Derakhshan, Mohammad HosseinOrdokhani, YadollahKumar, PushpendraGomez-Aguilar, J. F.2025-06-152025-06-1520250920-85421573-048410.1007/s11227-025-07287-72-s2.0-105005263692https://doi.org/10.1007/s11227-025-07287-7https://hdl.handle.net/20.500.14517/7972This manuscript proposes an efficient hybrid numerical approach that combines high accuracy with low computational cost to approximate solutions of the time-space diffusion model governed by the Caputo and Riesz fractional derivatives. Addressing the fractional time derivative in the Caputo sense, we employ a combination of quadratic and linear interpolations, achieving an accuracy of O{(Delta t)2-gamma)}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {O}\big \{\big (\Delta t\big )<^>{2-\gamma }\big )\big \}$$\end{document}. For the spatial discretisation of the Riesz space-fractional operator, we utilise a compact finite difference scheme with fourth-order accuracy. The stability and convergence properties of the proposed numerical method are rigorously analysed and verified. Finally, we provide with several numerical examples, supplemented by graphical and tabular illustrations, to demonstrate the accuracy, efficiency, and robustness of the proposed approach.eninfo:eu-repo/semantics/closedAccessDiffusion ModelCaputo And Riesz Fractional DerivativesFinite Difference MethodAlternating Direction Implicit MethodStability And Convergence AnalysisArticle