Fractional Partial Differential Equations for Conservation Laws and - PowerPoint PPT Presentation

Fractional Partial Differential Equations for Conservation Laws and Beyond George Em Karniadakis Division of Applied Mathematics, Brown University & Department of Mechanical Engineering, MIT The CRUNCH group: www.cfm.brown.edu/crunch

Tomorrow’s Science 21 st Century and Beyond 19 th & 20th Centuries

Once complexity enters by the door, Normal statistics leaves by the window!  Empirical PDFs for complex processes are non-Gaussian and non-Poisson.  Complexity in systems/networks: self-similarity , hence no characteristic space/time scales.  Boltzmann believed that microscopic dynamics should be described by continuous but not differentiable representations e.g. using Weierstrass function.  Jean Perrin (Nobel prize, Avogadro’s number): we need curves without tangents (derivatives), which are more common to the physical world.  Mandelbrot: “… the emperor had no clothes ” lightening does not come in straight lines… Clouds are not spheres…and most physical phenomena violate the underlying assumptions of Euclidian geometry, in agreement with Da Vinci’s observations and sketches.

Anomalous Transport Klafter (Physics World, 2005) The clear picture that has emerged over the last few decades is that although these phenomena are called anomalous, they are abundant in everyday life i.e., Anomalous is Normal! Fractional diffusion, Meerschaert et. al 2010 Fractional order tissue electrode , Ovadia, et al. 2006 Levy flights (Özarslan et al., JMR, 183;315, 2006)

Anomalous Dispersion of Particles: Mixing Layer Notion of Enhanced (super-diffusive) Mixing! standard-diffusion along x-axis Super-diffusion along y-axis

Experimental Evidence: Near-Wall Measurement Observation of Stochastic Levy Flights ! Single-Particle Quantum Tagging 28.4 μm Z Dimensions in nm 39.7 μm Z is not scaled as X,Y X Y z 25 20 15 10 5 20 0 -5 -10 0 Brownian -15 -20 z -20 -25 Levy Random Walk 0 -30 -35 2000 -40 -40 4000 -45 -400 6000 x -200 8000 0 S. Pooya and M. Koochesfahani , MSU y 10000 200

Fractional Modeling of Soft Materials 7

Continuous Time Random Walks - CTRWs Credit: B. Henry, UNSW

Standard Diffusion or Fractional Diffusion Credit: B. Henry, UNSW

Fractional calculus The basic mathematical ideas were developed in the 17th century by the mathematicians Leibniz (1695), Liouville (1834) and Riemann (1892). Later, it was brought to the attention of the engineering world by Oliver Heaviside in the 1890s.  The first published results are in a letter from L ’ Hospital to Leibniz in 1695! L ’ Hospital Leibniz Thus it follows that d 1/2 (x)/dx 1/2 = 2(x/pi) 1/2 , an What if in the general expression for the n th derivative, of x, d n (x)/dx n we apparent paradox, from which one day useful let n=1/2? consequences will be drawn. 10

Fractional calculus Riemann Liouville Riemann-Liouville Fractional integration of order Riemann-Liouville fractional derivative of order

Fractional calculus Liouville Riemann Caputo fractional derivative of order Gerasimov, 1948 Riemann Liouville vs. Caputo fractional derivative:

Fluid Mechanics: Stokes Problems Laplace T ce Transforms rms Laplace (ODE) Solving the ODE and eliminating the constants Time-Fractional Advection Equation This is exact! Inverse No assumptions! transform: No approximations ! : viscous property as the transport velocity! 13

Fluid Mechanics: Stokes Problems (continued) 1 st Stokes problem: ( U is constant) 2 nd Stokes problem: Periodic Steady Unsteady Fresnel function As time evolves, the unsteady part vanishes, and the known analytical shear stress is obtained (by expanding the sin term)  ASME JFE, Kulish and Lage (2002)

Singular Sturm-Liouville Problem (Integer-Order): Jacques François Sturm Joseph Liouville (1803-1855) (1809-1882) Local Operator , (Jacobi polynomials) , (Quadratic) Local Boundary Conditions

Singular Fractional Sturm-Liouville problem: M. Zayernouri and G.E. Karniadakis, Fractional Sturm-Liouville Eigen-Problems: Theory and Numerical Approximatio n , J. Comput. Phys. vol. 252, (2013), Pages 495–517 Global Operator Non-local Boundary Conditions i =1: SFSLP of Kind-I i =2: SFSLP of Kind-II

Singular Fractional Sturm-Liouville problem: Jacobi Poly-fractonomials Theorem: The exact eigenfunctions of SFSLP-I (i=1) and SFSLP-II (i=2) are given respectively as and corresponding the exact eigenvalues are given as

Eigen-solutions of RFSLP-I Eigen-solutions of RFSLP-II • The same number of zeros • Sharp gradient near Dirichlet end

Approximation Properties of the Poly-fractonomials

Fractional Differential Equations • Model Problem: Fractional Initial-Value Problem (RL derivative) Expansion : Diagonal Stiffness Matrix Test Functions:

• Zhiping Mao

 H-refinement + H-matrix, Xuan Zhao et al, 2016 (ICFDA Riemann-Liouville award)

Fractional Conservation Laws  Burgers with Non-Local Non-linearity: 1 β β + = ε < β β < 2 ( ) , 0 , 1 , u D D u u 1 2 1 2 t a x a x x x 2  Exact Solution – Hopf-Cole transformation! β = − ε ( , ) 2 log ( , ) β + β = u x t D 1 w x t 1 1 2 a x     2 − | | 1 x y 1 β ∞ − − 1 0 ( ) ∫ I u y β = − ε +     ε ε a y ( , ) 2 log 1 4 2 u x t D e t dy 1   a x πε −∞   4 t   

DISCONTINUOUS GALERKIN METHOD Zhiping Mao (2016)

Fractional Fluxes − β β ∀ > ±∞ = ± ∞ = 2 0, ( , ) ( , ) 0 t D u t D u t 1 2 a x a x = ∫ β ∂ ( ) ∞ I 1 β − β ( ) 1 = ( , ) 0 I D u x t dx 1 1 ∂ a x −∞ t

Fractional Phase-Field Modeling of Multi-Phase Flows Fractional Allen-Cahn equation in conserving form (1) Fractional Laplace operator Surface Tension effect (2 ) (3 The mixing energy density ) Variable density Variable viscosity

Interfaces and Singularities

One-Dimensional Modeling We solve the 1D fractional Allen-Cahn equation: in domain (-1, 1) × (0, T], the above equation is discretized in time by the following scheme Here, we use spectral method for space discretization. Inspired by the results in the extreme case 1, we conjecture that the equilibrium solution, denoted by , would behave like which coincides with the solution of Allen-Cahn equation in case of s=1.

Fractional Model Sharpens the Interface Numerical and Analytical Results of 1D problem agree very well.

Fractional Turbulence Modeling Yiqing Du, and George Em Karniadakis Science 2000;288:1230-1234

Fully-Developed Turbulent Channel Flow

Turbulent Channel Flow: Discretization The Grünwald–Letnikov Fractional-Order Derivative

Fractional Order: Universal Scaling?

Fractional Order: Universal Scaling!

Predictive Fractional Model – Law of the Wall Re ~ 100,000

Alternative Model:

Princeton Super-pipe Experiment Variable fractional order Re = 35,000,000

 That’s great! But how do I know the order?  Answer: BIG DATA/Regression  What about noisy data and uncertainty?  Answer: Distributed-Order Derivatives

Petrov-Galerkin Variational Form Distributed Fractional Sobolev Space  Petrov--Galerkin and Spectral Collocation Methods for Distributed Order Differential Equations, E Kharazmi, M Zayernouri, GE Karniadakis SIAM Journal on Scientific Computing 39 (3), A1003-A1037

arXiv:1808.00931

Machine Learning Discovered New Equations! Data of groundwater solute transport from Macro-dispersion Experimental (MADE) site at Columbus Air Force Base  Green: Tritium concentration data from MADE site  Red: Prediction in the literature using trial and error  Blue: Prediction from machine learning

Machine Learning Discovered New Equations! − ∂ ∂ ∂ 0.75 2 0.0028 ( , ) ( , ) x ( , ) Old fractional model u x t u x t u x t = − × + × 0.14 0.14 ∂ ∂ − 0.75 2 0.0028 ∂ x t x (One example) x ∂ ∂ ∂ − 0.73+0.00053 1.87 0.0029 x ( , ) ( , ) x ( , ) u x t u x t u x t = − × + × 0.14 0.14 New fractional model ∂ ∂ − 0.73+0.00053 1.87 0.0029 x ∂ x t x x Old fractional New fractional Comparison model model How to identify Trial and error Machine learning parameters Extension to a large Difficult Easy number of parameters Prediction accuracy Low High

When to Think Fractionally?

Fractional Modeling: A New Meta-Discipline? plasma physics fluid, turbulence heat conduction superstatistics, ergodic properties Levy flights dynamical stochastic deterministic fractional calculus systems diffusion theory theory socio-economics experiments nonlinear maps, biophysics: Hamiltonian systems cells, migration disordered reaction-diffusion disordered fractals systems glasses, gels porous material molecular diffusion, NMR spectroscopy Slide Credit: “Anomalous Transport”