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Introduction to PDE classification Numerical Methods for PDEs Spring 2007

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Typically describe time evolution towards a steady state. ... The point (x,t) is influenced only by initial conditions bounded by characteristic curves. ... – PowerPoint PPT presentation

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Title: Introduction to PDE classification Numerical Methods for PDEs Spring 2007


1
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

2
Partial Differential Equations (PDEs) 2nd order
model problems
  • PDE classified by discriminant b2-4ac.
  • Negative discriminant Elliptic PDE. Example
    Laplaces equation
  • Zero discriminant Parabolic PDE. Example Heat
    equation
  • Positive discriminant Hyperbolic PDE. Example
    Wave equation

3
Example Parabolic Equation (Finite Domain)
Heat equation
Typical Boundary Conditions
xL/2
x0
x-L/2
4
Example Parabolic Equation
Heat equation
Typical Boundary Conditions
Initial temperature profile in rod
Temperatures for end of rod
xL/2
x0
x-L/2
5
Example Parabolic Equation (Infinite Domain)
Heat equation
Dirac Delta Boundary Conditions
x0
6
Dirac Delta Function
The Dirac delta function is the limit of
Physically it corresponds to a localized intense
source of heat
7
Example Parabolic Equation (Infinite Domain)
Heat equation
Dirac Delta Boundary Conditions
Solution (verify)
8
Example Parabolic Equation (Infinite Domain)
t.1
t.01
t1
t10
9
Parabolic PDES
  • 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 x0 makes temperature nonzero
    throughout domain for all t gt 0.

10
Example Hyperbolic Equation (Infinite Domain)
Heat equation
Boundary Conditions
11
Example Hyperbolic Equation (Infinite Domain)
Heat equation
Boundary Conditions
Solution (verify)
12
Hyperbolic Equation characteristic curves
xctconstant
x-ctconstant
t
(x,t)
x
13
Example Hyperbolic Equation (Infinite Domain)
xctconstant
x-ctconstant
t
The point (x,t) is influenced only by initial
conditions bounded by characteristic curves.
(x,t)
x
14
Example Hyperbolic Equation (Infinite Domain)
Heat equation
Boundary Conditions
15
Example Hyperbolic Equation (Infinite Domain)
t.01
t.1
t1
t10
16
Hyperbolic PDES
  • 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.

17
Example Elliptic Equation (Finite Domain)
Laplaces equation
Typical Boundary Conditions
W
18
The Problem
PDE solution (verify)
19
Elliptic Solution
20
Elliptic PDES
  • 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

21
Partial Differential Equations (PDEs) 2nd order
model problems
  • PDE classified by discriminant b2-4ac.
  • Negative discriminant Elliptic PDE. Example
    Laplaces equation
  • Zero discriminant Parabolic PDE. Example Heat
    equation
  • Positive discriminant Hyperbolic PDE. Example
    Wave equation
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