A highly accurate method for multi-term time fractional diffusion equation in two dimensions with ψ-Caputo fractional derivative

In this study, the $\psi$-Caputo fractional derivative (as a generalization of the classical Caputo derivative where the fractional derivative is defined with respect to the function $\psi$) is considered to introduce a class of multi-term time fractional 2D diffusion equations. A numerical method based on the Chebyshev cardinal polynomials (CCPs) is proposed to solve this problem. In this way, a new operational matrix for the $\psi$-Caputo fractional derivative of the CCPs is provided. By approximating the solution of the problem by a finite series of the CCPs (with some unknown coefficients) and employing the derived fractional matrix, an algebraic system of equations is generated, which by solving it the expressed coefficients, and consequently, the problem’s solution are identified. The validity of the established method is investigated by solving some numerical examples.