Math 445: Applied PDEs: models, problems, methods
D. Gurarie
Models: processes
Transport 1-st order linear (quasi-linear) PDE in space-time
Heat-diffusion 1-st order in t, 2-nd order in x, called parabolic
, - density of traffic (gas, flu
0;
id);
- speed of motion, - source/sink
or ..., ,t x t xu cu u C u
f u x
u x t
c F
t
2 20 0
; ;
, - mass density, or temperature
- diffusivity
...,... ,.
; - sour
.
ce
.t xx t x y
u x t
u Du u D uu
D F
F u F
Similar equations apply to Stochastic Processes (Brownian motion): u(x,t) - Probability to find particle at point x time t
Wave equation
2-st order in x, t (hyperbolic) 2 2
- speed of propagation; - (external
0 0;
,.
) fo
.. ,.
c
.
e
.
r
tt xx tt
c
u c u u c uF u F u
F
Vibrating strings, membranes,…: u – vertical displacement (from rest)Elasticity: medium displacement components (P,S –waves)Acoustics: u – velocity/pressure/density perturbation in gas/fluidOptics, E-M propagation: u – component(s) of E-M field, or potentials
Laplace’s (elliptic) equation
0
.. - sou. ; rcexx yyu u u FF u
Stationary heat distributionPotential theory (gravitational, Electro-static, electro-dynamic, fluid,…)
Nonlinear models
1
0
0
t xx
t x xx
t x xxx
u au u u
u uu au
u uu au
- Fisher-Kolmogorov (genetic drift)
- Burgers (sticky matter)
- KdV (integrable Hamiltonian system)
PDE systems:Fluid dynamics Electro-magnetism:
2
- dislacement, , -Lame coefficients
;ttu u u
u
0 -mass
... -momentum
... - en
ergy
t
t
t
v
pv v v
e
1
40
0t
t
E
H
E H j
H E
Elasticity Acoustics
Basic Problems:
• Initial and Boundary value problems (well posedness)
• Solution methods: – exact; approximate;
– analytic/numeric;
– general or special solutions (equilibria, periodic et al)
• Analysis: stability, parameter dependence, bifurcations
• Applications– Prediction and control
– Mechanical (propagation of heat, waves/signals)
– Chemical, bio-medical,
• Other…
Solution methods1. Analytic
– Method of characteristics (1-st and higher order PDE)– Separation of variables, reduction to ODE – Expansion and transform methods (Fourier, Laplace et al);
special functions– Green’s functions and fundamental solutions (integral
equations)
2. Approximate and asymptotic methods3. Variational methods4. Numeric methods (Mathematica/Matlab)5. Other techniques (change of variables, symmetry
reduction, Integrable models,…)
Examples (with Mathematica) Half-plane potential of point charge
-1.5 -1 -0.5 0 0.5 1 1.5 2
0.25
0.5
0.75
1
1.25
1.5
1.75
2
-1.5 -1 -0.5 0 0.5 1 1.5 2
0.25
0.5
0.75
1
1.25
1.5
1.75
2
3D- radial wave snap-shot
Periodic heat source
-5-2.5
02.5
5 0
5
10
15
-0.5
0
0.5
-5-2.5
02.5
5
2D incompressible fluid Shear instability
Vorticity Stream f.
3 2 1 0 1 2 3
1
2
3
4
Time evolution of traffic jam for initial Gaussian profile
Analytic
Computational
