High-accuracy pseudospectral scheme for fractional diffusion-wave problems based on cardinal functions.

Abstract

Fractional diffusion-wave equations (FDWEs) are essential for modeling complex physical phenomena with memory and hereditary properties, such as anomalous diffusion and viscoelasticity, which classical integer-order models fail to capture accurately. In this paper, we introduce an efficient and high-accuracy pseudospectral scheme utilizing Legendre cardinal functions (LCFs) as basis functions to solve both second- and fourth-order FDWEs. By reformulating the governing equations into equivalent integral forms and developing direct matrix representations for the Caputo fractional derivative and fractional integral operators, we systematically transform the original problem into a solvable system of algebraic equations. Detailed convergence analysis and numerical experiments confirm that this method consistently achieves spectral convergence. A key novelty of this technique, and what significantly advances it beyond previous efforts in the literature, is its exploitation of the cardinal properties of LCFs to avoid numerical integral evaluations when computing basis coefficients entirely. Consequently, this approach dramatically reduces computational overhead while delivering superior accuracy and efficiency compared to existing finite difference and collocation methods.