The asymptotics of the solutions to the anomalous diffusion equations

Abstract

In this paper, we study the asymptotics of the solutions to the anomalous diffusion equations. The fractional anomalous diffusion equations are obtained from the existing anomalous diffusion and typical diffusion equations by replacing the first-order time derivative with fractional derivatives of order α∈(0,1) for the sub-diffusion and α∈(1,2) for the super-diffusion respectively. In most situations, fractional derivatives mean Riemann–Liouville derivative or Caputo derivative. In this paper, we use these two kinds of fractional derivatives. Using Laplace transform and Fourier transform, we obtain the asymptotics estimates of solutions to the anomalous diffusion equations.