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Partial Differential Equations

1

Next week

Tuesday: guest lecture (Prof. Keenan Crane)

Thursday: in-class test

2

Recap

ODEs: implicitly define a function through its time derivative

3

tqvtq ,)(

0q

time

q

Recap

What does an ODE for exponential decay look like?

• It could be a very crude model for how the temperature of a particle changes through time…

• But where does the heat go? It’s a PDE!

4

Partial Differential Equations

5

PDEs: implicitly define a function through its derivatives with respect to time and space

Most physical phenomena and processes can be described by partial differential equations

Partial Differential Equations

6

Partial Differential Equations

ODEs: want to solve for function of time

PDEs: want to solve for function of time and space

7

txF

dt

txdm

2

2

Example ODE:

2

2 ,,

x

xtuc

t

xtu

Example PDE:

Partial Differential Equations

Definition of PDE

Function implicitly given in terms of derivatives

8

),( xtu

,,,3

udt

duu

,,,21

2

2

2

1 x

u

xx

u

x

u

Any combination of time derivatives

Plus any combination of space derivatives

Partial Differential Equations

Abbreviation

Linear vs non-linear PDEs

Order of a PDE: how many derivatives in space and time?

- wave equation: 2nd order in time, 2nd order in space

- Burger’s equation: 1st order in time, 2nd order in space

,..),(,..),(2

2

2

yxuyx

utut

u xytt

Nonlinear example

Burgers’ equation

xxxt uuuu

Linear example

wave equation

xxtt cuu

PDEs: a first example

A linear, first order PDE: 1D advection

10

Weather forecast: simulate temperature evolution.

T

x

cT0(x)

Given: initial temperature distribution 𝑇0 𝑥 = 𝑇 𝑥, 0 , wind speed c.

Find: temperature distribution 𝑇(𝑥, 𝑡) for any 𝑡, 𝑥.

One space dimension + time

PDEs: a first example

11

• How does the temperature change over a time interval

Dt?T

x

cT(x,t)

T(x,t+Dt)T(x-cDt,t)

cDt-cDt

At a given location in space, some time

later, what will the temperature be?

Where was this air front at time t?

PDEs: a first example

12

1D advection equation

• How does the temperature change over a time interval

Dt?

DT

x

Tc

t

T

0Dt

T

x

cT(x,t)

T(x,t+Dt)T(x-cDt,t) ),(),( txTttxTT DD

),(),( ttcxTttxT DD

)(),(),( 2tOtcx

TtxTttcxT DD

D

x

Tc

t

T

D

D

),( ttxT D

cDt-cDt

PDEs: a first example

Analytical solution:

• Want a function T(x,t) that satisfies

13

x

Tc

t

T

• The solution also needs to satisfy the initial condition:

)()0,( 0 xTxT

• The solution is thus )(),( 0 ctxTtxT

Note: only simple PDEs can be solved analytically!

)(),( ctxftxT • Any T(x,t) of the form will work! How do

we choose one of the many options?

14

PDEs: a first example

Numerical solution:

• Want to approximate function T(x,t) by discretizing in space and time (estimate T(x,t) at different (xi,ti))

T

x

t

PDEs: a first example

• Sample temperature T(x,t) on

• Discretize derivatives with finite differences (space & time)

h

iTiTctiTiT

tttt ]1[][

][][1 D• Solving for Tt+1[i]

yields update rule

• Provide initial values T0[i]

• Need some boundary conditions Tt[0]!

x

Tc

t

T

…]0[0T ][0 nT

…]0[1T

x

t),(][ tthiTiT t D

..)2,1,0(),,..,1( tni

1D grids

with

h

iTiTc

t

iTiT tttt ]1[][][][1

D

PDEs: a first example

Eulerian vs Lagrangian viewpoint

16

T

x

cT0(x)T

x

cT0(x)

Monitor temperature at

fixed locations in space

Release weather balloons and see

where they end up (the temperature

they record does not change)!

Basic Recipe for solving PDEs

numerically

Pick a spatial discretization

Pick a time discretization (forward Euler, backward Euler, etc)

Run a time-stepping algorithm using resulting update rule

We will see a few more examples…

17

Mathematical Background

Many, many types of PDEs

2nd order linear PDEs are most important for us

They involve the Laplace operator (“average” curvature)

• Nabla operator t

xx d

,...,

1

• Laplace operator D 22

2

1 k

d

kx

(in d dimensions)

st

x

s

x

s

d

,...,

1

PDE classification

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• 2nd order linear PDEs are of highest practical

relevance dedicated classification

- Hyperbolic B2-AC > 0 (Wave equation)

- Parabolic B2-AC = 0 (Heat equation)

- Elliptic B2-AC < 0 (Laplace equation)

),,,,(2 yxyyxyxx uuuyxFCuBuAu

• A 2nd order linear PDE in 2 variables (x,y) has the form

• A 2nd order linear PDE in 2 variables is

Elliptic PDEs

Elliptic 2nd –order PDEs ( B2-AC < 0 ) describe static problems (systems in equilibrium)

Prototype: 0)( D xuLaplace Equation

Applications

• Steady-state solutions to parabolic PDEs

• Equilibrium problems

What is the smoothest function given boundary data? Think elastic membrane clamped at boundaries.

How do we solve it?

21

Laplace Equation

22

Laplace Equation

0),(),(),(2

2

2

2

D yxu

yyxu

xyxu

Discretizing on a grid with cell size h:

- Solution: each point must be equal to average of “neighbors”

- Can iteratively keep setting to average of neighbor values

(Jacobi method!) or set up a linear system. Very easy!

04

2

,1,1,,1,1

,

D

h

uuuuuu

jijijijiji

ji

Boundary Conditions

What happens at the “edges” of the simulation domain?

• Need additional information

• For ODEs, we had an initial value

• For PDEs, we also need boundary conditions

Two common choices:

• Dirichlet – boundary data specifies values

• Neumann – boundary data specifies derivatives

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Boundary Conditions

Dirichlet Boundary Conditions:

Many possible functions! The Laplace equation gives the smoothest one.

24

ba )1(,)0(

Boundary Conditions

Neumann Boundary Conditions:

Again, many possible functions!

Some combinations of PDEs + boundary conditions may not have solutions!

25

vu )1(',)0('

Parabolic PDEs

Applications

• Heat conduction and general diffusion processes

Parabolic 2nd –order PDEs ( B2-AC = 0 ) Are typically time dependent problems

Solutions smooth out as time increases

Prototype: Heat Equation

thermal diffusivitytemperature,u

),(),( 2 tuct

tu xx D

The Heat Equation

How does an initial distribution of heat spread out over time? After a long enough time, solution is the same as the Laplace equation.

27

The Heat Equation

Know how to discretize the Laplacian

Know how to discretize time (e.g. forward Euler)

Can easily put the two together…

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2

,1,1,,1,1

,

4

h

uuuuuu

jijijijiji

ji

D

n

ji

n

ji

n

ji utuu ,,

1

, DD

Hyperbolic PDEs

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Hyperbolic 2nd –order PDEs ( B2-AC > 0 ) time dependent problems

Retain & propagate disturbances present in initial data

Prototype: ),(),(

2

2

2

tuct

tux

xD

Wave Equation

Applications

• Simulate wave propagation for sound, light, and

water

• Mechanics (oscillatory motion, vibrating strings)

c propagation speedamplitude,u

The Wave Equation

How does a wave front propagate over time?

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