A numerical approach for multi-dimensional ψ-Hilfer fractional nonlinear Galilei invariant advection–diffusion equations

Abstract

In this paper, we introduce the $\psi$-Hilfer fractional version of nonlinear Galilei-invariant advection–diffusion equations in one and two dimensions. A new type of basic functions, namely the $\psi$-Chebyshev cardinal functions (CFs), is introduced to establish a hybrid numerical strategy to solve these equations. The key advantageous property of these functions is the simplicity of computing their $\psi$-Hilfer fractional derivative. Utilizing this property, a new operational matrix for the $\psi$-Hilfer fractional derivative of these functions is derived. Consequently, a hybrid numerical strategy based on the shifted Chebyshev polynomials (CPs) and $\psi$-Chebyshev CFs is proposed to solve these equations. More precisely, in the proposed strategy, a finite expansion for the solution of the equation under investigation is considered. The shifted CPs are used to approximate the solution in the spatial domain, while the $\psi$-Chebyshev CFs are utilized to approximate the solution in the temporal domain. By applying the $\psi$-Hilfer fractional derivative operational matrix of the $\psi$-Chebyshev CFs, the classical derivatives operational matrices of the shifted CPs, and employing the collocation method, the solution of the equation under consideration is obtained by solving a system whose elements are algebraic equations. The accuracy of the presented strategy is examined by numerous examples.