Numerical approximation of the space-time Caputo-Fabrizio fractional derivative and application to groundwater pollution equation

Recently, Caputo and Fabrizio proposed a new derivative with fractional order without singular kernel. The derivative has several interesting properties that are useful for modeling in many branches of sciences. For instance, the derivative is able to describe substance heterogeneities and configurations with different scales. In order to accommodate researchers dealing with numerical analysis, we propose a numerical approximation in time and space. We modified the advection dispersion equation by replacing the time derivative with the new fractional derivative. We solve numerically the modified equation using the proposed numerical approximation. The stability and convergence analysis of the numerical scheme were presented together with some simulations.