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

Area of Science:

• Computational Mathematics
• Applied Mathematics
• Numerical Analysis

Background:

• Fractional diffusion-wave equations (FDWEs) model complex phenomena like anomalous diffusion and viscoelasticity.
• Classical integer-order models are insufficient for systems with memory and hereditary properties.
• Existing numerical methods for FDWEs often face limitations in accuracy and efficiency.

Purpose of the Study:

• To introduce an efficient and high-accuracy pseudospectral scheme for solving second- and fourth-order FDWEs.
• To develop a method that avoids numerical integral evaluations by leveraging the cardinal properties of Legendre cardinal functions (LCFs).
• To provide a computationally less intensive and more accurate alternative to existing numerical techniques.

Main Methods:

• A pseudospectral scheme employing Legendre cardinal functions (LCFs) as basis functions.
• Reformulation of FDWEs into equivalent integral forms.
• Development of direct matrix representations for Caputo fractional derivative and integral operators.
• Systematic transformation of the problem into a system of algebraic equations.

Main Results:

• The proposed method achieves spectral convergence for both second- and fourth-order FDWEs.
• Numerical experiments confirm the high accuracy and efficiency of the LCF-based scheme.
• The technique successfully avoids numerical integral evaluations, reducing computational overhead.
• The method demonstrates superior performance compared to finite difference and collocation approaches.

Conclusions:

• The LCF-based pseudospectral scheme offers an efficient and accurate solution for FDWEs.
• This novel approach significantly reduces computational cost while maintaining high accuracy.
• The method represents a substantial advancement in the numerical solution of fractional differential equations.