Sturm Liouville Equations
A regular Sturm Liouville equation is a real second-order linear differential equation of the form
with boundary conditionswith at least one of and similarly forand with the functionsspecified.
In the simplest case all coefficients are continuous on the finite closed intervalandhas continuous derivative. In this case, this functionis a solution if it is continuously differentiable onand satisfies (1) at every point inIn addition,is typically required to satisfy some boundary conditions atandThe functionis called the weight function.
The value ofis not specified in the equation; finding the values offor which there exists a non-trivial solution of (1) satisfying the boundary conditions is part of the problem called the Sturm–Liouville problem.
Such values ofwhen they exist are called the eigenvalues of the boundary value problem defined above and the given boundary conditions. The corresponding solutions (for each) are the eigenfunctions of this problem. The solutions (eigenfunctions) of each Sturm - Liouville problem form an orthogonal basis for the set of continuous functions, so that ifandare solutions for distinctthen
We wish to find a functionwhich solves the following Sturm – Liouville problem:
In factis a solution with eigenvalue
The eigenvalues of Sturm – Liouville problems form an infinite set:with as
Any pair of eigenfunctions corresponding to a particular eigenvalue are non – zero multiples of one another.
Ifis a complete set of eigenfunctions for (1) then
a) Theform a basis forthe set of continuous functions with continuous first derivatives.
b)Letbe any function inand letbe the nth Fourier coefficient ofwith respect to the basisthen the Fourier seriesconverges pointwise toon