An efficient numerical method for solving a class of variable-order fractional mobile-immobile advection-dispersion equations and its convergence analysis

Abstract

This study provides a numerical scheme for solving a class of fractional partial differential equations with the variable order, referred to as fractional mobile-immobile advection-dispersion equations, using the sixth-kind Chebyshev polynomials. The fractional mobile-immobile advection-dispersion equations model complex systems involving heat diffusion, ocean acoustic propagation, and solute transport in rivers and streams. Integral operational matrices of integer and variable orders are derived and then the Collocation method is used to reduce the main equation to a system of algebraic equations. Appeared derivative and integral operators are in the Caputo and Riemann-Liouville sense, respectively. Two-variable Chebyshev polynomials of the sixth-kind are constructed and their operational matrices are derived using matrices obtained from the one-variable case. The existence and uniqueness of the solution of these equations are proved. A convergence analysis is performed in a two-dimensional Chebyshev-weighted Sobolev space and error bounds of approximate solutions are presented. It has been observed that approximation error tends to zero if the number of terms of the solution series is chosen sufficiently large. The proposed scheme is employed to solve three numerical examples in order to test its efficiency and accuracy. The convergence rate of the suggested method is numerically computed that confirms the exponential property of the convergence order of the method. Finally, the results are compared with the outcome of some existing methods to show the accuracy of the proposed method.