An Efficient Spectral–Time Discretization Method for Partial Fractional Differential Equations in Weighted Sobolev Space

This study presents a novel numerical framework for solving fractional differential equations, in which the spatial domain is approximated by an expansion in Chebyshev polynomials of the eighth kind. For time discretization, the L1-2 scheme is employed due to its high accuracy in approximating fractional derivatives with nonlocal memory effects. The spectral theorem for the chosen approximation is established, and the stability and convergence of the proposed scheme are examined in a weighted Sobolev space. To demonstrate its practical applicability and validate the theoretical results, several numerical experiments are conducted on problems involving fractional dynamics. The numerical results confirm the high accuracy, robustness, and efficiency of the combined spectral and time discretization approaches. These findings highlight the advantages of combining high-order spectral techniques with advanced fractional time-stepping methods, resulting in a robust and reliable computational framework for accurately and stably simulating fractional differential equations, with broad potential applications in physics, engineering, and the applied sciences. The novelty of this work lies in the first-time use of eighth-kind Chebyshev polynomials for the spatial approximation of fractional differential equations, coupled with the L1-2 time discretization scheme. Moreover, new spectral theorems are established, along with a rigorous stability and convergence analysis in weighted Sobolev spaces.