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

ISBN 0-471-31999-6

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U(r, t) = J2 [knun(r, 0 + cnvn(r, 0] n=1

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= Y2 [knJ0(knr) sinknat + cnJ0(knr) cosknat\. (17)

n=1

The initial conditions require that

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u(r, 0) = Y2 CnJ0(knr) = f (r) (18)

n=1

and

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Ut(r, 0) = ^2knaknJ0(knr) = 0. (19)

n=1

From Eq. (26) of Section 11.4 we obtain

f rf(r) J0(knr) dr kn = 0, Cn = ^-----------------; n = 1, 2,.... (20)

/ r[J0(knr)]2 dr J0

Thus, the solution of the partial differential equation (3) satisfying the boundary condition (4) and the initial conditions (5) and (6) is given by

?

u(r, t) = ^2, cnJ0(knr) cosknat (21)

n=1

with the coefficients cn defined by Eq. (20).

1. Consider Laplace’s equation uxx + uyy = 0 in the parallelogram whose vertices are (0, 0), (2, 0), (3, 2), and (1, 2). Suppose that on the side y = 2 the boundary condition is u(x, 2) = f(x) for 1 < x < 3, and that on the other three sides u = 0 (see Figure 11.5.1).

D '

(0, 2)

C

(2, 2)

B'(2, 0) (b)

FIGURE 11.5.1 The region in Problem 1.

11.5 Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion 667

(a) Show that there are no nontrivial solutions of the partial differential equation of the form u(x, y) = X(x)Y(y) that also satisfy the homogeneous boundary conditions.

(b) Let f = x — 1 y, n = /. Show that the given parallelogram in the xy-plane transforms into the square 0 < f < 2,0 < n < 2 in the fn-plane. Show that the differential equation transforms into

5 — uh u =0.

4 ff fn nn

How are the boundary conditions transformed?

(c) Show that in the fn-plane the differential equation possesses no solution of the form

u(f,n) = u (f)V (n).

Thus in the xy-plane the shape of the boundary precludes a solution by the method of the separation of variables, while in the fn-plane the region is acceptable but the variables in the differential equation can no longer be separated.

2. Find the displacement u (r, t) in a vibrating circular elastic membrane of radius 1 that satisfies the boundary condition

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

and the initial conditions

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

where g(1) = 0.

Hint: The differential equation to be satisfied is Eq. (3) of the text.

3. Find the displacement u(r, t) in a vibrating circular elastic membrane of radius 1 that satisfies the boundary condition

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

and the initial conditions

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

where f (1) = g(1) = 0.

4. The wave equation in polar coordinates is

urr + (1/r)ur + (1/r2)uee = a—2utr

Show that if u(r,6, t) = R(r)®(9) T(t), then R, ©, and T satisfy the ordinary differential equations

r2 R" + rR + (k2r2 — n2) R = 0,

©" + n2® = 0,

T" + k2a2 T = 0.

5. In the circular cylindrical coordinates r,9, z defined by

x = r cos 9, y = r sin 9, z = z,

Laplace’s equation is

urr + (1/r)ur + (1/r2)u99 + uzz = °.

(a) Show that if u(r, 9, z) = R(r)®(9)Z(z), then R, ®, and Z satisfy the ordinary differential equations

r2 R" + rR + (k2r2 — n2) R = 0,

®" + n2® = 0,

Z"- k2 Z= 0.

668

Chapter 11. Boundary Value Problems and SturmLiouville Theory

(b) Show that if u(r, Q, z) is independent of Q, then the first equation in part (a) becomes

r2 R" + rR + k2r2 R = 0,

the second is omitted altogether, and the third is unchanged.

6. Find the steady-state temperature in a semi-infinite rod 0 < z < to, 0 < r < 1, if the temperature is independent of Q and approaches zero as z ^ to. Assume that the temperature u(r, z) satisfies the boundary conditions

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

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

Hint: Refer to Problem 5.

7. The equation

v + v + k2 v = 0

xx 1 yy

is a generalization of Laplace’s equation, and is sometimes called the Helmholtz (18211894) equation.

(a) In polar coordinates the Helmholtz equation is

vrr + (1/r)vr + (1/r2)vee + k2v = 0.

If v(r,Q) = R(r)©(Q), show that R and © satisfy the ordinary differential equations r2R'+ rR + (k2r2 - k2)R = 0, ©" + k2© = 0.

(b) Consider the Helmholtz equation in the disk r < c. Find the solution that remains bounded at all points in the disk, that is periodic in Q with period 2n, and that satisfies the boundary condition v(c, Q) = f (Q), where f is a given function on 0 < Q < 2n.

Hint: The equation for R is a Bessel equation. See Problem 3 of Section 11.4.

8. Consider the flow of heat in an infinitely long cylinder of radius 1:0 < r < 1,0 < Q < 2n, —to < z < to. Let the surface of the cylinder be held at temperature zero, and let the initial temperature distribution be a function of the radial variable r only. Then the temperature u is a function of r and t only, and satisfies the heat conduction equation

u2[urr + (1/r)ur] = ut, 0 < r < 1, t > 0,

and the following initial and boundary conditions:

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

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

Show that

TO

v > 21 2 *

u(r, t) = J2 cnJ0(knr)e-a knt,

n=1

where J0(kn) = 0. Find a formula for cn.

9. In the spherical coordinates p, Q, 0 (p > 0, 0 < Q < 2n, 0 < 0 < n) defined by the equations

x = p cos Q sin 0, y = p sin Q sin0, z = p cos 0,

Laplace’s equation is

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