Anomalous Diffusion, Fractional Differential Equations,

High Order Discretization Schemes

Weihua DengLanzhou University

Email: dengwh@lzu.edu.cn

Jointed with WenYi Tian, Minghua Chen, Han Zhou

Robert Brown, 1827

DTRW Model, Diffusion EquationAlbert Einstein, 1905

Fick’s Laws hold here!

Examples of Subiffusion

I. Golding and E.C. Cox,, Phys. Rev. Lett., 96, 098102, 2006.

Trajectories of the motion of individual fluorescently labeled mRNA molecules inside live E. coli cells:

Simulation Results

Y. He, S. Burov, R. Metzler, and E. Barkai, Phys. Rev. Lett., 101, 058101, 2008.

Superdiffusion

The pdf of jump length: 20,~)( )1( xx

x

WK

t

W

1

2

~)(2 tKtx

Competition between Subdiffusion and Superdiffusion

The pdf of jump length: 20,~)( )1( xx

x

WK

t

W

1

2

~)(2 tKtx

The pdf of waiting time: 10,~)( )1( tt

Applications of SuperdiffusionN.E. Humphries et al, Nature, 465, 1066-1069, 2010;M. Viswanathan, Nature, 1018-1019, 2010;Viswanathan, G. M. et al. Nature, 401, 911-914, 1999.

1. Lévy walkers can outperform Brownian walkers by revisiting sites far less often.

2. The number of new visited sites is much larger for N Levy walkers than for N brownian walkers.

Where to locate N radar stations to optimize the search for M targets?

Definitions of Fractional CalculusFractional Integral

Fractional Derivatives

Riemann-Liouville Derivative

Caputo Derivative

Grunwald Letnikov Derivative

Hadamard Integral

Existing Discretization Schemes

Shifted Grunwald Letnikov Discretization (Meerschaert and Tadjeran, 2004, JCAM), most widely used Second Order Accuracy obtained by extrapolation (Tadjeran and Meerschaert, 2007, JCP)

Transforming into Caputo Derivative

Centralinzed Finite Difference Scheme with Piecewise Linear Approximation

Hadamard Integral

Fractional Centred Derivative for Riesz Potential Operators with Second Order Accuracy (Ortigueira, 2006, Int J Math Math Sci)

A Class of Second Order Schemes

Based on the Analysis in Frequency Domain by Combining the Different Shifted Grunwald Letnikov Discretizations

The shifted Grunwald Letnikov Discretization

which has first order accuracy, i.e.,

What happens if

Taking Fourier Transform on both Sides of above Equation, there exists

and there exists

We introduce the WSGD operator

Similarly, for the right Riemann-Liouville derivative

Third Order Approximation

Compact Difference Operator with 3rd Order Accuracy

Substituting

into

leads to

Further combining , there exists

We call Compact WSGD operator (CWSGD)

Acting the operator on both sides of above equation leads to

WSLD Operator: A Class of 4th Order Schemes

Based on the Analysis in Frequency Domain by Combining the Different Shifted Lubich‘s Discretizations (Lubich, 1986, SIAM J Math Anal)

Generating functions for (p+1) point backward difference formula of

Still valid when α<0, Not work for space derivative!

Based on the second order approximation

Simply shiftting it, the convergent order reduces to 1

Weighting and shiftting it, the convergent order preserves to 2

Works for space derivative when

Efficient 3rd order approximation

Works for space derivative when

Efficient 4th order approximation

Works for space derivative when

References: