An Efficient Numerical Approach for Fractional Heat Equations with Nonlocal Memory Terms

Résumé

This paper presents a numerical method for solving partial integro-differential equations with weakly singular kernels, using a tempered φ-Caputo fractional derivative of order α ∈ (0, 1). We apply a second-order time discretization and use a tempered fractional integral operator along with piecewise linear interpolation to handle the singularity in the kernel. The stability of the method is analyzed using Von Neumann stability analysis. Finally, numerical examples are provided to demonstrate the effectiveness of the approach.