Sturm-Liouville theory

In mathematics and its applications, a Sturm-Liouville problem, named after Charles Francois Sturm (1803-1855) and Joseph Liouville (1809-1882), is a second-order linear differential equation of the form

<math>{d\over dx}\left(p(x){dy\over dx}\right)+q(x)y=\lambda w(x)y,<math> (1)

often together with specified boundary values of y and dy/dx. The value of λ is not specified by the problem; finding the values of λ for which there exist solutions satisfying the boundary conditions is part of the problem. The function w(x) is the "weight" or "density" function.

The solutions are eigenfunctions of a Hermitian differential operator in some function space defined by boundary conditions.

Sturm-Liouville theory is important in applied mathematics, where S-L problems occur very commonly, particularly when dealing with linear partial differential equations which are separable.

Contents

1 Sturm-Liouville theorem

2 Sturm-Liouville form

2.1 Examples

3 Sturm-Liouville differential operators

3.1 Some highly technical details
3.2 Useful consequences of the preceding technicalities

4 Example

5 Application to normal modes

6 Bibliography

Sturm-Liouville theorem

The Sturm-Liouville theorem states:

The eigenvalues <math>\lambda_n<math> of a regular ( p(x) is differentiable, q(x) and w(x) are continuous, p(x) > 0 and q(x) > 0 over the interval ) Sturm-Liouville problem are real and well ordered such that

<math>\lambda_1 < \lambda_2 < \lambda_3 < ... < \lambda_n < ... \to \infty <math>.

Corresponding to each eigenvalue <math>\lambda_n<math> is a unique eigenfunction <math>y_n(x)<math>.

The eigenfunctions are mutually orthogonal and satisfy the orthogonality relation

<math> \int_{a}^{b}y_n(x)y_m(x)w(x)dx = 0 , m \ne n <math>, where <math>w(x)<math> is the weighting function.

If the set of eigenfunctions satisfy the orthogonality relation

<math> \int_{a}^{b}y_n(x)y_m(x)w(x)dx = \delta_{mn}<math>, then it's said to form an orthonormal set.

eigenvalues of the Sturm-Liouville problem are given by the Rayleigh quotient:

<math> \lambda_n = \frac{-p y_{n}(x) y' _{n}(x)|_a^b + \int_a^b (p y'_{n}(x)^2 - q y_{n}(x)^2)dx}{\int_a^b (y_{n}(x)^2 w(x))dx} <math>

Sturm-Liouville form

The differential equation

<math>{d\over dx}(p(x){d\over dx}y(x))+q(x)y(x)=\lambda w(x)y(x)<math>

is said to be in Sturm-Liouville form. The function w is known as the weight function. All second-order linear ordinary differential equations can be recast in the form to the left of "=" above by multiplying both sides of the equation by an appropriate "exponential multiplier" (although the same is not true of second-order partial differential equations, or if y is a vector.)

Examples

The Legendre equation,

can easily be put into Sturm-Liouville form, since D(1-x²)=-2x, so, the Legendre equation is equivalent to

Less simple is such a differential equation:

Divide throughout by x³:

Multiplying throughout by an integrating factor of

giving

which can be easily put into Sturm-Liouville form since

so the differential equation is equivalent to

In general, given a differential equation

dividing by P(x) and then multiply through by the integrating factor of

and then collect to give the Sturm-Liouville form.

Sturm-Liouville differential operators

The map

<math>L u ={d\over dx}\left(p(x){du\over dx}\right)+q(x)u<math>

can be viewed as a linear operator mapping a function u to another function Lu. We may study this linear operator in the context of functional analysis. If we put w=1 in equation (1), it can be written as

<math>L u = \lambda u \,<math>

This is precisely the eigenvalue problem; that is, we are trying to find the eigenvalues λ and eigenvectors u of the L operator. However, to be honest we must also include the boundary conditions. Let's say that we want to look at the problem over the interval [0,1] and that we pose the boundary conditions u(0) = u(1) = 0.

The importance of eigenvalue problems stems from the fact that they may help us to solve the associated inhomogeneous problem

<math>L u = f \,<math> in the interval (0,1)

<math>u = 0 \,<math> at 0 and 1.

Here, f is some function in L². If a solution u exists and is unique, we may write it as

because the mapping from f to u must be linear. Now observe that finding eigenvectors and eigenvalues of A is essentially the same as finding eigenvectors and eigenvalues of L. Indeed, if u is an eigenvector of L with eigenvalue λ it must be that u is also an eigenvector of A with eigenvalue 1/λ.

Some highly technical details

Under some assumptions on L, the map A will be continuous from L² to the Sobolev space H² of "twice differentiable" L² functions (differentiability must be understood in terms of Sobolev spaces.) This is for instance the case if p is in H¹, q is in L², p ≤ c for some negative constant c, and q ≥ 0. However, this is not a necessary condition: there are other L which make A continuous.

Here we use three very important theorems:

H² is a subset of L²; if B is the open unit ball in H² then the closure of B in L² is compact.
Hence the map A regarded as a linear map from L² to L² is a compact linear map. (See the spectral theorem.)
All hermitian compact linear maps have an orthonormal basis of eigenvectors; the eigenvalues form a sequence which must tend to zero.

The key words are not all that important, the only important conclusion is that A has an orthonormal basis of eigenvectors.

Useful consequences of the preceding technicalities

If we can find the eigenvectors of L, that is, find the solutions u_k of

<math> L u_{k} = \lambda_k u_k \,<math> in (0, 1)

<math> u = 0 \,<math> at 0 and 1,

along with the eigenvalues λ_k, we can attempt to solve the problem

<math> L u = f \,<math> in (0,1)

<math> u = 0 \,<math> at 0 and 1.

Indeed, from the technical property that the eigenvectors form an orthonormal basis and from Fourier series, we see that any solution u and data f can be written as

If we take the liberty of exchanging the summation sign and the operator L (which can be justified in Sobolev spaces) we obtain:

<math> \sum_k \lambda_k a_k u_k = \sum_k b_k u_k \,<math>

We must use another theorem of Fourier series, which tells us that there is only one way of representing a function as a Fourier series. Hence, we obtain that

<math> a_k = \frac{1}{\lambda_k} b_k <math> (2)

That is, given f (or equivalently its Fourier coefficients b_k) we may compute the Fourier coefficients a_k of u, which is almost as good as computing u directly. Also, as noted above, the coefficients 1/λ_k converge to zero hence (again by Fourier series) the vector u=∑a_ku_k is well defined as long as f=∑b_ku_k is well defined.

When implemented on a computer, this is the spectral method.

Example

We wish to find a function u(x) which solves the following Sturm-Liouville problem:

<math> L u = \frac{d^2u}{dx^2} = \lambda u<math>

where the unknowns are λ and u(x). As above, we must add boundary conditions, we take for example

Observe that if k is any integer, then the function

is a solution with eigenvalue λ = −k². We know that the solutions of a S-L problem form an orthogonal basis, and we know from Fourier series that this set of sinusoidal functions is an orthogonal basis. Since orthogonal bases are always maximal (by definition) we conclude that the S-L problem in this case has no other eigenvectors.

Given the preceding, let us now solve the inhomogeneous problem

with the same boundary conditions. In this case, we must write f(x)=x in a Fourier series. The reader may check, either by integrating ∫exp(ikx)xdx or by consulting a table of Fourier transforms, that we thus obtain

<math>L u =\sum_{k=1}^{\infty}-2\frac{(-1)^k}{k}\sin kx.<math>

This particular Fourier series is troublesome because of its poor convergence properties. It is not clear apriori whether the series converges pointwise. Because of Fourier analysis, since the Fourier coefficients are "square-summable", the Fourier series converges in L² which is all we need for this particular theory to function. We mention for the interested reader that in this case we may rely on a result that says that Fourier's series converges at every point of differentiability, and at jump points (the function x, considered as a periodic function, has a jump at π) converges to the average of the left and right limits (see convergence of Fourier series).

Therefore, by using formula (2), we obtain that the solution is

<math>u=\sum_{k=1}^{\infty}2\frac{(-1)^k}{k^3}\sin kx.<math>

In this case, we could have found the answer using antidifferentiation. This technique yields u=(x³-π²x)/6, whose Fourier series agrees with the solution we found. The antidifferentiation technique is no longer useful in most cases when the differential equation is in many variables.

Application to normal modes

Suppose we are interested in the modes of vibration of a thin membrane, held in a rectangular frame, 0 < x < L₁, 0 < y < L₂. We know the equation of motion for the vertical membrane's displacement, W(x, y, t) is given by the wave equation:

<math>\frac{\partial^2W}{\partial x^2}+\frac{\partial^2W}{\partial y^2} = \frac{1}{c^2}\frac{\partial^2W}{\partial t^2}.<math>

The equation is separable (substituting W = X(x) × Y(y) × T(t)), and the normal mode solutions that have harmonic time dependence and satisfy the boundary conditions W = 0 at x = 0, L₁ and y = 0, L₂ are given by

<math>W_{mn}(x,y,t) = A_{mn}\sin\left(\frac{m\pi x}{L_1}\right)\sin\left(\frac{n\pi y}{L_2}\right)\cos\left(\omega_{mn}t\right)<math>

where m and n are non-zero integers, A_mn is an arbitrary constant and

<math>\omega^2_{mn} = c^2 \left(\frac{m^2\pi^2}{L_1^2}+\frac{n^2\pi^2}{L_2^2}\right).<math>

Since the eigenfunctions W_mn form a basis, an arbitrary initial displacement can be decomposed into a sum of these modes, which each vibrate at their individual frequencies <math>\omega_{mn}<math>. Infinite sums are also valid, as long they converge.

Bibliography

A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, Chapman & Hall/CRC Press, Boca Raton, 2003 (2nd edition).

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Categories: Ordinary differential equations