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

In the present work, we investigate a distributed-order time-fractional partial integro-differential equation (DOTFPIDE) in one and higher dimensions, employing the graded temporal mesh for discretization. The distributed-order time-fractional derivative is approximated using the nonuniform \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_1$$\end{document} formula, while the fractional integral term is discretized via a product integration approach. For spatial discretization, Chebyshev nodes are utilized as collocation points, and a hybrid methodology combining the alternating direction implicit scheme with collocation techniques is applied to address multi-dimensional problems. In irregular computational domains, the finite block method is adopted for two-dimensional DOTFPIDEs. The stability and convergence of the numerical schemes are rigorously analyzed, and computational experiments are conducted to validate their theoretical accuracy and efficiency. These results demonstrate the robustness of the proposed framework in solving complex distributed-order fractional integro-differential equations across diverse geometries.