**Two-Dimensional Advection–Di**ff**usion Process with Memory and Concentrated Source**

#### **Najma Ahmed 1, Nehad Ali Shah 2 and Dumitru Vieru 3,\***


Received: 25 June 2019; Accepted: 3 July 2019; Published: 4 July 2019

**Abstract:** Two-dimensional advection–diffusion processes with memory and a source concentrated in the symmetry center of the domain have been investigated. The differential equation of the studied model is a fractional differential equation with short-tail memory (a differential equation with Caputo–Fabrizio time-fractional derivatives). An analytical solution of the initial-boundary value problem has been determined by employing the Laplace transform and double sine-Fourier transforms. A numerical solution of the studied problem has been determined using finite difference approximations. Numerical simulations for both solutions have been carried out using the software Mathcad.

**Keywords:** Advection–diffusion; fractional derivative; concentrated source; integral transform
