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

Abstract

In this work, the ψ-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, ψ, 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 ψ-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 ψ-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.

Keywords

• ψ-Caputo fractional derivative
• Fractional fourth-order 2D diffusion-wave equation
• Chebyshev polynomials
• Chebyshev–Gauss–Lobatto points