Partial Differential Equations Solution Manual
© Leon van Dommelen
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5.1 A simple example
5
. Separation of Variables
Subsections
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A simple example
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The physical problem
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The mathematical problem
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Outline of the procedure
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Step 1: Find the eigenfunctions
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Should we solve the other equation?
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Step 2: Solve the problem
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Comparison with D'Alembert
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Understanding the Procedure
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An ordinary differential equation as a model
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Vectors versus functions
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The inner product
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Matrices versus operators
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Some limitations
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An Example with Periodic Boundary Conditions
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The physical problem
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The mathematical problem
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Outline of the procedure
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Step 1: Find the eigenfunctions
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Step 2: Solve the problem
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Summary of the solution
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Finding the Green's function
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Inhomogeneous boundary conditions
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The physical problem
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The mathematical problem
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Outline of the procedure
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Step 0: Fix the boundary conditions
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Step 1: Find the eigenfunctions
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Step 2: Solve the problem
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Summary of the solution
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Finding the Green's functions
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An alternate procedure
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The physical problem
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The mathematical problem
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Step 0: Fix the boundary conditions
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Step 1: Find the eigenfunctions
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Step 2: Solve the problem
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Summary of the solution
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A Summary of Separation of Variables
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The form of the solution
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Limitations of the method
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The procedure
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More general eigenvalue problems
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More general eigenfunctions
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The physical problem
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The mathematical problem
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Step 0: Fix the boundary conditions
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Step 1: Find the eigenfunctions
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Step 2: Solve the problem
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Summary of the solution
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An alternative procedure
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A Problem in Three Independent Variables
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The physical problem
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The mathematical problem
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Step 1: Find the eigenfunctions
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Step 2: Solve the problem
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Summary of the solution
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5.1 A simple example
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