# The solution of direct and inverse fractional advection–dispersion problems by using orthogonal collocation and differential evolution

## Abstract

The advection–dispersion phenomenon can be observed in various fields of science. Mathematically, this process can be studied by considering empirical models, high-order differential equations, and fractional differential equations. In this paper, a fractional model considered to represent the transport of passive tracers carried out by fluid flow in a porous media is studied both in the direct and inverse contexts. The studied mathematical model considers a one-dimensional fractional advection–dispersion equation with fractional derivative boundary conditions. The solutions of both direct and inverse problems are obtained by using the orthogonal collocation method and the differential evolution optimization algorithm approaches, respectively. In this case, the source term along the spatial and time coordinates is taken as a design variable. The obtained results with the solution of the direct problem are compared with those determined by using an implicit finite difference scheme. The results indicate that the proposed approach characterizes a promising methodology to solve the direct and inverse fractional advection–dispersion problems.

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## Acknowledgements

The authors are thankful for the financial support provided to the present research effort by CAPES, CNPq (574001/2008-5, 304546/2018-8, 439126/2018-5 and 431337/2018-7), FAPEMIG (TEC-APQ-3076-09, TEC-APQ-02284-15, TEC-APQ-00464-16, and PPM-00187-18), and CAPES through the INCT-EIE.

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Correspondence to R. A. Borges.

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