Numerical Method for the Diffusion‐Wave Equation With Time Fractional ψ$\psi$‐Caputo Derivative

ABSTRACT This work is devoted to numerical analysis and computation of the time fractional diffusion‐wave equations with the ‐Caputo derivative of order . The ‐Caputo derivative, characterized by its adaptive integral kernel function , offers a unified framework for modeling complex memory effects in anomalous diffusion. We first discuss the existence, uniqueness, regularity, and decay of the solution to the considered model. Subsequently, an efficient fully discrete scheme is developed by combining the H2N2 discretization in time with finite element method in space. Stability and convergence of the proposed scheme are rigorously analyzed. The proposed scheme turns out to be of th order convergence in time and th order convergence in space. Numerical experiments are conducted to corroborate the theoretical findings.