Sturm-Liouville Theory

Without going in too deep, Sturm-Liouville theory is a method for expanding functions in terms of orthogonal eigenfunctions. The basic theory states that eigenfunctions of the Sturm-Liouville operator are orthogonal with respect to a weight function given certain boundary conditions. It is very important in the study of partial differential equations.

{(d/dx)p(x)(d/dx)+q(x)}y = λw(x)y

Is the Sturm-Liouville (S-L) equation where {(d/dx)p(x)(d/dx)+q(x)} is the S-L operator and w(x) is usually called the weight function. An important property of the S-L operator is that any linear second order ODEcan be written in S-L form.

The basis for S-L theory depends on having boundary conditions that allow for the S-L operator to be hermitian on the boundary (note that the eigenfunctions of a hermitian operator are always orthogonal). Boundary conditions that satisfy this on the interval {a,b} may be:

Homogeneous (any combination of one condition at a and one at b):

p(a) = 0 and the eigenvalues are finite

C₁y(a) + C₂y'(a) = 0 (where the values of C₁ and C₂ are the same for all eigenfunctions, this is essential a homogeneous Robin condition)

y(a) = 0 (Dirichlet condition)

y'(a) = 0 (Neumann condition)

p(b) = 0 and the eigenvalues are finite

C₁y(b) + C₂y'(b) = 0 (where the values of C₁ and C₂ are the same for all eigenfunctions)

y(b) = 0

y'(b) = 0

Symmetric Conditions:

p(a) = p(b)

y(a) = y(b)

y'(a) = y'(b)

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A piecewise smooth function can be expanded in terms of a S-L series as:

f(x) = ∑B_ny_n where B_n = ∫w(v)f(v)y_ndv/∫w(v)y_n²dv (with summation from some valid integer to another depending on y_n and integration over the interval {a,b})

An example:

Consider the operator L = d²/dx² on an interval {a,b}.

Suppose now that we want to solve the eigenvalue equation Ly = λy with homogeneous boundary conditions y(a) = y(b) = 0

This is a second order linear ODE and thus can be represented in S-L form with p(x) = 1, q(x) = 0, and w(x) = 1.

Since y(x) has homogeneous boundary conditions, we know by S-L theory that L is hermitian and thus the eigenfunctions must be orthogonal with respect to the weight function, 1. So the eigenfunctions which satisfy the differential equation, the boundary condition, and orthogonality is the fourier sine series, y_n(x) = C_nsin(nπx/(b-a))

Expanding a piecewise smooth function f(x) in terms of the S-L series we get: f(x) = ∑C_nsin(nπx/(b-a)) (with summation from 1 to infinity), where C_n = ∫w(v)f(v)y_ndv/∫w(v)y_n²dv = ∫(1)f(v)sin(nπv/(b-a))dv/∫(1){sin(nπv/(b-a))}²dv which is equal to the more familiar: 2/L∫f(v)sin(nπv/L)dv (where L = b-a), with integrations performed from a to b.

Source: Professor Charles Roth, Math 271 Course Notes, McGill University, 2008.