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This paper proposes a new numerical method based on the Chebyshev wavelets(CWs) to solve the variable-order time fractional mobile-immobileadvection-dispersion equation. To do this, a new operational matrix ofvariable-order fractional derivative in the Caputo sense for the CWs is derivedand is used to obtain an approximate solution for the problem under study.Along the way a new family of piecewise functions is introduced and employed toderive a general method to compute this matrix. The main advantage behind theproposed approach is that the problem under consideration is transformed into alinear system of algebraic equations. So, it can be solved simply to obtain anapproximate solution. The efficiency and accuracy of the proposed method areshown for some concrete examples. These results show that the proposed methodis very efficient and accurate.

A new approach of the Chebyshev wavelets for the variable-order time fractional mobile-immobile advection-dispersion model