Shifted Chebyshev Adapted Pseudo Collocation Method for Multidimensional Time-Space Fractional Diffusion Equations with Robin Boundary Conditions

This paper introduces a highly accurate spectral framework, termed the Shifted Chebyshev Adapted Pseudo Collocation (SCAPC) method, for solving multidimensional time-space fractional diffusion equations subject to Robin boundary conditions. The proposed approach integrates a spectral spatial discretization based on shifted Chebyshev polynomials with a Generalized Exponential Time Differencing (GETD) scheme for temporal integration. By leveraging the eigendecomposition of a modified Chebyshev discrete Laplacian, the SCAPC method efficiently approximates the spectral fractional Laplacian while strictly enforcing complex boundary conditions. We provide a rigorous theoretical analysis, establishing detailed theorems on derivative approximation, boundary condition enforcement, spectral convergence, global error bounds, and the stability and conditioning of the resulting algebraic systems. Comprehensive numerical experiments on benchmark problemsincluding non-smooth source terms and coupled fractional systemsdemonstrate that the SCAPC method achieves exponential (spectral) spatial convergence, vastly outperforming traditional second-order finite difference-based matrix transfer techniques in both accuracy and computational efficiency.