Numerical Solution of Diffusion Equation with Caputo Time Fractional Derivatives Using Finite-Difference Method with Neumann and Robin Boundary Conditions

Many problems in various branches of science, such as physics, chemistry, and engineering have been recently modeled as fractional ODEs and fractional PDEs. Thus, methods to solve such equations, especially in the nonlinear state, have drawn the attention of many researchers. The most important goal of researchers in solving such equations has been set to provide a solution with the possible minimum error. The fractional PDEs can be generally classified into two main types, spatial-fractional, and time-fractional differential equations. This study was designed to provide a numerical solution for the fractional-time diffusion equation using the finite-difference method with Neumann and Robin boundary conditions. The time fraction derivatives in the concept of Caputo were considered, also the stability and convergence of the proposed numerical scheme have been completely proven and a numerical test was also designed and conducted to assess the efficiency and precision of the proposed method. Eventually, it can be said that based on findings, the present technique can provide accurate results.