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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