Analysis Solution Manual
© Leon van Dommelen
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22.1 A simple example
22
. Separation of Variables
Subsections
22
.
1
A simple example
22
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1
.
1
The physical problem
22
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1
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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
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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
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2
Comparison with D'Alembert
22
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3
Understanding the Procedure
22
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3
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1
An ordinary differential equation as a model
22
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3
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2
Vectors versus functions
22
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3
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3
The inner product
22
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3
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4
Matrices versus operators
22
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3
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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
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2
The mathematical problem
22
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4
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3
Outline of the procedure
22
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4
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4
Step 1: Find the eigenfunctions
22
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4
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5
Step 2: Solve the problem
22
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4
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6
Summary of the solution
22
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5
Finding the Green's function
22
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6
Inhomogeneous boundary conditions
22
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6
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1
The physical problem
22
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6
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2
The mathematical problem
22
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6
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3
Outline of the procedure
22
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6
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4
Step 0: Fix the boundary conditions
22
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6
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5
Step 1: Find the eigenfunctions
22
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6
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6
Step 2: Solve the problem
22
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6
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7
Summary of the solution
22
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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
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2
The mathematical problem
22
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8
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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
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5
Step 2: Solve the problem
22
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8
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6
Summary of the solution
22
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9
A Summary of Separation of Variables
22
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9
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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
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1
The physical problem
22
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10
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2
The mathematical problem
22
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10
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3
Step 0: Fix the boundary conditions
22
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10
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4
Step 1: Find the eigenfunctions
22
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10
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5
Step 2: Solve the problem
22
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10
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6
Summary of the solution
22
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10
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7
An alternative procedure
22
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11
A Problem in Three Independent Variables
22
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11
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1
The physical problem
22
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11
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2
The mathematical problem
22
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11
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3
Step 1: Find the eigenfunctions
22
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11
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4
Step 2: Solve the problem
22
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11
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5
Summary of the solution
Next:
22.1 A simple example
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