George Em KarniadakisDivision of Applied Mathematics, Brown University

& Department of Mechanical Engineering, MITThe 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-axisstandard-diffusion along x-axis

Notion of Enhanced (super-diffusive) Mixing!

x

0

2000

4000

6000

8000

10000y

-400-200

0200

z

-40

-20

0

20

X Y

Z

z

2520151050

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

Dimensions in nmZ 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 !

BrownianLevy 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 Transforms

1st Stokes problem: (U is constant)

2nd Stokes problem:

UnsteadyPeriodic Steady

Fresnel function

ASME JFE, Kulish and Lage (2002)

As time evolves, the unsteady part vanishes, and the known analytical shear stress is obtained (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 MatrixExpansion:

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 21 2, 0 , 1

1 ( ) 2

,t x x xa a xu D uD uβ β ε β β= <

Zhiping Mao (2016)

DISCONTINUOUS GALERKIN METHOD

Fractional Fluxes1 22 ,0, ( ) ( ) 0,a ax xD t Dt u tuβ β− ±∞ ±> ∞∀ = =

1 1( ) 1 ( , )xaI D u x t dxβ β∞ −

−∞= ∫

1( )

0It

β∂=

∂

Fractional Phase-Field Modeling of Multi-Phase Flows

Fractional Allen-Cahn equation in conserving form

Surface Tension effect

Variable viscosityVariable density

Fractional Laplace operator(1)

(2)

(3)

The mixing energy density

Interfaces and Singularities

One-Dimensional ModelingWe 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

https://link.springer.com/article/10.1007/s00009-015-0525-3

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 KarniadakisSIAM Journal on Scientific Computing 39 (3), A1003-A1037

arXiv:1808.00931

https://arxiv.org/abs/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 0.14x

x

u x t u x t u x txt x

−

−

∂ ∂ ∂= − × + ×

∂∂ ∂

0.73+0.00053 1.87 0.0029

0.73+0.00053 1.87 0.0029

( , ) ( , ) ( , )0.14 0.14x x

x x

u x t u x t u x txt x

−

−

∂ ∂ ∂= − × + ×

∂∂ ∂

Old fractional model (One example)

New fractional model

Comparison Old fractional modelNew fractional

modelHow 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?

stochastictheory

dynamicalsystemstheory

disorderedsystems

experiments

plasma physicsheat conduction

ergodic properties

deterministicdiffusion

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

Slide Number 1Slide Number 2Slide Number 3Slide Number 4Slide Number 5Slide Number 6Fractional Modeling of Soft MaterialsSlide Number 8Slide Number 9 Fractional CalculusFractional CalculusFractional CalculusFluid Mechanics: Stokes ProblemsSlide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18Slide Number 19Approximation Properties of the Poly-fractonomials�Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28Slide Number 29Slide Number 30Slide Number 31Slide Number 32Slide Number 33Slide Number 34Slide Number 35Fractional Conservation LawsSlide Number 37Fractional FluxesFractional Phase-Field Modeling of Multi-Phase FlowsSlide Number 40One-Dimensional ModelingSlide Number 42Slide Number 43Slide Number 44Slide Number 45Slide Number 46Slide Number 47Slide Number 48Slide Number 49Slide Number 50That’s great! But how do I know the order?Slide Number 52Slide Number 53Slide Number 54Slide Number 55Slide Number 56Slide Number 57Slide Number 58Slide Number 59Slide Number 60Slide Number 61Slide Number 62