Solving a fractional diffusion PDE using some standard and nonstandard finite difference methods with conformable and Caputo operators - Appadu, Appanah R.; Kelil, Abey S.; Nyingong, Ndifon Wikocho (Frontiers in Applied Mathematics and Statistics, 2024)
Fractional diffusion equations offer an effective means of describing transport phenomena exhibiting abnormal diffusion patterns, often eluding traditional diffusion models. In this paper, we construct four finite difference methods whereby the fractional derivative is approximated using either conformable or Caputo operators and then finite difference approximations are used. To study the stability of the proposed numerical schemes, we employ von Neumann stability analysis or establish conditions under which the schemes replicate/preserve the positivity of the continuous model. Furthermore, consistency analysis is done for all the four methods. Numerical results are presented with fractional parameter (α) set to 0.75, 0.90, 0.95, and 1.0. We also compute the rate of convergence in time for the four methods.