Return to computing page for the first course APMA0330
Return to computing page for the second course APMA0340
Return to computing page for the fourth course APMA0360
Return to Mathematica tutorial for the first course APMA0330
Return to Mathematica tutorial for the second course APMA0340
Return to Mathematica tutorial for the fourth course APMA0360
Return to the main page for the first course APMA0330
Return to the main page for the second course APMA0340
Return to the main page for the fourth course APMA0360
Return to Part II of the course APMA0360
Introduction to Linear Algebra with Mathematica

Glossary

Preface

In its original formulation, the Sturm--Liouville boundary value problem consists of a linear second-order ordinary differential equation expressible in the form \eqref{EqSingular.1} together with suitable separated or periodic boundary conditions at the endpoints of a finite interval [𝑎, b]. The spectrum of the associated self-adjoint operator consists of an increasing sequence of isolated real eigenvalues accumulating at infinity, the corresponding eigenfunctions being non-trivial solutions of \eqref{EqSingular.1} that satisfy the endpoint conditions \eqref{EqSingular.2}. Extension to the case where one of the endpoints of [𝑎, b] is singular was achieved by /Hermann Weyl in 1910. If [𝑎, b] = [0, ∞) or (−∞, ∞), then (1) is often referred to as the one-dimensional time independent Schrödinger equation, following subsequent recognition of its importance in the mathematical description of quantum phenomena.

In general, well-posed Sturm--Liouville boundary value problems generate self- adjoint differential operators in 𝔏²(𝑎, b) for which the generalized Parseval identity holds. However, if a pointwise expansion theorem is required for the same boundary value problem, then the function in 𝔏²(𝑎, b), to be expanded, has to satisfy additional smoothness conditions equivalent to the function belonging to the domain of the corresponding self-adjoint differential operator.

The definition of the spectrum of singular Sturm-Liouville boundary value problems is best seen from the operator theoretic viewpoint; for self-adjoint operators this definition concerns the resolution of the identity of the operator.

Singular Sturm--Liouville problems

Let us consider a homogeneous differential equation for Sturm--Liouville self-adoint operator
\begin{equation} \label{EqSingular.1} L\left[ x, \texttt{D} \right] y = q(x)\,y - \frac{\text d}{{\text d}x} \left( p(x)\,\frac{{\text d}y}{{\text d}x} \right) = \lambda\,w(x)\,y(x) , \qquad a < x < b, \end{equation}
where λ is a parameter, p(x) ∈ ℭ¹(𝑎, b), q(x), and w(x) > 0 are given continuous function on the open interval (𝑎, b); moreover, p(x) is assumed to be continuously differentiable.

A Sturm--Liouville problem for self-adjoint operator \eqref{EqSingular.1} include boundary conditions (of third kind)

\begin{equation} \label{EqSingular.2} \alpha_0 y(a) - \alpha_1 y'(a) =0 , \qquad \beta_0 y(b ) + \beta_1 y' (b ) =0 , \qquad |\alpha_0 | + |\alpha_1 | \ne 0 \quad\mbox{and} \quad |\beta_0 | + |\beta_1 | \ne 0. \end{equation}
These boundary conditions can be reformulated in the following equivalent form:
\[ \begin{split} \sin\alpha \,y(a) - \cos\alpha \,p(a)\,y' (a) &= 0, \\ \cos\beta \,y(b ) + \sin\beta\,p(b )\,y' (b ) &= 0, \end{split} \tag{2a} \]
where 0 ≤ α, β < π. Then the differential equation \eqref{EqSingular.1} together with boundary conditions of the third kind \eqref{EqSingular.2} 0r (2a) form a self-adjoint operator in space 𝔏².
A Sturm--Liouville problem consists of a differential equation \eqref{EqSingular.1} containing a parameter λ, subject to some additional conditions. These additional conditions could be homogeneous boundary conditions, periodic conditions or of some other type. The Sturm--Liouville problem asks to find nontrivial (not identically zero) solutions, called eigenfunctions and corresponding values of parameter λ, called eigenvalues. A Sturm--Liouville problem is called singular if either the coefficient p(x) vanishes at least at one end point x = 𝑎 or/and x = b, or interval (𝑎, b) becomes infinite.
If a leading coefficient p(x) vanishes at a point, then this point is called singular and conditions at this point are usually not specified because it may cause nonexistence of solutions. When singular Sturm--Liouville problem is given on interval [𝑎, b], there are four possible kinds of boundary conditions.
  • p(𝑎) = 0    and boundary condition at x = 𝑎 is dropped, but assumed that the solution is bounded at this point. At another end point x = b regular boundary condition of the third kind is imposed.
  • p(b) = 0    and boundary condition at x = b is dropped, but assumed that the solution is bounded at this point. At another end point x = 𝑎 regular boundary condition of the third kind is imposed.
  • p(𝑎) = p(b) = 0    so there are no boundary conditions; however, the solution of the Sturm--Liouville problem is assumed to be a square integrable function.
  • If 𝑎 = −∞ or/and b = ∞, then no boundary condition is imposed to the infinite point; however, a solution to the Sturm--liouville problem is assumed to be square integrable.

Accordingly, there are four different situations each arising from the zero of p(x). The corresponding differential equations may contain two real parameters α and β along with a nonnegative integer n, used to identify the eigenvalue.

  1. If the function p(x) has a single zero, the Sonin equation emerges:
    \[ x\,y'' + \left( \alpha + 1 -x \right) y' + n\,y = 0. \]
  2. If the function p(x) has two distinct zeroes at end points x = 𝑎 and x = b, then by appropriate translation and scaling, the Jacobi differential equation appears:
    \[ \left( 1 - x^2 \right) y'' + \left[ \left( \beta - \alpha\right) - \left( 2 + \alpha + beta \right) x \right] y' + \left( n + \alpha + \beta + 1 \right) y = 0. \]
    For α = β = 0, these are called the Legendre polynomials.
    For α = β = ± ½, one obtains the Chebyshev polynomials (of the second and first kind, respectively).
  3. If the function p(x) has a double zero, The Bessel equation is discovered:
    \[ x^2 y'' \left( \alpha x + \beta \right) y' + n \left( n+\alpha -1 \right) y = 0. \]
  4. If there is no zero but the interval is infinite, we find the Hermite equation:
    \[ y'' - 2x\, y' + 2n\,y = 0 \qquad (-\infty < x < \infty ). \]

Note that it is convenient to transfer the interval [𝑎, b], we consider, without any loss of generality, into interval [0, 1].

In the following, we discuss these four cases along with some of their applications in partial differential equations. However, more applications can be found in the following sections.

Example 1: Let us consider a standard Sturm--Liouville problem on finite interval of length ℓ with Dirichlet boundary conditions:

\[ - y'' (x) = \lambda\, y(x), \qquad y(0) = 0, \quad y(\ell ) = 0. \]
We discussed this problem previously (section ii), so we know eigenvalues and eigenfunctions:
\[ \lambda_n = \left( \frac{n\pi}{\ell} \right)^2 , \qquad \phi_n (x) = \sin \frac{n\pi x}{\ell} , \qquad n=1,2,3,\ldots . \]
Now we turn ℓ approaches infinity. Onviously, neither eigenvalues nor eigenfunctions have limits, and we need another approach.

Example 2:

End of Example 2

Example 3: The following problem was solved by Daniel Bernoulli (1700–1782) in 1732 and involved the first use of a Bessel function. Small transverse displacements from equilibrium were assumed. The problem was discussed further by Leonhard Euler in 1781. Friedrich Bessel (1784–1846) investigated the functions that now bears his name.

In Bernoulli’s treatment, the chain is a one-dimensional continuum that has constant density. We formulate a slightly more general model that permits variable density. Suppose the length of the chain is ℓ. We set up coordinates so that the x-axis is directed vertically upward with the origin at the free end of the chain when the chain hangs in its vertical equilibrium position. Let ρ0(x) be the density of the chain when it is hanging in equilibrium and u(x, t) be the transverse displacement at time t of the point on the chain that is located at position x when the chain hangs in equilibrium. The only external force acting on the chain is gravity, with constant acceleration g, and the tension at a cross section of the chain acts tangentially and is due to the part of the chain that lies below the cross section.

Under these assumptions the initial boundary value problem for the chain is

\begin{align*} &\mbox{Wave equation:} \qquad &\rho_0 (x)\,u_{tt} &= \left( p(x)\,u_x \right)_x \qquad\mbox{for } x > 0 \mbox{ and } t > 0, \\ &\mbox{Initial condition:} \qquad &u(x,0) &= f(x) , \qquad u_t (x,0) = v(x) , \qquadx > 0, \\ &\mbox{Boundary condition:} \qquad &|u(0,t)| &< \infty , \qquad u(\ell ,t) &= 0 , \qquad t > 0, \\ &\mbox{Regularity condition:} \qquad & \lim_{x,t \to +\infty}\,|u(x,t)| &= 0 , \quad\qquad u, u_t , u_{xx} \in 𝔏²(R), \\ \end{align*}
where
\[ p(x) = g \int_0^x \rho_0 (\xi )\,{\text d}\xi \]
for 0 ≤ x ≤ ℓ, f(x) specifies the initial shape of the chain, and v(x) is its initial velocity profile. Observe that the differential equation is singular because p(0) = 0. Typically such equations can have both bounded and unbounded solutions. Physically realistic solutions for the displacement u(x, t) must be bounded. This leads to the boundary condition |u(0, t)u(0, t)| < ∞, which means that the displacement is bounded for x > 0 and near 0 for all time t. It follows that u(x, t) is bounded in space and time.

We apply separation of ariables upon substitution u(x, t) = X(x) T(t) into the wave equation. This yields

\[ \rho_0 (x)\,T'' (t)\,X(x) = \left( p(x)\,T\,X' \right)' \qquad \Longrightarrow \qquad \frac{T'' (t)}{T(t)} = \frac{(p(x)\,X' (x))'}{\rho_0 (x)\,X(x)} = -\lambda , \]
where λ is a separation constant. Indeed, two functions of distinct independent variables (in our case, they are time t and spacial variable x) can be equal only when both functions are constants. This leads to two differential equations
\[ T'' (t) + \lambda\, T(t) = 0 \qquad \mbox{or} \qquad \frac{{\text d}^2 T(t)}{{\text d} t^2} + \lambda\, T(t) = 0 \]
and
\[ \frac{\text d}{{\text d} x} \left( p(x)\,\frac{{\text d}X(x)}{{\text d} x} \right) + \lambda\,\rho_0 (x)\, X(x) = 0 . \]
Similarly, we get boundary conditions for X(x):
\[ | X(0) | < \infty , \qquad X(\ell ) = 0. \]
Since the differential equation for X(x) has a singular point at the origin, we cannot impose any specifi condition at this point.

In Bernoulli’s original problem ρ0(x) = ρ0, given positive constant, p(x) = gρ0x, the wave equation for the chain becomes

\[ u_{tt} = g \left( x\,u_x \right)_x , \]
and the separated solutions u(x, t) = T(t)X(x) are determined from equations
\[ T'' (t) + \lambda\, T(x) = 0, \]
and
\[ g \left( x\,X' \right)' + \lambda\, X = 0, \qquad X(0) < \infty , \quad X(\ell ) = 0. \]
It turns out that the X-equation is reducible to a Bessel’s equation of order 0 and, hence, that the spatial component of a normal mode is a multiple of a bounded solution of that equation---so called Bessel function of the first kind.
End of Example 3
Now we analyze the difference between regular Sturm--Liouville problem and singular one. First observation tells us that the Sturm--Liouville operator \eqref{EqSingular.1} is linear independently whether it is singular or regular because it satisfies two conditions:
  • \( L \left[ u + v \right] = L[u] + L[v] \quad \) for any two vectors u, v ∈ ℭ²;
  • \( L \left[ k\,u \right] = k\,L[u] \quad \) for any vector u ∈ ℭ² and any constant k.
We remind the following definition:
For given differential operator of the second order \( M\left[ x, \texttt{D} \right] = a(x)\, \texttt{D}^2 + b(x)\, \texttt{D} + c(x) \, \texttt{I} , \) where \( \texttt{D} = {\text d}/{\text d}x , \) then the differential operator span class="math">\( M^{\ast} \left[ x, \texttt{D} \right] = a(x)\, \texttt{D}^2 + \left( 2 a'(x) -b(x)\right) \texttt{D} + \left( a'' (x) - b' (x) + c(x) \right) \texttt{I} , \) is called adjoint to operator M. The operator \( M\left[ x, \texttt{D} \right] \) is called self-adjoint if M = M*.
However, the definition of adjoint operator (and self-adjoint as well) makes sense only for operators acting in the space of functions on whole line, ℝ. When the problem involves functions defined on interval [𝑎, b] and homogeneous boundary conditions are imposed as, for instance, \eqref{EqSingular.2}, this definition needs an adjustment in order to incorporate boundary conditions.

ℱ Let D(L) denote the domain of Sturm--Liouville operator \eqref{EqSingular.1}, with consists of twice continuously differentiable functions hat satisfy the homogeneous boundary conditions of third kind specified in Eq.\eqref{EqSingular.2}. We embed the domain into the Hilbert space D(L) ⊂ 𝔏²[𝑎, b]. In this Hilbert space, the inner product is known to be

\[ \left\langle f, g \right\rangle = \int_a^b f(x)^{\ast} \,g(x)\,{\text d}x , \qquad \mbox{with} \quad \| u \|^2 = \langle u , u \rangle , \]
where asterisk represents complex conjugate (also denoted by overline). Then the Sturm--Liouville operator becomes self-adjoint if the following relation holds for any two functions u, v from the domain D(L):
\[ \left\langle L\left[ x, \texttt{D} \right] u, v \right\rangle = \left\langle u, L\left[ x, \texttt{D} \right] v \right\rangle = \int_a^b \left[ q(x)\,u(x) - \frac{\text d}{{\text d}x} \left( p(x)\, \frac{{\text d}u}{{\text d}x} \right) \right] v(x)\,{\text d}x = \int_a^b u \left[ q(x)\,v(x) - \frac{\text d}{{\text d}x} \left( p(x)\, \frac{{\text d}v}{{\text d}x}\right) \right] {\text d}x . \]
To prove this identity, we integrate by parts and use Lagrange's formula to obtain
\begin{equation} \label{EqSingular.3} \left\langle L\left[ x, \texttt{D} \right] u, v \right\rangle - \left\langle u, L\left[ x, \texttt{D} \right] v \right\rangle = \lim_{x\to b} \left[ v(x)\,p(x)\, \frac{{\text d}u}{{\text d}x} - u(x) \,p(x)\, \frac{{\text d}v}{{\text d}x}\right] - \lim_{x\to a} \left[ v(x)\,p(x)\, \frac{{\text d}u}{{\text d}x} - u(x) \,p(x)\, \frac{{\text d}v}{{\text d}x}\right] \end{equation}
For regular Sturm--Liouville problem, the limit terms in the right-hand side of Eq.\eqref{EqSingular.3} vanish due to homogeneous boundary conditions \eqref{EqSingular.2}. So the Sturm--Liouville operator becomes self-adjoint in this case. Is it true for a singular Sturm--Liouville operator? The answer is positive because its domain includes all functions that satisfy the boundary conditions \eqref{EqSingular.2}. However, this domain is too narrow for our purposes and it is not dense in the Hilbert space 𝔏²[𝑎, b]. In order to understand why, you need to refresh the material from tutorial I about power series solutions to second order differential equations with variable coefficients.

A Sturm--Liouville operator \eqref{EqSingular.1} with a leading term p(x) vanishing at a point defines a differential equation with a singular point. This singular point (either 𝑎, b, or ±&infinl) becomes a regular singular point only when the leading coefficient approaches the singular point no faster than square of the distance to this point. Otherwise, the singular point will be irregular and solutions of such equations are hard or impossible to identify. Moreover, one of the linearly independent solutions is unbounded in a neighborhood of the singular point. Therefore, the requirement to satisfy the boundary conditions \eqref{EqSingular.2} cuts off these solutions, so a one dimensional space is eliminated from the domain. In order to avoid it, we enlarge the domain D(L) to include such twice differentiable functions that annihilate the right-hand side of Eq.\eqref{EqSingular.3}. So this domain includes functions that increase near the singular point, but they grow not too fast to be compensated by p(x).

Remark: A singular Sturm--Liouville problem inckudes only such Sturm--Liouville operator \eqref{EqSingular.2} for which the leading term p(x) vanishing at an endpoint no faster than the second power of the distance to it. Correspondingly, the domain D(L) of this operator includes all twice differentiable functions that annihilate the right-had side of Eq.\eqref{EqSingular.3}.

  1. Weyl, H., Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen, Math. Ann. 68 (1910), 220– 269.

 

Return to Mathematica page)
Return to the main page (APMA0360)
Return to the Part 1 Basic Concepts
Return to the Part 2 Fourier Series
Return to the Part 3 Integral Transforms
Return to the Part 4 Parabolic Differential Equations
Return to the Part 5 Hyperbolic Differential Equations
Return to the Part 6 Elliptic Equations
Return to the Part 7 Numerical Methods