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Sturm—Liouville Problems and Boundary Value Problems
Published in Steven G. Krantz, Differential Equations, 2015
In a sense, Sturm–Liouville theory is a generalization of the Fourier theory that we learned in the last chapter. One of the key ideas in the Fourier theory is that most any function can be expanded in a series of sine and cosine functions. And of course these two families of functions are the eigenfunctions for the problem y″ + λy = 0. The eigenvalues (values of λ) that arise form an infinite increasing sequence, and they tend to infinity. And only one eigenfunction corresponds to each eigenvalue. We shall learn now that these same phenomena take place for a Sturm–Liouville problem.
Sign-changing points of solutions of homogeneous Sturm–Liouville equations with measure-valued coefficients
Published in Applicable Analysis, 2022
When 1/p and q are real-valued locally integrable functions on , , and when p>0 on , zeros of every nontrivial real-valued solution of the famous homogeneous Sturm–Liouville differential equation are isolated in . This fact is the key to establish two big results in Sturm–Liouville theory, namely the Sturm separation theorem and the Sturm comparison theorem. These two celebrated results are due to Sturm [1] and they date back to 1836. The Sturm separation theorem states that zeros of two linearly independent solutions of Equation (1) are interlaced. The Sturm comparison theorem states that if u and v are nontrivial solutions of (1) and , respectively, u vanishes at s and t, and but different on a set of positive Lebesgue measure, then v vanishes at some point between s and t.
Transfer function models for distributed-parameter systems with impedance boundary conditions
Published in International Journal of Control, 2018
Rudolf Rabenstein, Maximilian Schäfer, Christian Strobl
For systems which depend on space and time, suitable frequency domain transformations have to be established. Again, the Laplace transformation turns the time dynamics into a function of a complex temporal frequency variable. There is, however, no such standard transformation for the spatial behaviour. The number of spatial dimensions, the geometry of the problem, and also the kind of boundary conditions call for a spatial transformation which is custom designed for the problem at hand. For a large class of problems, Sturm– Liouville theory allows to expand the spatial behaviour into orthogonal or bi-orthogonal basis functions. The name Sturm– Liouville-transformation has been coined for this procedure (Eringen, 1954).
Trigonometric and cylindrical polynomials and their applications in electromagnetics
Published in Applicable Analysis, 2020
The general results of the classical Sturm–Liouville theory concerning the existence and distribution of (real or complex) spectrum for this type of problems are not applicable because the boundary (transmission) conditions depend on the spectral parameter; this dependence which is specified by concrete functions obtained explicitly governs the presence or absence of spectrum.