Numerical Solution of Nonlinear Reaction-Advection-Diffusion Equation in Sense of Caputo-Fabrizio Derivative with Finite Difference and Collocation Method

In this paper, we consider a nonlinear reaction-diffusion equation with a Caputo-Fabrizio derivative and its solution is obtained by the finite difference collocation method. First, we approximate the Caputo-Fabrizio derivative with the aid of shifted Legendre polynomials. To deal with the time derivative, a finite difference scheme is applied, and to deal with the spatial Caputo-Fabrizio derivative, the shifted Legendre spectral collocation method is used. After using spectral method to the problem, the problem reduces to the system of PDE with time fractional derivative. This system of PDEs is reduced to a system of algebraic equations by applying the finite difference scheme, and the resulting algebraic system is solved with the support of initial conditions. To signify the efficiency and validity of the developed scheme, a few numerical examples are solved whose absolute error between exact and numerical results is presented in tabular form.