Anomalous Diffusion, Fractional Differential Equations, High Order Discretization Schemes

1. Anomalous Diffusion, Fractional Differential Equations,High Order Discretization Schemes Weihua Deng Lanzhou University Jointed with WenYi Tian, Minghua Chen, Han Zhou Email:dengwh@lzu.edu.cn

2. Robert Brown, 1827

3. DTRW Model, Diffusion Equation Albert Einstein, 1905 Fick’s Laws hold here!

4. 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.

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

6. Superdiffusion The pdf of jump length:

7. Competition between Subdiffusion and Superdiffusion The pdf of waiting time: The pdf of jump length:

8. 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? • Lévy walkers can outperform Brownian walkers by revisiting sites far less often. • The number of new visited sites is much larger for N Levy walkers than for N brownian walkers.

9. Definitions of Fractional Calculus Fractional Integral Fractional Derivatives Riemann-Liouville Derivative Caputo Derivative Grunwald Letnikov Derivative Hadamard Integral

10. Existing Discretization Schemes Shifted Grunwald Letnikov Discretization (Meerschaert and Tadjeran, 2004, JCAM), most widely used Second Order Accuracy obtained by extrapolation (Tadjeran andMeerschaert, 2007, JCP) Transforming intoCaputo 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)

11. 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

13. We introduce the WSGD operator and there exists Similarly, for the right Riemann-Liouville derivative

16. Third Order Approximation

17. Compact Difference Operator with 3rd Order Accuracy Substituting into leads to Further combining , there exists

18. We call Compact WSGD operator (CWSGD) Acting the operator on both sides of above equation leads to

19. 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!

20. Based on the second order approximation

21. Simply shiftting it, the convergent order reduces to 1

22. Weighting and shiftting it, the convergent order preserves to 2 Works for space derivative when

23. Efficient 3rd order approximation Works for space derivative when

24. Efficient 4th order approximation Works for space derivative when

25. References: