Fractional dynamics. Applications to the study of some transport phenomena Dana Constantinescu

1. Fractional dynamics. Applications to the study of some transport phenomena Dana Constantinescu Department of Applied Mathematics University of Craiova, 13 A. I. Cuza Street, 200585, Craiova Romania dconsta@yahoo.com • Outline • Basic elements of Fractional Calculus • 2. Fractional Diffusion Equation • 3. Applications in Physics • 4. Applications in Economics • 5. Numerical method to solve fractional diffusion equations

2. The derivation operator is local (Local spatial effect) (Local temporal effect-loss of memory) 1.1 Basic definitions. Local and non-local operators The models described by ordinary or partially differential equations involve only local properties of the system

3. Using fractional derivatives: The integration operator is non-local The models described by integral or integro-differential equations involve non-local properties of the system Volterra integral equation How could be included non-local effects in mathematical models similar with ordinary or partially differential equations?

4. Global spatial and global memory effects included in any position and in any moment Spatial effects for any position Memory-effects at any time

5. 1695 correspondence l’Hospital ( ) Leibniz (One day very useful consequences will be drawn from this paradox, since there are little paradoxes without usefulness) …………………………………………………. • 1783 L. Euler introduces the Gamma-function as generalization of factorials (makes some comments on something which is “more curious than useful”) • 1812 P.S. Laplace – Theorie Analitique des Probabilites • 1819 F. S. Lacrois – Traite du calcul Differentiel et du calcul integral (use the Gamma-function in the definition of derivatives of non-integer order) • 1823 N.H. Abel uses the derivative of order ½ for solving the “Tautochrone problem” • 1832-1837 J. Liouville (8 papers, one of them dedicated to fractional derivatives) • 1847 G. F. B. Riemann proposed a formula • for the fractional integral 1.2 History (memory effect?) • 1695-1697 correspondence Leibniz-Johann Bernoulli, Leibniz-J Wallis …………………………………………………. • 1867 A.K. Grunwald (Ueber “begrenzte”Derivationen und deren Anverdung) • 1868 A.V. Letnikov (Theory of differentiation with an arbitrary index )(Russian)

6. 1869 N. Y. Sonin used the Cauchy’s Integral formula as starting point to reach differentiation of arbitrary order • 1872 A. V. Letnikov extended the idea on Sonin • 1884 H. Laurent obtained the fractional Riemann-Liouville integral and derivatives • 1927 A. Marchaud developed an integral version of Grundwald-Letnikov derivatives • 1938 M. Riesz considered the “Riesz potential” which can be expressed in terms of Riemann-Liouville derivatives • …………………………………………………………… • 1967 M. Caputo proposed the formula (Caputo derivative) • 1974 The first conference dedicated to FRACTIONAL CALCULUS (New Haven) • 1974 The first book about FRACTIONAL CALCULUS (Oldham and Spanier) • 1998 First issue of “Fractional Calculus and Applied Mathematics”

7. Integrals of order “n” (Cauchy) Riemann-Liouville integrals of any non-integer (fractional) order (1847) 1.3 Fractional integrals

8. 1.4 Fractional derivatives Riemann (1847)-Liouville (1832-1837) H Laurent (1884) Left derivative (left spatial effect) Right derivative (right spatial effect) Riesz (1938) R-L derivatives are not compatible with Laplace transform

9. Caputo (1967) Left derivative (left spatial effect) Right derivative (right spatial effect) Relations with Riemann_Liouville derivatives

10. Marchaud(1927) Grundwald (1867) Letnikov(1868) Riemann-Liouville Caputo Grundwald-Letnikov

11. Example

12. 4 Linear fractional differential equations The direct Laplace transform Mittag-Lefler Functions (1927) The inverse Laplace transform

13. = probability density function of waiting times = probability density function of jumps = probability of finding the particle in the position “x” at the moment “t” 2. Fractional Diffusion Equation Continuous Time Random Walk (CTRW) (Stochastic processes, disordered systems-the position of a particle is influenced by its (random) interaction with other particles and/or sources) Montroll-Weiss equation Particles that did not move from “x” in the time interval [0,t] Particles that came in “x” in the time interval [0,t]

14. Poisson distribution for waiting time Gaussian distribution for the jumps Laplace Transform in time Fourier Transform in space Levy flight distribution for waiting times Levy flight distribution for the jumps CTRW are equivalent to Brownian motion on large spatio-temporal scales

15. Numerical solution n=number of time steps (of length h) m=number of spatial discretization intervals (of length k) Analytical solution Mittag-Lefler Function (1927) Matrix approach to discrete fractional calculus I. Podlubni, A. Chechkin, T Skovranek, Y. Chen, Journal of Computational Physics 228 (2009) 3137-3153

16. Atomic nucleus Atomic nucleus Atomic nucleus + + Energy 3. Applications in Fusion Plasma Physics (Tokamaks) The tokamaks are toroidal devices used for producing energy through controlled thermonuclear fusion The fuel is heated to temperatures in excess of 150 millions °C The hot plasma is confined in the core region using a magnetic field poloidal field (traveling in circles orthogonal to the toroidal field) electrical current driven through the plasma toroidal field (traveling around the torus) superconducting coils surrounding the vessel Section of the ITER Project Tokamak reactor

17. Tokamaks were invented in 1950s in Soviet Union tokamak=toroidal’naya kamera s magnitnymi katushkami (toroidal chamber with magnetic coils) 215 tokamaks over the world ITER (France, Cadarache, the world’s largest tokamak, in construction) joint program of China, European Union, India, Japan, Korea, Russia, USA http://en.wikipedia.org/wiki/Tokamakwww.tokamak.infowww.iter.org A main problem: to study the heat propagation (transport) -core region- diffusive transport -edge region-nondiffusive transport

18. T(x,t)=averaged temperature on the surface at the moment t Fractional diffusion model of non-local transport (D. Del Castillo Negrete, Fractional diffusion models of nonlocal transport, Physics of Plasmas 13 (2006), 082308) Finite size model Non-local fluxes Standard diffusion term Fokker-Planck Fractional diffusion Classical diffusion Balistic transport Superdifusive scaling Difusive transport Collisional dominated transport Free streaming regime

19. Power modulation Experiment dTe [keV] Fractional model dTe [keV] Electron temperature time [s] Source: D. Del Castillo Negrete, Physics of Plasmas 13 (2006), 082308

20. Experiment Model Consistent with the experiment, the fractional model gives a delay of the order of 4ms for cold pulses

21. Comparison of a radial fractional transport model with tokamak experiments (A. Kullberg, G. J. Morales, and J. E. MaggsPHYSICS OF PLASMAS 21, 032310 (2014)) Radial diffusion model was derived by azymuthally averaging an isotropic Laplacian operator Kullberg, D. del Castillo-Negrete, G. J. Morales, and J. E. Maggs, Phys. Rev. E 87, 052115 (2013). Temperature profile in Rijnhuizen Tokamak Project (RTP device) for the plateau in core temperature Temperature profile in RTP device for different locations of the heating source (Black dark symbols are experimental temperature measurements)

22. 4. Applications in Economics Fractional diffusion models of option prices in markets with jumps A. Cartea et al, Physica A 374 (2007), 749-763 An important problem in finance is pricing the financial instruments that derive their value from financially traded assets( stocks for example) S(t)=stock price/unit B(t)=price/unit of the bond Financial portofolio Main assumption: the stock price follows a Brownian process r = risk-free rate the volatility of the returns from holding S(t) increment of Brownian motion average growth of the stock (direct computation) (Ito Lemma)

23. Nobel Prize in Economic Sciences (1997) Black-Scholes (BS) model Financial portofolio Assumption: the stock price follows a Levy process with the density r = risk-free rate v = convexity adjustment Fractional Black-Sholes model

24. Numerical integration using Grundwald-Letnikov approximation (matrix approach) Option prices: European call The owner has the right (but not obligation) to buy/sell a unit of stock at the future time T for a pre-specified price K Restriction imposed in the “down and out” call Interpretation of the numerical results (K=50, S(0)=50 and barrier bellow the strike price K) : - BS options are more expensive that LS options for S<K - LS options are more expensive for S>53 Differences between classical BS and fractional BS in down-out model

25. 3D financial model Ma JH, Chen YS, Study for the bifurcation topological structure and the global complicated character of nonlinear financial systems , Appl Math Mech 22 (2001, 1375-1382 x=interest rate a=saving amount b = cost per investment y=investment c = elasticity of demand of commercial markets z=inflation Wei-Ching Chen, Nonlinear dynamics and chaos in a fractional-order, Chaos, Solitons and Fractals 36 (2008) 1305–1314 fixed points, periodic orbits, and chaotic dynamics

26. Parameters Initial point Conclusion Chaos exists in fractional systems with order In this case The largest Lyapunov exponent

28. 5. Numerical method to solve fractional diffusion equations 1D symmetric diffusion equation n=number of time steps (of length h) m=number of spatial discretization intervals (of length k) 1D non-symmetric diffusion equation

29. 2D-non-symmetric diffusion equation 3D grid the Caputo derivative in time at these nodes can be approximated using discretized Grundwald-Letnikov operators The spatial derivative at these nodes can be also approximated using discretized Grundwald-Letnikov operators

30. Constantinescu D., Negrea M., Petrisor I., Theoretical and numerical aspects of fractional 2D transport equation. Applications in fusion plasma theory (to be published in Romanian Journal of Physics)

31. Radial heat transport (Average on poloidal direction) Heat transport in radial and poloidal directions