A numerical approach for multi-dimensional <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg" display="inline" id="d1e4206"><mml:mi>ψ</mml:mi></mml:math>-Hilfer fractional nonlinear Galilei invariant advection–diffusion equati

In this paper, we introduce the ψ -Hilfer fractional version of nonlinear Galilei-invariant advection–diffusion equations in one and two dimensions. A new type of basic functions, namely the ψ -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 ψ -Hilfer fractional derivative. Utilizing this property, a new operational matrix for the ψ -Hilfer fractional derivative of these functions is derived. Consequently, a hybrid numerical strategy based on the shifted Chebyshev polynomials (CPs) and ψ -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 ψ -Chebyshev CFs are utilized to approximate the solution in the temporal domain. By applying the ψ -Hilfer fractional derivative operational matrix of the ψ -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. • ψ -Hilfer fractional nonlinear Galilei invariant advection–diffusion equations are defined. • A new family of basic functions, namely the ψ -Chebyshev cardinal functions are introduced. • A novel matrix for the ψ -Hilfer fractional derivative of these functions is obtained. • Chebyshev polynomials and ψ -Chebyshev cardinal functions are simultaneously used for these equations. • The accuracy and correctness of the developed method are checked by solving some examples.