Sturm-Liouville theory

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

-\frac{d}{dx}\left[p(x)\frac{dy}{ dx}\right]+q(x)y=\lambda w(x)y, \qquad (1).

Here the functions p(x), q(x), and w(x) are specified at the outset, and in the simplest of cases are continuous on the finite closed interval [a,b]. The problem formulation is often together with specified values (boundary values) of y and dy/dx at a and b. The function w(x) is called the "weight" or "density" function.

The value of λ is not specified in the equation; finding the values of λ for which there exist a non-trivial solution of (1) satisfying the boundary conditions is part of the problem called the Sturm-Liouville problem (S-L).

Such values of λ when they exist are called the eigenvalues of the boundary value problem defined by (1) and the prescribed set of boundary conditions. The corresponding solutions (for such a λ) are the eigenfunctions of this problem. Under normal assumptions on the coefficient functions p(x), q(x), and w(x) above, they induce a Hermitian differential operator in some function space defined by boundary conditions. The resulting theory of the existence and asymptotic behavior of the eigenvalues, the corresponding qualitative theory of the eigenfunctions and their completeness in a suitable function space became known as Sturm-Liouville theory. This theory is important in applied mathematics, where S-L problems occur very commonly, particularly when dealing with linear partial differential equations which are separable.

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[edit] Sturm-Liouville theory

The main tenet of Sturm-Liouville theory states that: In the case of regular separated boundary conditions of the form

y(a)\cos \alpha - p(a)y^{\prime}(a)\sin \alpha = 0, \qquad (2)
y(b)\cos \beta - p(b)y^{\prime}(b)\sin \beta = 0, \qquad (3)
where \alpha, \beta \in [0, \pi),
  • The eigenvalues λn of the regular Sturm-Liouville problem (1)-(2)-(3) where (p(x) is differentiable, q(x) and w(x) are continuous, p(x) > 0 and w(x) > 0 over the interval) are real and well ordered such that
\lambda_1 < \lambda_2 < \lambda_3 < \cdots < \lambda_n < \cdots \to \infty;
  • Corresponding to each eigenvalue λn is a unique eigenfunction yn(x) and yn(x) has exactly n − 1 zeros in (a,b);
  • The eigenfunctions are mutually orthogonal and satisfy the orthogonality relation
\int_{a}^{b}y_n(x)y_m(x)w(x)\,dx = 0 , m \ne n,
where w(x) is the weight function.
  • The eigenvalues of the Sturm-Liouville problem (1)-(2)-(3) can be characterized by the Rayleigh quotient
\lambda_n = \frac{-p(x) 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}.

[edit] Sturm-Liouville form

The differential equation

- {d\over dx}\left[p(x){d\over dx}y(x)\right]+q(x)y(x)=\lambda w(x)y(x)

is said to be in Sturm-Liouville form or self-adjoint form. 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 integrating factor (although the same is not true of second-order partial differential equations, or if y is a vector.)

[edit] Examples

Bessel's equation:

x^2y''+xy'+(\lambda^2x^2-\nu^2)y=0\,

can be written in Sturm-Liouville form as

(xy')'+(\lambda^2 x-\nu^2/x)y=0.\,

The Legendre equation,

(1-x^2)y''-2xy'+\nu(\nu+1)y=0\;\!

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

[(1-x^2)y']'+\nu(\nu+1)y=0\;\!

Less simple is such a differential equation as

x^3y''-xy'+2y=0.\,

Divide throughout by x3:

y''-{x\over x^3}y'+{2\over x^3}y=0

Multiplying throughout by an integrating factor of

e^{\int -{x / x^3}\,dx}=e^{\int -{1 / x^2}\, dx}=e^{1 / x},

gives

e^{1 / x}y''-{e^{1 / x} \over x^2} y'+ {2 e^{1 / x} \over x^3} y = 0

which can be easily put into Sturm-Liouville form since

D e^{1 / x} = -{e^{1 / x} \over x^2}

so the differential equation is equivalent to

(e^{1 / x}y')'+{2 e^{1 / x} \over x^3} y =0.

In general, given a differential equation

P(x)y''+Q(x)y'+R(x)y=0\,

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

e^{\int {Q(x) / P(x)}\,dx},

and then collect to give the Sturm-Liouville form.

[edit] Sturm-Liouville differential operators

The map

L u =-{d\over dx}\left[p(x){du\over dx}\right]+q(x)u

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

L u = \lambda u \,

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

L u = f \, in the interval (0,1)
u = 0 \, at 0 and 1.

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

u = A f \,

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.

[edit] 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:

  1. H2 is a subset of L2; if B is the open unit ball in H2 then the closure of B in L2 is compact.
  2. Hence the map A regarded as a linear map from L2 to L2 is a compact linear map. (See the spectral theorem.)
  3. 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.

[edit] Useful consequences of the preceding technicalities

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

L u_{k} = \lambda_k u_k \, in (0, 1)
u = 0 \, at 0 and 1,

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

L u = f \, in (0,1)
u = 0 \, 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

u = \sum_k a_k u_k \,
f = \sum_k b_k u_k \,

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

\sum_k \lambda_k a_k u_k = \sum_k b_k u_k \,

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

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

That is, given f (or equivalently its Fourier coefficients bk) we may compute the Fourier coefficients ak of u, which is almost as good as computing u directly. Also, as noted above, the coefficients 1/λk converge to zero hence (again by Fourier series) the vector u = ∑akuk is well-defined as long as f = ∑bkuk is well defined.

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

[edit] Example

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

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

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

u(0) = u(\pi) = 0 \,

Observe that if k is any integer, then the function

u(x) = \sin kx \,

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

L u =x, x\in(0,\pi)

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)x dx or by consulting a table of Fourier transforms, that we thus obtain

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

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

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

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.

[edit] 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:

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

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

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)

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

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

Since the eigenfunctions Wmn form a basis, an arbitrary initial displacement can be decomposed into a sum of these modes, which each vibrate at their individual frequencies ωmn. Infinite sums are also valid, as long they converge.

[edit] See also

[edit] References

  • A. Zettl, Sturm-Liouville Theory, American Mathematical Society, 2005. ISBN 0-8218-3905-5.
  • A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, Chapman & Hall/CRC Press, Boca Raton, 2003 (2nd edition). ISBN 1-58488-297-2
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