Solving a fractional diffusion PDE using some standard and nonstandard finite difference methods with conformable and Caputo operators

Abstract

Introduction Fractional diffusion equations offer an effective means of describing transport phenomena exhibiting abnormal diffusion pat-terns, often eluding traditional diffusion models. Methods We construct four finite difference methods where fractional derivatives are approximated using either conformable or Caputo operators. Results Stability of the proposed schemes is analyzed using von Neumann stability analysis, and conditions are established to preserve positivity. Consistency analysis is performed for all methods, and numerical results with fractional parameters (α) set to 0.75, 0.90, 0.95, and 1.0 are presented. Discussion The rate of convergence in time for the four methods is computed.