Fundamental solutions to advection-diffusion equation with time-fractional Caputo-Fabrizio derivative

The advection-diffusion equation with time-fractional derivatives without singular kernel and two space-variables is considered. The fundamental solutions in a half-plane are obtained by using the Laplace transform with respect to temporal variable t and Fourier transform with respect to the space coordinates x and y. The Dirichlet problem and the source problem are investigated. The general solution for the fractional case is particularized for the ordinary case of the normal advection-diffusion phenomena and for the fractional/normal diffusion process. It should be noted that, it is more advantageous to use the time-fractional derivative without singular kernel (Caputo-Fabrizio time-fractional derivative), instead of Caputo time-fractional derivative. The advantages are reflected both in the handling of computations, but especially in expressing convenient of solutions. Some numerical calculations are carried out and the results are discussed and illustrated graphically.