Numerical approximation of Atangana-Baleanu Caputo derivative for space-time fractional diffusion equations

In this study, we attempt to obtain the approximate solution for the time-space fractional linear and nonlinear diffusion equations. A finite difference approach is given for the solution of both linear and nonlinear fractional order diffusion problems. The Riesz fractional derivative in space is specifically approximated using the centered difference scheme. A system of Atangana-Baleanu Caputo equations that have been converted through spatial discretization is solved using a newly developed modified Simpson's 1/3 formula. A study of the proposed scheme is done to ascertain its stability and convergence. It has been shown that for mesh size h and time steps $\delta t$ the recommended method converges at a rate of $O(\delta t^2+h^2)$. Based on graphic results and numerical examples, the application of the model is also examined.