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Unformatted text preview: 6.5 The Sturm ‐ Liouville Problem We have now considered several Separation of Variables (SofV) examples: • Wave equation 6.1 • Diffusion equation 6.3 • Laplace equation 6.4 In all of these examples, we had to solve an eigenvalue problem of the following form (*) 2 2 2 = + ∂ φ λ φ x d with two homogeneous BCs, for example, ) ( and ) ( = = L φ φ Therefore, eigenvalues and eigenfunctions we have seen so far were quite similar. However, as we will see later in the course, eigenvalue problems for other SofV solutions can be quite different. In general, we have to solve All possible cases are covered by the Sturm-Liouville problem. The general form of an ODE that comes from separating a PDE is given by (1) ) ( ) ( ) ( 2 = + − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ φ λ φ φ x p x q dx d x s dx d where s(x), q(x) , and p(x) are functions that come from a given PDE, and λ 2 is related to the separation constant k . The boundary conditions, which come from separating the BCs for the unknown dependent ....
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- Spring '10
- Laplace, Partial differential equation, wave equation, dx, eigenvalue problems, eigenfunctions φ1