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(c) Show that p(x) W(y1, y2)(x) is a constant, by showing that its derivative is zero.
(d) Using Eq. (iv) and the result of part (c), show that G(x, s) = G(s, x).
(e) Verify that y = $(x) from Eq. (iii) with G(x, s) given by Eq. (iv) satisfies the differential equation (i) and the boundary conditions (ii).
In each of Problems 31 through 34 solve the given boundary value problem by determining the appropriate Green’s function and expressing the solution as a definite integral. Use Eqs. (i) to (iv) of Problem 30.
31. — y = f (x), y(0) = 0, y(1) = 0
32. — y = f (x), y(0) = 0, y(1) + /(1) = 0
33. —(y + y) = f(x), y (0) = 0, y(1) = 0
34. — y = f (x), y(0) = 0, /(1) = 0
35. Consider the boundary value problem
L [y] = -[ p(x)y\ + q (x) y = mr (x) y + f(x), (i)
a1 y(0) + a2 y(0) = 0, b1 y(1) + b2/(1) = 0. (ii)
According to the text, the solution y = 0(x) is given by Eq. (13), where cn is defined by
Eq. (9), provided that p, is not an eigenvalue of the corresponding homogeneous problem.
In this case it can also be shown that the solution is given by a Green’s function integral of the form
y = $(x) = ( G(x, s, p) f (s) ds. (iii)
Note that in this problem the Green’s function also depends on the parameter p.
Chapter 11. Boundary Value Problems and Sturm-Liouville Theory
(a) Show that if these two expressions for 0 (x) are to be equivalent, then
^ ^ ^ 0 (x)0t (s) (. )
G(x, s,p)=y —---------------, (iv)
kj — p
where kj and 0 are the eigenvalues and eigenfunctions, respectively, of Eqs. (3), (2) of the text. Again we see from Eq. (iv) that p cannot be equal to any eigenvalue kj.
(b) Derive Eq. (iv) directly by assuming that G(x, s, p) has the eigenfunction expansion
G(x, s,p) = Y2 at (x, p)0j (s); (v)
determine at (x, p) by multiplying Eq. (v) by r(s)0j(s) and integrating with respect to s from s = 0 to s = 1.
Hint: Show first that k} and 0 satisfy the equation
0 (x) = (kt — p) G(x, s, p)r(s)0. (s) ds. (vi)
36. Consider the boundary value problem
— d2 y/ds2 = S(s — x), y(0) = 0, y(1) = 0,
where s is the independent variable, s = x is a definite point in the interval 0 < s < 1, and S is the Dirac delta function (see Section 6.5). Show that the solution of this problem is the Green’s function G(x, s) obtained in Problem 28.
In solving the given problem, note that S(s — x) = 0 in the intervals 0 < s < x and x < s < 1. Note further that —dy/ds experiences a jump of magnitude 1 as s passes through the value x.
This problem illustrates a general property, namely, that the Green’s function G(x, s) can be identified as the response at the point s to a unit impulse at the point x. A more general nonhomogeneous term f on 0 < x < 1 can be regarded as a continuous distribution of impulses with magnitude f (x) at the point x. The solution of a nonhomogeneous boundary value problem in terms of a Green’s function integral can then be interpreted as the result of superposing the responses to the set of impulses represented by the nonhomogeneous term f (x).
11.4 Singular Sturm-Liouville Problems
In the preceding sections of this chapter we considered Sturm-Liouville boundary value problems: the differential equation
L[y] = -[p(x)/]' + q(x)y = kr(x)y, 0 < x < 1, (1)
together with boundary conditions of the form
a1 y(0) + a2/(0) = 0, (2)
b1 y(1) + b2/(1) = 0. (3)
Until now, we have always assumed that the problem is regular; that is, p is differentiable, q and r are continuous, and p(x) > 0 and r(x) > 0 at all points in the closed
11.4 Singular Sturm-Liouville Problems
interval. However, there are also equations of physical interest in which some of these conditions are not satisfied.
For example, suppose that we wish to study Bessel’s equation of order v on the interval 0 < x < 1. This equation is sometimes written in the form6
— (x/)' + — y = Xxy, (4)
so that p(x) = x, q(x) = v2/x, and r(x) = x. Thus p(0) = 0, r(0) = 0, and q(x) is unbounded and hence discontinuous as x ^ 0. However, the conditions imposed on regular Sturm-Liouville problems are met elsewhere in the interval.
Similarly, for Legendre’s equation we have
— [(1 — x2)y] = Xy, —1 < x < 1, (5)
where X = a(a + 1), p(x) = 1 — x2, q(x) = 0, and r(x) = 1. Here the required conditions on p, q, and r are satisfied in the interval 0 < x < 1 except at x = 1 where p
We use the term singular Sturm-Liouville problem to refer to a certain class of boundary value problems for the differential equation (1) in which the functions p, q, and r satisfy the conditions stated earlier on the open interval 0 < x < 1, but at least one of these functions fails to satisfy them at one or both of the boundary points. We also prescribe suitable separated boundary conditions, of a kind to be described in more detail later in this section. Singular problems also occur if the interval is unbounded,
for example, 0 < x < to. We do not consider this latter kind of singular problem in
As an example of a singular problem on a finite interval, consider the equation