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

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In three-dimensional problems the separation of variables in Laplace’s operator uxx + u + uzz leads to the equation X" + X X = 0 in rectangular coordinates, to Bessel’s equation in cylindrical coordinates, and to Legendre’s equation in spherical coordinates. It is this fact that is largely responsible for the intensive study that has been made of these equations and the functions defined by them. It is also noteworthy that two of the three most important situations lead to singular, rather than regular, Sturm-Liouville problems. Thus, singular problems are by no means exceptional and may be of even greater interest than regular ones. The remainder of this section is devoted to an example involving an expansion of a given function as a series of Bessel functions.

The Vibrations of a Circular Elastic Membrane. In Section 10.7 [Eq. (7)] we noted that the transverse vibrations of a thin elastic membrane are governed by the twodimensional wave equation

To study the motion of a circular membrane it is convenient to write Eq. (1) in polar coordinates:

We will assume that the membrane has unit radius, that it is fixed securely around its circumference, and that initially it occupies a displaced position independent of the angular variable â, from which it is released at time t = 0. Because of the circular symmetry of the initial and boundary conditions, it is natural to assume also that u is independent of â; that is, u is a function of r and t only. In this event the differential equation (2) becomes

a2(uxx + uyy) = utt

(1)

(2)

0 < r < 1, t > 0.

(3)

The boundary condition at r = 1 is

u(1, t) = 0, t > 0,

(4)

and the initial conditions are

u(r, 0) = f(r), 0 < r < 1,

ut(r, 0) = 0, 0 < r < 1,

(5)

(6)

11.5 Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion 665

where f(r) describes the initial configuration of the membrane. For consistency we also require that f (1) = 0. Finally, we state explicitly the requirement that u(r, t) is to be bounded for 0 < r < 1.

Assuming that u(r, t) = R(r)T(t), and substituting for u(r, t) in Eq. (3), we obtain

where J0 and Y0 are Bessel functions of the first and second kinds, respectively, of order zero (see Section 11.4). In terms of r we have

The boundedness condition on u(r, t) requires that R remain bounded as r ^ 0. Since Y0(kr) ^ — ñþ as r ^ 0, we must choose c2 = 0. The boundary condition (4) then requires that

Consequently, the allowable values of the separation constant are obtained from the roots of the transcendental equation (14). Recall from Section 11.4 that J0(k) has an infinite set of discrete positive zeros, which we denote by ê1, ê2, ê3,. ¦ ¦ ,kn, ¦ ¦¦, ordered in increasing magnitude. Further, the functions J0(knr) are the eigenfunctions of a singular Sturm-Liouville problem, and can be used as the basis of a series expansion for the given function f. The fundamental solutions of this problem, satisfying the partial differential equation (3), the boundary condition (4), and boundedness condition, are

R" + (1/r) R 1 T" 2

r = a t=~k-

(7)

We have anticipated that the separation constant must be negative by writing it as -ê2 with ê > 0.9 Then Eq. (7) yields the following two ordinary differential equations:

r2 R' + rR + k2r2 R = 0, T" + k2a2 T = 0^

(8)

(9)

Thus, from Eq. (9),

T (t) = k1 sin kat + k2cos kat¦

Introducing the new independent variable f = kr into Eq. (8), we obtain

(10)

(11)

(12)

R = C1 J0(kr) + c2Y0(kr )¦

(13)

J0(k) = 0-

(14)

un (r, t) = J0(knr) sin knat, n = 1, 2,¦¦¦,

vn (r, t) = J0(knr) cos knat, n = 1, 2,¦¦¦¦

(15)

(16)

9By denoting the separation constant by —k2, rather than simply by —k, we avoid the appearance of numerous radical signs in the following discussion.

666

Chapter 11. Boundary Value Problems and Sturm-Liouville Theory

PROBLEMS

Next we assume that u(r, t) can be expressed as an infinite linear combination of the fundamental solutions (15), (16):

TO

u(r, t) = ? [knun(r, t) + cnvn(r, 0]

n= 1

TO

= Y2 [knJ0(Ynr) sin Ynat + cnJ0(Ynr) cos Ynat]¦ (17)

n= 1

The initial conditions require that

TO

u(r, 0) = Y2 cnMYnr) = f (r) (18)

n=1

and

TO

ut(r> 0) = Y2 YnaknMYnr) = °- (19)

n=1

From Eq. (26) of Section 11.4 we obtain

¥ rf(r) J0(Ynr) dr

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