Analyzing Fractional Diffusion Equations: Cauchy and Signaling Problems Explained

1. FOC Reading GroupProbability Distributions Generated by Fractional Diffusion Equations Prepared by: Yiding Han CSIOS Dept. of Electrical and Computer Engineering Utah State University E: yiding.h@aggiemail.usu.edu 4/9/2008

2. Outline • Standard Diffusion Equation • Fundamental solution for Cauchy problem. • Fundamental solution for Signalling problem. • Time-fractional Diffusion Equation. • The Cauchy problem for the time-fractional diffusion equation. • The Signallingproblem of the time-fractional diffusion equation. • Symmetric, space-fractional diffusion equation. • Conclusions FOC reading group

3. Standard diffusion equation • Linear partial differential equation where denotes a positive constant with dimensions x and t are space-time variables, is the field variable, which is a causal function of time. • The typical physical phenomenon related to such an equation is heat conduction in a thin solid rod extended along x, and u is the temperature. FOC reading group

4. Boundary conditions • Cauchy problem • Space-time domain • Data are assigned at t=0+ on the whole space axis. • Signalling problem • Space-time domain • Data are assigned both at t=0+ on semi-infinite space x>0 and x=0+on semi-infinite space t>0 FOC reading group

5. Initial values for both problems • Cauchy problem • Signalling problem FOC reading group

6. Cauchy problem • Denote then the Fourier transform of the initial condition is: • The Fourier transform of the standard diffusion equation: • Solving the above ODE, we have, FOC reading group

7. Continued… • Letting , then , therefore • is the Fourier transform of the fundamental solution or Green function • Thus, This interprets the Gaussian pdf. FOC reading group

8. Continued… • Furthermore, the moments of even order of the Gaussian pdf turns out to be: or FOC reading group

9. Signalling problem • Using Laplace transform, we have • The transformed solution of standard diffusion equation satisfies FOC reading group

10. Continued… • The fundamental solution (or Green function) of the Signalling problem: • It can be interpreted as: This is the pdf of one-sided Levy pdf. FOC reading group

11. Time-Fractional Diffusion Equation • By replacing the first-order time derivative by a fractional derivative of order 0<α≤2 (in Caputo sense), it reads: where denotes a positive constant with dimensions • The definition of the Caputo fractional derivative of order α>0 for a causal function is given as: where m=1,2,… and 0 ≤m-1<α≤m FOC reading group

12. Continued… • Thus we need to distinguish the cases and • Then the time-fractional diffusion equation reads: FOC reading group

13. Cauchy Problem for the time-fractionalDiffusion equation • Cauchy problem • For the time-fractional diffusion equation subject to the above condition, the Fourier transformation leads to the ODE of order • The transformed solution is Where denotes the Mittag-Leffler function of order 2v FOC reading group

14. Continued… • The Fourier transformation of Green function • The inverse Fourier transformation cannot be obtained but by turning it to be Laplace transform pair, the inverse can be obtained: FOC reading group

15. The pdf plots FOC reading group

16. Continued • Furthermore, the pdf can be written as: which is a symmetric pdf in space. • The absolute moments of positive order of the Green function are finite, in particular: • Which can be regarded as a generalization of the moments of order function. • the anomalous diffusion is said to be sub-diffusion when v<1/2, and super-diffusion when v>1/2. FOC reading group

17. The signalling Problem for the Time-Fractional diffusion Equation • Signalling problem • For the time-fractional diffusion equation subject to the above condition, the application of the Laplace transform leads to 2nd order ODE: • The transformed solution reads: FOC reading group

18. Continued … • The Laplace transform of Green function • Using the same method, we have: Which is a one-sided stable distribution in time FOC reading group

19. Pdf plots FOC reading group

20. Cauchy Problem for symmetric space-fractional diffusion equation • The symmetric space-fractional diffusion equation is obtained by replacing the 2nd order space derivative with order α. where denotes a positive constant with dimensions • The Fourier transformation of Green function reads: FOC reading group

21. Continued … • Therefore, we have pα (x;0,γ,0) is the pdf of Symmetric (β=0) α-stable distribution with placement variable a=0 and scaling factor • For α=1 and α=2, we can easily obtain the corresponding Green functions by inverse Fourior transformation: Cauchy Distribution Gaussian Distribution FOC reading group

22. PDfs The fundamental solutions against x of the Cauchy problem for the symmetric space-fractional diffusion equation. a) α=1/2 (bold line), α=1(dashed line) b) α=3/4 (bold line), α=2(dashed line) FOC reading group

23. Thanks FOC reading group