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Anomalous Diffusion, Fractional Differential Equations,
High Order Discretization Schemes
Weihua DengLanzhou University
Email: [email protected]
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: