|Sturm-Liouville Boundary Value Problem|
A differential equation defined on the interval having the form of
and the boundary conditions
is called as Sturm-Liouville boundary value problem or Sturm-Liouville system, where , ; the weighting function are given functions; , , , are given constants; and the eigenvalue is an unspecified parameter.
|Orthogonality and General Fourier Series|
The non-trivial (non-zero) solutions , , of the Sturm-Liouville boundary value problem only exist at certain , . is called eigenvalue and is the eigenfunction.
The eigenvalues of a Sturm-Liouville boundary value problem are non-negative real numbers. In addition, the associated eigenfunctions are orthogonal to each other with respect to the weighting function ,
The complete set of the solutions forms a complete orthogonal set of functions defined on the interval . Therefore, a piecewise continuous function can be expressed in terms of , , such that
The completeness allows us to express any piecewise continuous function in terms of these eigenfunctions while the orthogonality makes the expression unique and compact (no redundant terms). In addition, it can be shown that the orthogonal series is the best series available, i.e., each additional term fine tunes but not overhauls the sum of the existing terms. These properties generalize the conventional Fourier series and to any complete orthogonal series . The series is hence called the generalized Fourier series. The method of forming solutions by the general Fourier series is called the method of eigenfunction expansion which is an important technique in solving partial differential equations.
Examples of generalized Fourier series can be found in Bessel functions, Legendre polynomials, and other orthogonal polynomials such as Laguerre polynomials, Hermite polynomials, and Chebyshev polynomials. Each of these polynomials represents a complete orthogonal set in different coordinates or circumstances and can be considered as a special case of the Sturm-Liouville boundary value problem.