Numerical Solution of Caputo-Fabrizio Time Fractional Distributed Order Reaction-diffusion Equation via Quasi Wavelet based Numerical Method

In this paper, we derive a novel numerical method to find out the numerical solution of fractional partial differential equations (PDEs) involving Caputo-Fabrizio (C-F) fractional derivatives. We first find out the approximation formula of C-F derivative of function tk. We approximate the C-F derivative in time with the help of the Legendre spectral method and approximation formula of tk. The unknown function and their derivatives in spatial direction are approximated with the quasi wavelet-based numerical method. We apply this newly derived method to solve the nonlinear distributed order reaction-diffusion in which time-fractional derivative is of C-F type. The accuracy and validity of the proposed method is exhibited by giving a solution to some numerical examples. The obtained numerical results are compared with the analytical results and conclude that our proposed numerical method achieves accurate results. On the other hand, the method is easy to apply on higher-order fractional partial differential equations and variable-order fractional partial differential equations.

Keywords:

Fractional PDE , distributed order reaction-diffusion equation , Caputo-Fabrizio fractional derivative , quasi wavelet , Legendre polynomial