Numerical Solution of Some Fractional Diffusion Equations

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This Demonstration shows numerical solutions of the fractional diffusion equation by means of weighted average methods (or methods). The boundary conditions specify that the solution equals zero at and . Four different initial conditions can be chosen. When the weight parameter 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.

Contributed by: Santos Bravo Yuste (March 2011)
Open content licensed under CC BY-NC-SA


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Details

The fractional diffusion equation is , where the is the Riemann–Liouville derivative of order , , is solved numerically by means of the fractional difference weighted-average methods (or methods) discussed in [1]. When , the equation is just the normal diffusion equation and these methods become normal weighted-average methods [2]. Depending on the value of the weight factor , the method can be fully implicit when , or fully explicit when . The fractional Crank–Nicolson method for , or for other values of , is a generic -weighted-average method. It is also determined whether the stability criterion of equation (35) of [1] is also satisfied. The size of the timestep and the number of timesteps can be chosen. The diffusion constant and the space discretization are fixed to 1 and with , respectively. When the "normal solution" checkbox is checked, the normal diffusion solution () is also plotted (orange points). The broken lines are to guide the eye. The values of the numerical solutions are shown when the mouse pointer is over the points.

References

[1] S. B. Yuste, "Weighted Average Finite Difference Methods for Fractional Diffusion Equations," Journal of Computational Physics, 216, 2006 pp. 264–274.

[2] K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations, Cambridge: Cambridge University Press, 1994.



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