Definition:Sturm-Liouville Equation
 | This page has been identified as a candidate for refactoring of basic complexity. In particular: Extract "Weight Function" and "Eigenvalues" into separate pages and transclude Until this has been finished, please leave {{Refactor}} in the code.
New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Refactor}} from the code. |
Definition
A classical Sturm-Liouville equation is a real second order ordinary linear differential equation of the form:
- $\ds (1): \quad - \map {\frac \d {\d x} } {\map p x \frac {\d y} {\d x} } + \map q x y = \lambda \map w x y$
where $y$ is a function of the free variable $x$.
The functions $\map p x$, $\map q x$ and $\map w x$ are specified.
In the simplest cases they are continuous on the closed interval $\closedint a b$.
In addition:
- $(1a): \quad \map p x > 0$ has a continuous derivative
- $(1b): \quad \map w x > 0$
- $(1c): \quad y$ is typically required to satisfy some boundary conditions at $a$ and $b$.
Weight Function
The function $\map w x$, which is sometimes called $\map r x$, is called the weight function or density function.
Eigenvalues
The value of $\lambda$ is not specified in the equation.
Finding the values of $\lambda$ for which there exists a non-trivial solution of $(1)$ satisfying the boundary conditions is part of the problem called the Sturm-Liouville problem (S-L).
Such values of $\lambda$ when they exist are called the eigenvalues of the boundary value problem defined by $(1)$ and the prescribed set of boundary conditions.
The corresponding solutions (for such a $\lambda$) are the eigenfunctions of this problem.
Source of Name
This entry was named for Jacques Charles François Sturm and Joseph Liouville.
Sources
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Sturm-Liouville equation