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June 1, 20260 citationsOpen Access

Numerical methods for the time-fractional diffusion equation: A review

AGAmin GhoreyshiMAMostafa AbbaszadehMDMehdi Dehghan

Key Points

  • This review aims to summarize various numerical methods for solving the time-fractional diffusion equation.
  • Discusses L-type approximations and Grünwald-Letnikov-based formulas for time-fractional derivatives.
  • Reviews spatial methods including compact finite difference, finite element, spectral element, meshless, Chebyshev spectral, and finite block methods.
  • Presents stability and convergence theorems with supporting numerical examples.
  • Numerical examples demonstrate the effectiveness of various discretization techniques.
  • Stability and convergence theorems confirm the reliability of the methods employed.

Abstract

This review paper focuses on the numerical solution of the time-fractional diffusion equation using various discretization techniques. For the time-fractional derivative, we consider methods such as L-type approximations and Grünwald-Letnikov-based formulas, while for the spatial diffusion term, we utilize the compact finite difference method, finite element method, spectral element method, meshless method, Chebyshev spectral method, and finite block method. In addition, stability and convergence theorems are presented, accompanied by numerical examples that confirm the theoretical results.

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Cite This Study

Ghoreyshi et al. (2026) studied this question.

synapsesocial.com/papers/6a1d224302fbce91306380d0https://doi.org/10.22060/ajmc.2026.24592.1441
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