Aplicação das equações diferenciais fracionárias em problemas de geociências: soluções analíticas usando o método da decomposição

Abstract

In this work, analytical solutions of fractional differential equations are obtained for three traditional problems in geophysics: (i) the case of a point source located on the surface injecting electric current into the ground; (ii) the case of a plane electromagnetic wave, and iii) the case of a point source releasing atmospheric pollutants. To obtain the analytical solutions of the proposed problems, the method of decomposition by Laplace (MDL) was used, which provides a solution in fast convergence series. In particular, for the problem of dispersion of pollutants in the planetary boundary layer, a solution of the three-dimensional diffusion-advection equation was obtained using the CDM method in two variables, considering the Caputo derivative (non-local) and the conformable derivative (local ). The proposed mathematical modeling represents an important advance in the area, considering that the fractional calculus is the generalization of the traditional whole order calculus. The 2D transient difusion-advection equation was also semi-analytically solved. For this solution, the double Laplace transform and the numerical inverse Fixed-Talbot were used.