An Efficient Spectral Technique Based on Operational Matrices to Solve Variable Coefficients Time-Space Fractional Advection-Diffusion Equations

This investigation presents a robust spectral technique for accurately solving timespace fractional advection-diffusion equations (FADEs) characterized by Caputo-type derivatives with coefficients that vary in both time and space. Such equations inherently model nonlocality and memory effects prevalent in various complex transport and physical systems. The proposed strategy reformulates the underlying problem as a system of fractional-order ordinary differential equations (FODEs), leveraging operational matrices derived from shifted Jacobi polynomials (SJPs) within the spatial discretization framework. The initial conditions of the resulting FODEs are directly inherited from those of the original equation. The construction of explicit particular solutions for each FODE relies on the introduction of auxiliary initial value problems. Subsequently, the particular solutions are assembled into a linear expansion designed to minimize the residual error across the domain. We employ a weighted residual procedure to perform the minimization, ensuring average error suppression and greater solution precision. Unlike conventional spectral operational matrix methods that rely on full discretization, the present approach constructs explicit particular solutions for the resulting FODEs and determines the temporal coefficients through a residual minimization procedure. A rigorous theoretical validation is conducted via residual-based error estimation, confirming the convergence behavior of the scheme. Performance is assessed using test problems, where empirical convergence rates are calculated and compared with those obtained via other numerical methods. The comparative analysis underscores the enhanced accuracy and robustness of the method under consideration. These results demonstrate the method’s potential as a reliable tool for solving FADEs in scientific and mathematical applications.