Transcript Works for space derivative when

Anomalous Diffusion,
Fractional Differential Equations,
High Order Discretization Schemes
Weihua Deng
Lanzhou University
Jointed with WenYi Tian, Minghua Chen, Han Zhou
Email: [email protected]
Robert Brown, 1827
DTRW Model, Diffusion Equation
Albert Einstein, 1905
Fick’s Laws hold here!
Examples of Subiffusion
Trajectories of the motion of individual fluorescently labeled mRNA molecules
inside live E. coli cells:
I. Golding and E.C. Cox,, Phys. Rev. Lett., 96, 098102, 2006.
Simulation Results
Y. He, S. Burov, R. Metzler, and E. Barkai, Phys. Rev. Lett., 101, 058101, 2008.
Superdiffusion
The pdf of jump length:
 ( x) ~ x (1  ) , 0    2

W
 W
 K1
t
x 
2
x (t ) ~ K  t
2

Competition between Subdiffusion
and Superdiffusion
The pdf of waiting time:
 (t ) ~ t (1 ) , 0    1
The pdf of jump length:
 ( x) ~ x (1  ) , 0    2


 W
 W
 K1


t
x
2
x (t ) ~ K  t
2

Applications of Superdiffusion
N.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.
Where to locate N radar
stations to optimize the
search for M targets?
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.
Definitions of Fractional Calculus
Fractional 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
We introduce the WSGD operator
and there exists
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
Acting the operator
Compact WSGD operator (CWSGD)
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: