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An improved meshless finite integration method for the time fractional diffusion and high order equations

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  • Liu, Pengyuan
  • Lei, Min
  • Hon, Yiu-Chung

Abstract

In this paper, an Improved Finite Integration Method (IFIM) is developed for the numerical solution of time-fractional diffusion equations (TFDEs) and time fractional partial differential equations (TFPDEs) of the fourth order in space. For the temporal discretization, the Hadamard finite-part integral is employed and approximated using piecewise Quadratic Interpolation Polynomials (QIP), which exhibit superior accuracy compared to the classical first-order finite difference scheme (L1). For spatial discretization, a multilayer Modified Shifted Chebyshev Integration (M-SCI) scheme is explicitly constructed to compute high-order integration matrices analytically. This modification significantly improves computational performance in solving high-order spatial differential operators. Furthermore, the error estimates for the IFIM incorporating the QIP-M-SCI scheme are established in the context of TFDEs. Several numerical experiments are presented to verify the theoretical results and demonstrate the effectiveness of the proposed method. Comparative studies are conducted among three numerical schemes – L1-M-SCI, QIP-SCI, and QIP-M-SCI – highlighting the superior accuracy and efficiency of the IFIM with QIP-M-SCI in solving both TFDEs and TFPDEs.

Suggested Citation

  • Liu, Pengyuan & Lei, Min & Hon, Yiu-Chung, 2026. "An improved meshless finite integration method for the time fractional diffusion and high order equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 241(PA), pages 15-39.
    • Handle: RePEc:eee:matcom:v:241:y:2026:i:pa:p:15-39
    DOI: 10.1016/j.matcom.2025.08.014
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    References listed on IDEAS

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    1. Yun, D.F. & Wen, Z.H. & Hon, Y.C., 2015. "Adaptive least squares finite integration method for higher-dimensional singular perturbation problems with multiple boundary layers," Applied Mathematics and Computation, Elsevier, vol. 271(C), pages 232-250.
    2. Roul, Pradip & Prasad Goura, V.M.K., 2020. "A high order numerical method and its convergence for time-fractional fourth order partial differential equations," Applied Mathematics and Computation, Elsevier, vol. 366(C).
    3. Ampol Duangpan & Ratinan Boonklurb & Tawikan Treeyaprasert, 2019. "Finite Integration Method with Shifted Chebyshev Polynomials for Solving Time-Fractional Burgers’ Equations," Mathematics, MDPI, vol. 7(12), pages 1-24, December.
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