Hahn polynomials method for ψ-Caputo fractional diffusion-wave equation involving damping and reaction terms

In this paper, the $\psi$-Caputo derivative is employed to develop a novel formulation of the fractional 2D diffusion-wave equation, incorporating damping and reaction terms. To solve this equation efficiently, a collocation algorithm is designed using the orthonormal discrete Hahn polynomials (ODHPs). A key component of this approach is the construction of a fractional integral matrix for the ODHPs, which plays a crucial role in the numerical solution process. By representing the fractional derivative term through an ODHPs finite expansion, including several undetermined coefficients, and integrating the fractional matrix, the problem is transformed into an algebraic system of equations. More specifically, the undetermined coefficients are computed by solving this algebraic system, ultimately yielding the solution to the primary equation. The accuracy and effectiveness of the developed method are validated through three illustrative examples, demonstrating its reliability in handling fractional diffusion-wave equation.