A discrete spectral method for time fractional fourth-order 2D diffusion-wave equation involving ψ-Caputo fractional derivative

In this work, the $\psi$-Caputo fractional derivative, as a generalization of the classical Caputo fractional derivative in which the fractional derivative of a sufficiently differentiable function is defined with respect to another strictly increasing function, $\psi$, is utilized to define the time fractional fourth-order 2D diffusion-wave equation. A Chebyshev–Gauss–Lobatto scheme is developed to solve this equation. In this way, a new operational matrix for the $\psi$-Riemann–Liouville fractional integration of the Chebyshev polynomials is derived. The solution of the equation is obtained by determining the solution of the algebraic system extracted from approximation the $\psi$-Caputo fractional derivative term via a finite discrete Chebyshev series and employing the expressed operational matrix. The validity of the established approach is examined by solving two examples.