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PPT – Introduction to PDE classification Numerical Methods for PDEs Spring 2007 PowerPoint presentation | free to download - id: 250b5d-MzAyO

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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

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

Example Parabolic Equation (Finite Domain)

Heat equation

Typical Boundary Conditions

xL/2

x0

x-L/2

Example Parabolic Equation

Heat equation

Typical Boundary Conditions

Initial temperature profile in rod

Temperatures for end of rod

xL/2

x0

x-L/2

Example Parabolic Equation (Infinite Domain)

Heat equation

Dirac Delta Boundary Conditions

x0

Dirac Delta Function

The Dirac delta function is the limit of

Physically it corresponds to a localized intense

source of heat

Example Parabolic Equation (Infinite Domain)

Heat equation

Dirac Delta Boundary Conditions

Solution (verify)

Example Parabolic Equation (Infinite Domain)

t.1

t.01

t1

t10

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.

Example Hyperbolic Equation (Infinite Domain)

Heat equation

Boundary Conditions

Example Hyperbolic Equation (Infinite Domain)

Heat equation

Boundary Conditions

Solution (verify)

Hyperbolic Equation characteristic curves

xctconstant

x-ctconstant

t

(x,t)

x

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

Example Hyperbolic Equation (Infinite Domain)

Heat equation

Boundary Conditions

Example Hyperbolic Equation (Infinite Domain)

t.01

t.1

t1

t10

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.

Example Elliptic Equation (Finite Domain)

Laplaces equation

Typical Boundary Conditions

W

The Problem

PDE solution (verify)

Elliptic Solution

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

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