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

George Em Karniadakis Division of Applied Mathematics, Brown University & Department of Mechanical Engineering, MIT The CRUNCH group: www.cfm.brown.edu/crunch

Fractional Partial Differential Equations for Conservation Laws and Beyond

Tomorrow’s Science

19th & 20th Centuries 21st Century and Beyond

 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.

Once complexity enters by the door, Normal statistics leaves by the window!

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 Transport

Anomalous Dispersion of Particles: Mixing Layer

Super-diffusion along y-axis standard-diffusion along x-axis Notion of Enhanced (super-diffusive) Mixing!

x

2000 4000 6000 8000 10000

y

• 400
• 200

200

z

• 40
• 20

20 X Y Z z 25 20 15 10 5

• 5
• 10
• 15
• 20
• 25
• 30
• 35
• 40
• 45

Dimensions in nm Z is not scaled as X,Y

• S. Pooya and M. Koochesfahani, MSU

39.7 μm 28.4 μm

Experimental Evidence: Near-Wall Measurement Observation of Stochastic Levy Flights ! Brownian Levy Random Walk

Single-Particle Quantum Tagging

7

Fractional Modeling of Soft Materials

Continuous Time Random Walks - CTRWs

Credit: B. Henry, UNSW

Standard Diffusion or Fractional Diffusion

Credit: B. Henry, UNSW

Fractional calculus

• The first published results are in a letter from L’Hospital to Leibniz in 1695!

L’Hospital

What if in the general expression for the nth derivative, of x, dn(x)/dxn we let n=1/2?

Leibniz

Thus it follows that d1/2 (x)/dx1/2 = 2(x/pi)1/2, an apparent paradox, from which one day useful consequences will be drawn. 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.

10

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

Riemann Liouville

Fractional calculus

Caputo fractional derivative of order Riemann Liouville vs. Caputo fractional derivative:

Riemann Liouville

Fractional calculus

Gerasimov, 1948

Fluid Mechanics: Stokes Problems

13

Solving the ODE and eliminating the constants Inverse transform:

Laplace (ODE) Time-Fractional Advection Equation : viscous property as the transport velocity! This is exact! No assumptions! No approximations!

Laplace T ce Transforms rms

1st Stokes problem: (U is constant) 2nd Stokes problem:

Unsteady Periodic Steady Fresnel function

• ASME JFE, Kulish and Lage (2002)

As time evolves, the unsteady part vanishes, and the known analytical shear stress is

• btained (by expanding the sin term)

Fluid Mechanics: Stokes Problems

(continued)

, (Jacobi polynomials) , (Quadratic)

Local Operator Local Boundary Conditions

Singular Sturm-Liouville Problem (Integer-Order):

Jacques François Sturm (1803-1855) Joseph Liouville (1809-1882)

Singular Fractional Sturm-Liouville problem:

i =1: SFSLP of Kind-I i =2: SFSLP of Kind-II

Global Operator Non-local Boundary Conditions

• M. Zayernouri and G.E. Karniadakis, Fractional Sturm-Liouville Eigen-Problems:

Theory and Numerical Approximation, J. Comput. Phys. vol. 252, (2013), Pages 495–517

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

Jacobi Poly-fractonomials

Singular Fractional Sturm-Liouville problem:

• The same number of zeros
• Sharp gradient near Dirichlet end

Eigen-solutions of RFSLP-I Eigen-solutions of RFSLP-II

Approximation Properties of the Poly-fractonomials

• Model Problem: Fractional Initial-Value Problem

(RL derivative)

Diagonal Stiffness Matrix Expansion: Test Functions:

Fractional Differential Equations

• Zhiping Mao

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

Fractional Conservation Laws

1 2

2 1 2

, 0 , 1 1 ( ) 2 ,

t x x x a a x

u D u D u

β β

ε β β = < < +

2 1 0 1

| | 1 ( ) 4 2

1 ( , ) 2 log 1 4

a y

x y I u y t a x

u x t D e dy t

β

β ε ε

ε πε

− − − ∞ −∞

  = − +             

∫

1 2

1 β β + =

1

( , ) 2 log ( , )

a x

u x t D w x t

β

ε = −

 Burgers with Non-Local Non-linearity:  Exact Solution – Hopf-Cole transformation!

Zhiping Mao (2016) DISCONTINUOUS GALERKIN METHOD

Fractional Fluxes

1 2

2

, 0, ( ) ( ) ,

a a x x

D t D t u t u

β β −

±∞ ± > ∞ ∀ = =

1 1

( ) 1

( , )

x a

I D u x t dx

β β ∞ − −∞

= ∫

1

( )

I t

β

∂ = ∂

Fractional Phase-Field Modeling of Multi-Phase Flows

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

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.

Numerical and Analytical Results of 1D problem agree very well.

Fractional Model Sharpens the Interface

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

Fractional Turbulence Modeling

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

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 0.75 2 0.0028

( , ) ( , ) ( , ) 0.14 0.14

x x

u x t u x t u x t x t x

− −

∂ ∂ ∂ = − × + × ∂ ∂ ∂

0.73+0.00053 1.87 0.0029 0.73+0.00053 1.87 0.0029

( , ) ( , ) ( , ) 0.14 0.14

x x x x

u x t u x t u x t x t x

− −

∂ ∂ ∂ = − × + × ∂ ∂ ∂

Old fractional model (One example) New fractional model

Comparison Old fractional model New fractional model How to identify parameters Trial and error Machine learning

Extension to a large number of parameters

Difficult Easy Prediction accuracy Low High Machine Learning Discovered New Equations!

When to Think Fractionally?

stochastic theory dynamical systems theory disordered systems experiments plasma physics heat conduction ergodic properties deterministic diffusion nonlinear maps, Hamiltonian systems disordered fractals porous material molecular diffusion, NMR spectroscopy glasses, gels reaction-diffusion biophysics: cells, migration socio-economics fractional calculus superstatistics, Levy flights fluid, turbulence

Slide Credit: “Anomalous Transport”

Fractional Modeling: A New Meta-Discipline?

http://www.brown.edu/research/projects/muri-fractional-pde/

• Integer-Order Calculus
• Fractional-Order Calculus

Discrete gears vs. constantly-variable transmission