Chebyshev spectral method for the variable-order fractional mobile–immobile advection–dispersion equation arising from solute transport in heterogeneous media

Abstract

This study focuses on the numerical solution of the space–time variable-order fractional derivative mobile–immobile advection–dispersion equation. These equations are preferred for dynamical system modeling, which includes determining the solute transport mechanism. We take into account the Caputo class of fractional derivatives. A Chebyshev collocation method in both the space and time directions is utilized for the approximate solution of the equation. The convergence analysis of the method is shown by using the Chebyshev-weighted Sobolev space. Finally, some examples are provided to illustrate the accuracy and efficiency of the present approach. Comparisons between the results obtained by our method and those obtained by the existing methods are presented to demonstrate the superiority of the proposed methodology. The spectral or exponential convergence of the suggested approach is one of its essential characteristics, which is verified through numerical results and provides additional support for the effectiveness of the suggested technique.

Appendix A

Kronecker product: If A and B are two matrices of dimension $r_1 \times s_1$ and $r_2 \times s_2$, respectively, then Kronecker product of A and B, denoted as $A \otimes B$, is

$$\begin{aligned}A \otimes B = \left[ {\begin{array}{*{20}{c}} {{a_{11}}B}&{}{{a_{12}}B}&{} \cdots &{}{{a_{1{s_1}}}B}\\ {{a_{21}}B}&{}{{a_{22}}B}&{} \cdots &{}{{a_{2{s_1}}}B}\\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ {{a_{{r_1}1}}B}&{}{{a_{{r_1}2}}B}&{} \cdots &{}{{a_{{r_1}{s_1}}}B} \end{array}} \right] \in {\mathbb {R}^{{r_1}{r_2} \times {s_1}{s_2}}}. \end{aligned}$$

Hadamard product: The Hadamard product, also known as Schur product, of two matrices M and N with same dimension is the element-wise product, i.e.,

$$\begin{aligned} {\left( {M \circ N} \right) _{ij}} = {\left( M \right) _{ij}}{\left( N \right) _{ij}}. \end{aligned}$$

Vectorization of a matrix: If A is a matrix of dimension $r_1 \times s_1$ defined as

$$\begin{aligned}A = \left[ {\begin{array}{*{20}{c}} {{a_{11}}}&{}{{a_{12}}}&{} \cdots &{}{{a_{1{s_1}}}}\\ {{a_{21}}}&{}{{a_{22}}}&{} \cdots &{}{{a_{2{s_1}}}}\\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ {{a_{{r_1}1}}}&{}{{a_{{r_1}2}}}&{} \cdots &{}{{a_{{r_1}{s_1}}}} \end{array}} \right] , \end{aligned}$$

then vectorization of matrix A, denoted by $\textbf{A}$, is the column vector of dimension $r_1s_1 \times 1$ obtained by putting the columns of matrix A on top of one another, i.e.,

$$\begin{aligned} \textbf{A} = \left[ {{a_{11}},{a_{21}},....,{a_{{r_1}1}},{a_{12}},{a_{22}},....,{a_{{r_1}2}},.....,{a_{1{s_1}}},{a_{2{s_1}}},....,{a_{{r_1}{s_1}}}} \right] ^T. \end{aligned}$$