Analytical solution of the time fractional diffusion equation and fractional convection-diffusion equation

Francisco Gomez, Victor Morales, Marco Taneco


In this paper, we obtain analytical solutions for the time-fractional diffusion and time-fractional convection-diffusion equations. These equations are obtained from the standard equations by replacing the time derivative with a fractional derivative of order $\alpha$. Fractional operators of type Liouville-Caputo, Atangana-Baleanu-Caputo, fractional conformable derivative in Liouville-Caputo sense and Atangana-Koca-Caputo were used to model diffusion and convection-diffusion equation. The Laplace and Fourier transforms were applied to obtain the analytical solutions for the fractional order diffusion and convection-diffusion equations. The solutions obtained can be useful to understand the modeling of anomalous diffusive, subdiffusive systems and super-diffusive systems, transport processes, random walk and wave propagation phenomenon.


Fractional calculus; Mittag-Leffler kernel; fractional conformable derivative; Diffusion equation.

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Revista Mexicana de Física

ISSN: 0035-001X

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