Analysis in Mechanical Engineering
© Leon van Dommelen
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18.1 Basic Concepts
18
. Introduction
Subsections
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
.
3
.
1
The conditions for properly posedness
18
.
3
.
2
An improperly posed parabolic problem
18
.
3
.
3
An improperly posed elliptic problem
18
.
3
.
4
Improperly posed hyperbolic problems
18
.
4
Energy methods
18
.
4
.
1
The Poisson equation
18
.
4
.
2
The heat equation
18
.
4
.
3
The wave equation
18
.
5
Variational methods [None]
18
.
6
Classification
18
.
6
.
1
Introduction
18
.
6
.
2
Scalar second order equations
18
.
7
Changes of Coordinates
18
.
7
.
1
Introduction
18
.
7
.
2
The formulae for coordinate transformations
18
.
7
.
3
Rotation of coordinates
18
.
7
.
4
Explanation of the classification
18
.
8
Two-Dimensional Coordinate Transforms
18
.
8
.
1
Characteristic Coordinates
18
.
8
.
2
Parabolic equations in two dimensions
18
.
8
.
3
Elliptic equations in two dimensions
Next:
18.1 Basic Concepts
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