Amirkabir University of Technology AUT Journal of Mathematics and Computing 2783-2449 7 2 2026 04 01 Numerical methods for the time-fractional diffusion equation: A review 213 269 5980 10.22060/ajmc.2026.24592.1441 EN Amin Ghoreyshi Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology (Tehran Polytechnic), No. 424, Hafez Ave., 15914 Tehran, Iran 0009-0001-7239-5342 Mostafa Abbaszadeh Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology (Tehran Polytechnic), No. 424, Hafez Ave., 15914 Tehran, Iran 0000-0001-6954-3896 Mehdi Dehghan Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology (Tehran Polytechnic), No. 424, Hafez Ave., 15914 Tehran, Iran 0000-0002-2573-9755 Journal Article 2025 08 18 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 Grunwald-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. Time-fractional diffusion equation L-type approximations Weighted and shifted Grunwald-Letnikov formulas Finite difference methods finite element method Meshless Method Spectral Element Method Chebyshev spectral method Finite block method Theoretical analysis https://ajmc.aut.ac.ir/article_5980_63dfdeb1ff9ff09ecc3f05d2d7221ffa.pdf