Sturm-Liouville Applications

In a Nut Shell: The Sturm-Liouville Problem involves solving a differential equation

having special properties along with associated boundary conditions defined in the

table below.

The Sturm-Liouville D.E. and associated boundary conditions:

d/dx [ p(x) dy/dx] - q(x) y + λ r(x) y = 0 where a < x < b

A₁ y(a) - A₂ y’(a) = 0 and B₁ y(b) + B₂ y’(b) = 0

where neither A₁ and A₂ nor B₁ and B₂ are both zero.

The objective is to find the eigenvalues, λ, that yield solutions to the differential equation

satisfying the prescribed boundary conditions.

Sturm-Liouville Problems have several useful properties that apply to their solution.

The table below lists these properties.

Property 1: If p(x), dp/ dx, q(x), and r(x) are continuous in [a,b] and if p(x) > 0

ad r(x) > 0, then the eigenvalues are nonnegative.

Property 2: Eigenfunctions of the Sturm-Liouville problem are orthogonal on the

interval with respect to the weight function r(x) .

[a,b] . i.e. if y_n(x) and y_m(x) are eigenfunctions, then

b

∫ y_n(x) y_m(x) r(x) dx = 0 for m ≠ n

a

Property 3: A function f(x) can be represented in the interval [a,b] by an eigenfunction

series. i.e.

∞

f(x) = ∑ c_n y_n(x)

n = 1

Click here for an example.

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