A Laplace-Chebyshev Spectral Method for Multi-Dimensional Anomalous Transport

Abstract

Anomalous transport processes, such as subsurface contaminant spread or wave attenuation in viscoelastic materials, are governed by time-fractional diffusion-wave equations (TFDWEs). The non-local nature of fractional operators and high computational cost of addressing multi-dimensional spaces pose significant challenges for numerical simulations. To overcome this, we develop a novel hybrid spectral method combining the Laplace transform (LT) technique with the Chebyshev spectral collocation method (CSCM) for solving TFDWEs featuring the  modified Atangana-Baleanu-Caputo derivative, chosen for its non-singular kernel and efficiency in modeling complex memory effects. Our numerical scheme, temporal and spatial discretizations are decoupled. The LT handles the fractional time derivative exactly in Laplace domain, removing time-stepping restrictions and convolution costs, while the CSCM enures the exponential convergence in the spatial domain. The numerical inversion of LT is achieved using the improved Talbot method, guaranteeing rapid $O(e^{-c\mathrm{N}})$ convergence. This work offers not only a robust computational technique but also rigorous mathematical analysis, establishing conditions for the solution existence, uniqueness, and Ulam-Hyers stability. The dimensional flexibility of our technique is demonstrated through 1D,~2D, and 3D numerical examples, which confirm its computational efficiency and high accuracy. This work provides a robust and stable numerical approach that can be extended to model complex multi-scale transport problems across applied mathematics and engineering.