# Elementary differential equations 7th edition - Boyce W.E

ISBN 0-471-31999-6

**Download**(direct link)

**:**

**287**> 288 289 290 291 292 293 .. 486 >> Next

/

{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(e)v/(e)]. (19)

I

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)

e^0

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

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

e^0

660

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 k, or for every value of k 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.

A systematic discussion of singular Sturm-Liouville problems is quite sophisticated8 indeed, requiring a substantial extension of the methods presented in this book. We restrict ourselves to some examples related to physical applications; in each of these examples it is known that there is an infinite set of discrete eigenvalues.

If a singular Sturm-Liouville problem does have only a discrete set of eigenvalues and eigenfunctions, then Eq. (17) can be used, just as in Section 11.2, to prove that the eigenvalues of such a problem are real, and that the eigenfunctions are orthogonal with respect to the weight function r. The expansion of a given function in terms of a series of eigenfunctions then follows as in Section 11.2.

Such expansions are useful, as in the regular case, for solving nonhomogeneous boundary value problems. The procedure is very similar to that described in Section

11.3. Some examples for ordinary differential equations are indicated in Problems 1 to 4, and some problems for partial differential equations appear in Section 11.5.

For instance, the eigenfunctions (x) = J0^^/k~n x) of the singular Sturm-Liouville

problem

— (x/)' = kxy, 0 < x < 1,

y, y bounded as x ^ 0, y(1) = 0

8See, for example, Chapter 5 of the book by Yosida.

11.4 Singular Sturm-Liouville Problems

661

PROBLEMS

satisfy the orthogonality relation

-1

(x)0n(x) dx = 0, m = n (23)

I ‘"r m'

Jo

with respect to the weight function r(x) = x. Then, if f is a given function, we assume that

n=1

f(x) = 1] cnJoUXnx)- (24)

Multiplying Eq. (24) by xJo^^fk^ x) and integrating term by term from x = 0to x = 1 yield

jo xf(x) x) dx=? cnjo xu^Knx) x) dx? (25)

Because of the orthogonality condition (23), the right side of Eq. (25) collapses to a single term; hence

J xf(x)Jij^x) dx

Jq x4(jKnx) dx

(26)

which determines the coefficients in the series (24).

The convergence of the series (24) is established by an extension of Theorem 11.2.4 to cover this case. This theorem can also be shown to hold for other sets of Bessel functions, which are solutions of appropriate boundary value problems, for Legendre polynomials, and for solutions of a number of other singular Sturm-Liouville problems of considerable interest.

It must be emphasized that the singular problems mentioned here are not necessarily typical. In general, singular boundary value problems are characterized by continuous spectra, rather than by discrete sets of eigenvalues. The corresponding sets of eigenfunctions are therefore not denumerable, and series expansions of the type described in Theorem 11.2.4 do not exist. They are replaced by appropriate integral representations.

**287**> 288 289 290 291 292 293 .. 486 >> Next