Analysis of a Laplace Spectral Method for Time-Fractional Advection-Diffusion Equations Incorporating the Atangana-Baleanu Derivative

In this article, we develop the Laplace transform (LT) based Chebyshev spectral collocation method (CSCM) to approximate the time fractional advection-diffusion equation, incorporating the Atangana-Baleanu Caputo (ABC) derivative. The advection-diffusion equation, which governs the transport of mass, heat, or energy through combined advection and diffusion processes, is central to modeling physical systems with nonlocal behavior. Our numerical scheme employs the LT to transform the time-dependent time-fractional PDEs into a time-independent PDE in LT domain, eliminating the need for classical time-stepping methods that often suffer from stability constraints. For spatial discretization, we employ the CSCM, where the solution is approximated using Lagrange interpolation polynomial based on the Chebyshev collocation nodes, achieving exponential convergence that outperforms the algebraic convergence rates of finite difference and finite element methods. Finally, the solution is reverted to the time domain using contour integration technique. We also establish the existence and uniqueness of the solution for the proposed problem. The performance, efficiency, and accuracy of the proposed method are validated through various fractional advection-diffusion problems. The computed results demonstrate that the proposed method has less computational cost and is highly accurate.