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Sturm-Liouville theory - Sturm-Liouville form | A Wisdom Archive on Sturm-Liouville theory - Sturm-Liouville form | | Sturm-Liouville theory - Sturm-Liouville form A selection of articles related to Sturm-Liouville theory - Sturm-Liouville form | |
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Sturm-Liouville theory, Sturm-Liouville theory - Application to normal modes, Sturm-Liouville theory - Example, Sturm-Liouville theory - Examples, Sturm-Liouville theory - Some highly technical details, Sturm-Liouville theory - Sturm-Liouville differential operators, Sturm-Liouville theory - Sturm-Liouville form, Sturm-Liouville theory - Sturm-Liouville theory, Sturm-Liouville theory - Useful consequences of the preceding technicalities, normal mode
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ARTICLES RELATED TO Sturm-Liouville theory - Sturm-Liouville form | |
| | |  Sturm-Liouville theory - Sturm-Liouville form: Encyclopedia II - Sturm-Liouville theory - Sturm-Liouville differential operatorsThe map
can be viewed as a linear operator mapping a function u to another function Lu. We may study this linear operator in the context of functional analysis. If we put w = 1 in equation (1), it can be written as
This is precisely the eigenvalue problem; that is, we are trying to find the eigenvalues λ and eigenvectors u of the L operator. However, to be honest we must also include the boundary conditions. Let's say that we want to look at the problem over the inter ...
See also:Sturm-Liouville theory, Sturm-Liouville theory - Sturm-Liouville theory, Sturm-Liouville theory - Sturm-Liouville form, Sturm-Liouville theory - Examples, Sturm-Liouville theory - Sturm-Liouville differential operators, Sturm-Liouville theory - Some highly technical details, Sturm-Liouville theory - Useful consequences of the preceding technicalities, Sturm-Liouville theory - Example, Sturm-Liouville theory - Application to normal modes Read more here: » Sturm-Liouville theory: Encyclopedia II - Sturm-Liouville theory - Sturm-Liouville differential operators |
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| | | | Sturm-Liouville theory - Sturm-Liouville form: Encyclopedia II - Sturm-Liouville theory - Application to normal modes Suppose we are interested in the modes of vibration of a thin membrane, held in a rectangular frame, 0 < x < L1, 0 < y < L2. We know the equation of motion for the vertical membrane's displacement, W(x, y, t) is given by the wave equation:
The equation is separable (substituting W = X(x) × Y(y) × T(t)), and the normal mode solutions that have harmonic time dependence and sati ...
See also:Sturm-Liouville theory, Sturm-Liouville theory - Sturm-Liouville theory, Sturm-Liouville theory - Sturm-Liouville form, Sturm-Liouville theory - Examples, Sturm-Liouville theory - Sturm-Liouville differential operators, Sturm-Liouville theory - Some highly technical details, Sturm-Liouville theory - Useful consequences of the preceding technicalities, Sturm-Liouville theory - Example, Sturm-Liouville theory - Application to normal modes Read more here: » Sturm-Liouville theory: Encyclopedia II - Sturm-Liouville theory - Application to normal modes |
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| | | | Sturm-Liouville theory - Sturm-Liouville form: Encyclopedia II - Sturm-Liouville theory - Example We wish to find a function u(x) which solves the following Sturm-Liouville problem:
where the unknowns are λ and u(x). As above, we must add boundary conditions, we take for example
Observe that if k is any integer, then the function
is a solution with eigenvalue λ = −k2. We know that the solutions of a S-L problem form an orthogonal basis, and we know from Fourier series that this set of sinusoi ...
See also:Sturm-Liouville theory, Sturm-Liouville theory - Sturm-Liouville theory, Sturm-Liouville theory - Sturm-Liouville form, Sturm-Liouville theory - Examples, Sturm-Liouville theory - Sturm-Liouville differential operators, Sturm-Liouville theory - Some highly technical details, Sturm-Liouville theory - Useful consequences of the preceding technicalities, Sturm-Liouville theory - Example, Sturm-Liouville theory - Application to normal modes Read more here: » Sturm-Liouville theory: Encyclopedia II - Sturm-Liouville theory - Example |
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