fractional partial differential equations for conservation ......fractional partial differential...
TRANSCRIPT
-
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