Analysis in Mechanical Engineering
© Leon van Dommelen
Next:
18. Introduction
IV
. Partial Differential Equations
Subsections
18
. Introduction
18
.
1
Basic Concepts
18
.
1
.
1
The prevalence of partial differential equations
18
.
1
.
2
Definitions
18
.
1
.
3
Typical boundary conditions
18
.
2
The Standard Examples
18
.
2
.
1
The Laplace equation
18
.
2
.
2
The heat equation
18
.
2
.
3
The wave equation
18
.
3
Properly Posedness
18
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3
.
1
The conditions for properly posedness
18
.
3
.
2
An improperly posed parabolic problem
18
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3
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3
An improperly posed elliptic problem
18
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3
.
4
Improperly posed hyperbolic problems
18
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4
Energy methods
18
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4
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1
The Poisson equation
18
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4
.
2
The heat equation
18
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4
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3
The wave equation
18
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5
Variational methods [None]
18
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6
Classification
18
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6
.
1
Introduction
18
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6
.
2
Scalar second order equations
18
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7
Changes of Coordinates
18
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7
.
1
Introduction
18
.
7
.
2
The formulae for coordinate transformations
18
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7
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3
Rotation of coordinates
18
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7
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4
Explanation of the classification
18
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8
Two-Dimensional Coordinate Transforms
18
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8
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1
Characteristic Coordinates
18
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8
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2
Parabolic equations in two dimensions
18
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8
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3
Elliptic equations in two dimensions
19
. Green’s Functions
19
.
1
Introduction
19
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1
.
1
The one-dimensional Poisson equation
19
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1
.
2
More on delta and Green’s functions
19
.
2
The Poisson equation in infinite space
19
.
2
.
1
Overview
19
.
2
.
2
Loose derivation
19
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2
.
3
Rigorous derivation
19
.
3
The Poisson or Laplace equation in a finite region
19
.
3
.
1
Overview
19
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3
.
2
Intro to the solution procedure
19
.
3
.
3
Derivation of the integral solution
19
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3
.
4
Boundary integral (panel) methods
19
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3
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5
Poisson’s integral formulae
19
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3
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6
Derivation
19
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3
.
7
The integral formula for the Neumann problem
19
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3
.
8
Smoothness of the solution
20
. First Order Equations
20
.
1
Classification and characteristics
20
.
2
Numerical solution
20
.
3
Analytical solution
20
.
4
Using the boundary or initial condition
20
.
5
The inviscid Burgers’ equation
20
.
5
.
1
Wave steepening
20
.
5
.
2
Shocks
20
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5
.
3
Conservation laws
20
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5
.
4
Shock relation
20
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5
.
5
The entropy condition
20
.
6
First order equations in more dimensions
20
.
7
Systems of First Order Equations (None)
21
. D'Alembert Solution of the Wave equation
21
.
1
Introduction
21
.
2
Extension to finite regions
21
.
2
.
1
The physical problem
21
.
2
.
2
The mathematical problem
21
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2
.
3
Dealing with the boundary conditions
21
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2
.
4
The final solution
22
. Separation of Variables
22
.
1
A simple example
22
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1
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1
The physical problem
22
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1
.
2
The mathematical problem
22
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1
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3
Outline of the procedure
22
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1
.
4
Step 1: Find the eigenfunctions
22
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1
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5
Should we solve the other equation?
22
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1
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6
Step 2: Solve the problem
22
.
2
Comparison with D'Alembert
22
.
3
Understanding the Procedure
22
.
3
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1
An ordinary differential equation as a model
22
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3
.
2
Vectors versus functions
22
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3
.
3
The inner product
22
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3
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4
Matrices versus operators
22
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3
.
5
Some limitations
22
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4
Handling Periodic Boundary Conditions
22
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4
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1
The physical problem
22
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4
.
2
The mathematical problem
22
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4
.
3
Outline of the procedure
22
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4
.
4
Step 1: Find the eigenfunctions
22
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4
.
5
Step 2: Solve the problem
22
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4
.
6
Summary of the solution
22
.
5
Finding the Green's function
22
.
6
Inhomogeneous boundary conditions
22
.
6
.
1
The physical problem
22
.
6
.
2
The mathematical problem
22
.
6
.
3
Outline of the procedure
22
.
6
.
4
Step 0: Fix the boundary conditions
22
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6
.
5
Step 1: Find the eigenfunctions
22
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6
.
6
Step 2: Solve the problem
22
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6
.
7
Summary of the solution
22
.
7
Finding the Green's functions
22
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8
An alternate procedure
22
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8
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1
The physical problem
22
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8
.
2
The mathematical problem
22
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8
.
3
Step 0: Fix the boundary conditions
22
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8
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4
Step 1: Find the eigenfunctions
22
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8
.
5
Step 2: Solve the problem
22
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8
.
6
Summary of the solution
22
.
9
A Summary of Separation of Variables
22
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9
.
1
The form of the solution
22
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9
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2
Limitations of the method
22
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9
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3
The procedure
22
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9
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4
More general eigenvalue problems
22
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10
More general eigenfunctions
22
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10
.
1
The physical problem
22
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10
.
2
The mathematical problem
22
.
10
.
3
Step 0: Fix the boundary conditions
22
.
10
.
4
Step 1: Find the eigenfunctions
22
.
10
.
5
Step 2: Solve the problem
22
.
10
.
6
Summary of the solution
22
.
10
.
7
An alternative procedure
22
.
11
A Problem in Three Independent Variables
22
.
11
.
1
The physical problem
22
.
11
.
2
The mathematical problem
22
.
11
.
3
Step 1: Find the eigenfunctions
22
.
11
.
4
Step 2: Solve the problem
22
.
11
.
5
Summary of the solution
23
. Fourier Transforms [None]
24
. Laplace Transforms
24
.
1
Overview of the Procedure
24
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1
.
1
Typical procedure
24
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1
.
2
About the coordinate to be transformed
24
.
2
A parabolic example
24
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2
.
1
The physical problem
24
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2
.
2
The mathematical problem
24
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2
.
3
Transform the problem
24
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2
.
4
Solve the transformed problem
24
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2
.
5
Transform back
24
.
3
A hyperbolic example
24
.
3
.
1
The physical problem
24
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3
.
2
The mathematical problem
24
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3
.
3
Transform the problem
24
.
3
.
4
Solve the transformed problem
24
.
3
.
5
Transform back
24
.
3
.
6
An alternate procedure
Next:
18. Introduction
FAMU-FSU College of Engineering
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