Sturm Liouville Equations

A regular Sturm Liouville equation is a real second-order linear differential equation of the form with boundary conditions with at least one of and similarly for and with the functions specified.

In the simplest case all coefficients are continuous on the finite closed interval and has continuous derivative. In this case, this function is a solution if it is continuously differentiable on and satisfies (1) at every point in In addition, is typically required to satisfy some boundary conditions at and The function is called the weight function.

The value of is not specified in the equation; finding the values of for 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 of when 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 if and are solutions for distinct then We wish to find a function which solves the following Sturm – Liouville problem: with In fact is a solution with eigenvalue Properties:

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.

If is a complete set of eigenfunctions for (1) then

a) The form a basis for the set of continuous functions with continuous first derivatives.

b)Let be any function in and let be the nth Fourier coefficient of with respect to the basis then the Fourier series converges pointwise to on  