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

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kn = 0, cn = ^-; n = 1, 2,.... (20)

/ r[JQ(Xnr)f dr 0

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

TO

u(r, t) = ^2 cnJ0(Ynr) cos Ynat (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).

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 ? = x - 1 y, n = y. Show that the given parallelogram in the xy-plane transforms into the square 0 < ? < 2, 0 < n < 2 in the ?n-plane. Show that the differential equation transforms into

5 — ø + u =0.

4 ?? ?n ÏÏ

How are the boundary conditions transformed?

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

u(?,n) = U (?)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 ? n-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 + (y/rl')uee = ar-'utf

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

r2 R' + rR + (X2r2 - n2) R = 0,

&" + n2& = 0,

T" + X2a2 T = 0.

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

x = r cos â, y = r sin â, z = z,

Laplace’s equation is

urr + (1/r )ur + (1/r2)Uee + uzz = °.

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

r2R' + rR + (X2r2 - n2) R = 0,

®" + n2& = 0,

Z"- X2Z= 0.

668

Chapter 11. Boundary Value Problems and Sturm Liouville Theory

(b) Show that if u(r, 9, z) is independent of 9, 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 < æ, 0 < r < 1, if the temperature is independent of 9 and approaches zero as z ^ æ. 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 + k2v = 0

xx 1 ÓÓ

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

(a) In polar coordinates the Helmholtz equation is

vrr + (1 j r)v r + (1/r2)v99 + k2v = 0^

If v(r,9) = R(r)©(9), 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

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