Introduction to PDE classification

Numerical Methods for PDEs Spring 2007

Jim E. Jones

References: •Partial Differential Equations of Applied Mathematics, Zauderer•Wikopedia, Partial Differential Equation

• PDE classified by discriminant: b2-4ac.– Negative discriminant = Elliptic PDE. Example

Laplace’s equation

– Zero discriminant = Parabolic PDE. Example Heat equation

– Positive discriminant = Hyperbolic PDE. Example Wave equation

Partial Differential Equations (PDEs) :2nd order model problems

0 yyxx uu

0 xxt uu

0 ttxx uu

gfueuducubuau yxyyxyxx

Example: Parabolic Equation (Finite Domain)

0,0 xxt uu

)(),2/(),(),2/(

)()0,(

tbtLutatLu

xfxu

Heat equation

Typical Boundary Conditions

x=0 x=L/2x=-L/2

),0()2/,2/(),( TLLtx

Example: Parabolic Equation

0 xxt uu

Heat equation

Typical Boundary Conditions

)(),2/(),(),2/(

)()0,(

tbtLutatLu

xfxu

x=0 x=L/2x=-L/2

Initial temperature profile in rod

Temperatures for end of rod

Example: Parabolic Equation (Infinite Domain)

0 xxt uu

)()0,( xxu

Heat equation

Dirac Delta Boundary Conditions

x=0

),0(),(),( tx

Dirac Delta Function

||,0

||,2

1)(

x

xx

The Dirac delta function is the limit of

Physically it corresponds to a localized intense source of heat

Example: Parabolic Equation (Infinite Domain)

0 xxt uu

)()0,( xxu

),0(),(),( tx

Heat equation

Dirac Delta Boundary Conditions

t

x

ttxu

4exp

2

1),(

2

Solution (verify)

Example: Parabolic Equation (Infinite Domain)

t=.01 t=.1

t=1 t=10

• Typically describe time evolution towards a steady state.– Model Problem: Describe the temperature evolution of

a rod whose ends are held at a constant temperatures.

• Initial conditions have immediate, global effect– Point source at x=0 makes temperature nonzero

throughout domain for all t > 0.

Parabolic PDES

Example: Hyperbolic Equation (Infinite Domain)

02 xxtt ucu

)()0,(

)()0,(

xgxu

xfxu

t

Heat equation

Boundary Conditions

),0(),(),( tx

Example: Hyperbolic Equation (Infinite Domain)

02 xxtt ucu

)()0,(

)()0,(

xgxu

xfxu

t

),0(),(),( txHeat equation

Boundary Conditions

ctx

ctx

dyygc

ctxfctxftxu )(2

1)]()([

2

1),(

Solution (verify)

Hyperbolic Equation: characteristic curves

x-ct=constantx+ct=constant

x

t

(x,t)

Example: Hyperbolic Equation (Infinite Domain)

x-ct=constantx+ct=constant

x

t

(x,t) The point (x,t) is influenced only by initial conditions bounded by characteristic curves.

ctx

ctx

dyygc

ctxfctxftxu )(2

1)]()([

2

1),(

Example: Hyperbolic Equation (Infinite Domain)

0 xxtt uu

0)0,(

)exp()0,( 2

xu

xxu

t

Heat equation

Boundary Conditions

),0(),(),( tx

Example: Hyperbolic Equation (Infinite Domain)

t=.01 t=.1

t=1 t=10

• Typically describe time evolution with no steady state.– Model problem: Describe the time evolution of the

wave produced by plucking a string.

• Initial conditions have only local effect – The constant c determines the speed of wave

propagation.

Hyperbolic PDES

Example: Elliptic Equation (Finite Domain)

0 yyxx uu

),(),(),( yxxfyxu

2),( Ryx

Laplace’s equation

Typical Boundary Conditions

PDE solution (verify)

The Problem

21),2ln(2),2(,ln),1(

21),2ln(2)2,(,ln)1,(

21,21,

yyyyuyyyu

xxxxuxxxu

yxx

y

y

xuu yyxx

)ln(),( xyxyyxu pde

Elliptic Solution

• Typically describe steady state behavior.– Model problem: Describe the final temperature profile

in a plate whose boundaries are held at constant temperatures.

• Boundary conditions have global effect

Elliptic PDES

• PDE classified by discriminant: b2-4ac.– Negative discriminant = Elliptic PDE. Example

Laplace’s equation

– Zero discriminant = Parabolic PDE. Example Heat equation

– Positive discriminant = Hyperbolic PDE. Example Wave equation

Partial Differential Equations (PDEs) :2nd order model problems

0 yyxx uu

0 xxt uu

0 ttxx uu

gfueuducubuau yxyyxyxx