# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

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11.4 Singular Sturm Liouville Problems

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determined only up to a multiplicative constant. The boundary value problem (7), (13), and (14) is an example of a singular Sturm-Liouville problem. This example illustrates that if the boundary conditions are relaxed in an appropriate way, then a singular Sturm-Liouville problem may have an infinite sequence of eigenvalues and eigenfunctions, just as a regular Sturm-Liouville problem does.

Because of their importance in applications, it is worthwhile to investigate singular boundary value problems a little further. There are two main questions that are of concern.

1. Precisely what type of boundary conditions can be allowed in a singular SturmLiouville problem?

2. To what extent do the eigenvalues and eigenfunctions of a singular problem share the properties of eigenvalues and eigenfunctions of regular Sturm-Liouville problems? In particular, are the eigenvalues real, are the eigenfunctions orthogonal, and can a given function be expanded as a series of eigenfunctions?

Both these questions can be answered by a study of the identity

>1

{L[u]v — uL[v]} dx = 0, (17)

1

0

which played an essential part in the development of the theory of regular Sturm-Liouville problems. We therefore investigate the conditions under which this relation holds for singular problems, where the integral in Eq. (17) may now have to be examined as an improper integral. To be definite we consider the differential equation (1) and assume that x = 0 is a singular boundary point, but that x = 1 is not. The boundary condition (3) is imposed at the nonsingular boundary point x = 1, but we leave unspecified, for the moment, the boundary condition at x = 0. Indeed, our principal object is to determine what kinds of boundary conditions are allowable at a singular boundary point if Eq. (17) is to hold.

Since the boundary value problem under investigation is singular at x = 0, we choose

ˆ > 0 and consider the integral J L[u]v dx, instead of J L[u]v dx, as in Section

11.2. Afterwards we let ˆ approach zero. Assuming that u and v have at least two continuous derivatives on ˆ < x < 1, and integrating twice by parts, we find that

ˆ

{L[u]v — uL[v]} dx = — p(x)[u'(x)v(x) — u(x)v'(x)]

(18)

The boundary term at x = 1 is again eliminated if both u and v satisfy the boundary condition (3), and thus,

1

{ L [u]v — uL[v]} dx = p(ˆ)[u' (ˆ)v(ˆ) — u(ˆ)v;(ˆ)]. (19)

ˆ

Taking the limit as ˆ — 0 yields

1

1

0

{L[u]v — uL[v]} dx = lim p(ˆ)[u'(ˆ)v(ˆ) — u(ˆ)v'(ˆ)]. (20)

ˆ —>0

Hence Eq. (17) holds if and only if, in addition to the assumptions stated previously,

lim p(ˆ)[u'(ˆ)v(ˆ) — u(ˆ)v'(ˆ)] = 0 (21)

ˆ— 0

1

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Chapter 11. Boundary Value Problems and Sturm-Liouville Theory

for every pair of functions u and v in the class under consideration. Equation (21) is therefore the criterion that determines what boundary conditions are allowable at x = 0 if that point is a singular boundary point. A similar condition applies at x = 1 if that boundary point is singular, namely,

lim p(1 — e)[u'(1 — e)v(1 — e) — u(1 — e)v'(1 — e)] = 0. (22)

e^0

In summary, as in Section 11.2, a singular boundary value problem for Eq. (1) is said to be self-adjoint if Eq. (17) is valid, possibly as an improper integral, for each pair of functions u and v with the following properties: They are twice continuously differentiable on the open interval 0 < x < 1, they satisfy a boundary condition of the form (2) at each regular boundary point, and they satisfy a boundary condition sufficient to ensure Eq. (21) if x = 0 is a singular boundary point, or Eq. (22) if x = 1 is a singular boundary point. If at least one boundary point is singular, then the differential equation (1), together with two boundary conditions of the type just described, are said to form a singular Sturm-Liouville problem.

For example, for Eq. (7) we have p(x) = x. If both u and v satisfy the boundary condition (14) at x = 0, it is clear that Eq. (21) will hold. Hence the singular boundary value problem, consisting of the differential equation (7), the boundary condition (14) at x = 0, and any boundary condition of the form (3) at x = 1, is self-adjoint.

The most striking difference between regular and singular Sturm-Liouville problems is that in a singular problem the eigenvalues may not be discrete. That is, the problem may have nontrivial solutions for every value of X, or for every value of X in some interval. In such a case the problem is said to have a continuous spectrum. It may happen that a singular problem has a mixture of discrete eigenvalues and also a continuous spectrum. Finally, it is possible that only a discrete set of eigenvalues exists, just as in the regular case discussed in Section 11.2. For example, this is true of the problem consisting of Eqs. (7), (13), and (14). In general, it may be difficult to determine which case actually occurs in a given problem.

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