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A novel efficient and high accuracy numerical method for the time-fractional differential equations (TFDEs) is proposed in this work. We show the equivalence between TFDEs and the integer-order extended parametric differential equations (EPDE) by dimensional expanding, and establish the stability of EPDE. We apply BDF-k formula for the temporal discretization, while we use the Jacobi spectral collocation method for the discretization of the extended direction. We analyze the stability of the proposed method and give rigorous error estimates with order O(Δtk + M-m), where Δt and M are time step size and number of collocation nodes in extended direction, respectively. Also, we point out that the computational cost and the storage requirement is essentially the same as the integer problems, namely, the computational cost and the storage of the present algorithm are O(N) and O(1), respectively, where N is the total number of time step. We present several numerical examples, including both linear and nonlinear problems, to demonstrate the effectiveness of the proposed method and to validate the theoretical results