Numerical Solution of Some Fractional Diffusion Equations

weight parameter λ

0.5

anomalous diffusion exponent γ

0.5

size of the timestep Δt

1.×

-6

10

number of timesteps

0

normal solution

u(x, 0)

double tent

sin(π x)

4 x(1-x)

δ(x-1/2)

fractional Crank–Nicolson method

The method is stable: the solution will remain finite.

This Demonstration shows numerical solutions

u(x,t)

of the fractional diffusion equation by means of weighted average methods (or

θ

methods). The boundary conditions specify that the solution

u

equals zero at

x=0

and

x=1

. Four different initial conditions can be chosen. When the weight parameter

λ=1-θ

equals 1/2, the numerical method is the fractional Crank–Nicolson method. When the "normal solution" checkbox is checked, the normal diffusion solution is also plotted.