A Finite Block Method Framework for Nonlinear Fractional Integro-Differential Equations Mostafa Abbaszadeh, Amin Ghoreyshi, Mehdi Dehghan Mathematical Methods in the Applied Sciences, 2026 This paper introduces a robust and efficient numerical framework for solving nonlinear time‐fractional partial integro‐differential equations (NLTFPIDEs). Temporal discretization is performed using the weighted and shifted Grünwald–Letnikov formula, incorporating the fractional trapezoidal rule and the second‐order backward differentiation formula (BDF2). Spatial discretization leverages Chebyshev nodes as discretization points, with the Lagrange‐collocation method applied to approximate partial derivatives. For irregular computational domains, the framework utilizes the finite block method (FBM) in two dimensions. Nonlinearities in the equations are handled through the quasilinearization technique. A comprehensive stability and convergence analysis using the energy method confirms the reliability of the proposed schemes. Numerical experiments validate the theoretical findings, highlighting the accuracy and efficiency of the approach.
Numerical methods for the time-fractional diffusion equation: A review Amin Ghoreyshi, Mostafa Abbaszadeh, Mehdi Dehghan Aut Journal of Mathematics and Computing, 2026 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.
Numerical methods for the time-fractional diffusion equation: A review A Ghoreyshi, M Abbaszadeh, M Dehghan AUT Journal of Mathematics and Computing 7 (2), 213-269 , 2026 2026
A Finite Block Method Framework for Nonlinear Fractional Integro‐Differential Equations M Abbaszadeh, A Ghoreyshi, M Dehghan Mathematical Methods in the Applied Sciences 49 (4), 2170-2188 , 2026 2026 Citations: 4
An accurate and robust numerical method for solving distributed-order space–time fractional PDEs A Ghoreyshi, M Abbaszadeh, MA Zaky, M Dehghan Zeitschrift für Angewandte Mathematik und Physik 76 (6), 242 , 2025 2025 Citations: 2
A Finite Difference–Finite Block Scheme for Nonlinear Fractional Partial Integro–Differential Equations A Ghoreyshi, M Abbaszadeh, M Dehghan 56th Annual Iranian Mathematics Conference, 787-791 , 2025 2025
Finite block method for nonlinear time-fractional partial integro-differential equations: stability, convergence, and numerical analysis A Ghoreyshi, M Abbaszadeh, MA Zaky, M Dehghan Applied Numerical Mathematics 214, 82-103 , 2025 2025 Citations: 21
Two high-order numerical schemes based on the Lagrange polynomials for solving a distributed-order time-fractional partial integro-differential equation on non-rectangular domains A Ghoreyshi, M Abbaszadeh, MA Zaky, M Dehghan Journal of Applied Mathematics and Computing 71, 7271-7311 , 2025 2025 Citations: 7
MOST CITED SCHOLAR PUBLICATIONS
Finite block method for nonlinear time-fractional partial integro-differential equations: stability, convergence, and numerical analysis A Ghoreyshi, M Abbaszadeh, MA Zaky, M Dehghan Applied Numerical Mathematics 214, 82-103 , 2025 2025 Citations: 21
Two high-order numerical schemes based on the Lagrange polynomials for solving a distributed-order time-fractional partial integro-differential equation on non-rectangular domains A Ghoreyshi, M Abbaszadeh, MA Zaky, M Dehghan Journal of Applied Mathematics and Computing 71, 7271-7311 , 2025 2025 Citations: 7
A Finite Block Method Framework for Nonlinear Fractional Integro‐Differential Equations M Abbaszadeh, A Ghoreyshi, M Dehghan Mathematical Methods in the Applied Sciences 49 (4), 2170-2188 , 2026 2026 Citations: 4
An accurate and robust numerical method for solving distributed-order space–time fractional PDEs A Ghoreyshi, M Abbaszadeh, MA Zaky, M Dehghan Zeitschrift für Angewandte Mathematik und Physik 76 (6), 242 , 2025 2025 Citations: 2
Numerical methods for the time-fractional diffusion equation: A review A Ghoreyshi, M Abbaszadeh, M Dehghan AUT Journal of Mathematics and Computing 7 (2), 213-269 , 2026 2026
A Finite Difference–Finite Block Scheme for Nonlinear Fractional Partial Integro–Differential Equations A Ghoreyshi, M Abbaszadeh, M Dehghan 56th Annual Iranian Mathematics Conference, 787-791 , 2025 2025
