Definition:Sturm-Liouville Equation

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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 Fran├žois Sturm and Joseph Liouville.


Sources