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

ISBN 0-471-31999-6

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®(6) = cleie + c2e—ie.

(24)

&(e) = c1 + c2e.

(25)

r2 R" + rR' = 0. This equation is of the Euler type, and has the solution

R(r) = kx + k2ln r.

(26)

(27)

u0(r,e) = 1.

(28)

r2R” + rR - j2R = 0

(29)

and

©" + j2© = 0,

respectively. Equation (29) is an Euler equation and has the solution

R(r ) = kjr1 + k2r —j,

while Eq. (30) has the solution

©(e) = c1 sin je + c2 cos je.

(30)

(31)

(32)

(34)

10.8 Laplace’s Equation

611

The boundary condition (18) then requires that

c “

u(a,9) = -2 + ^an(cn cosnd + kn sinnd) = f (9) (35)

n=1

for 0 < 9 < 2n. The function f may be extended outside this interval so that it is periodic with period 2n, and therefore has a Fourier series of the form (35). Since the extended function has period 2n, we may compute its Fourier coefficients by integrating over any period of the function. In particular, it is convenient to use the original interval (0, 2n); then

1 f2n

anc =— I f (9) cos n9 d9, n = 0, 1, 2,...; (36)

n Jo

1 f2x

ank = —1 f (9) sinn9 d9, n = 1,2,.... (37)

n Jo

With this choice of the coefficients, Eq. (34) represents the solution of the boundary value problem of Eqs. (18) and (19). Note that in this problem we needed both sine

and cosine terms in the solution. This is because the boundary data were given on

0 < 9 < 2n and have period 2n. As a consequence, the full Fourier series is required, rather than sine or cosine terms alone.

PROBLEMS

? 1. (a) Find the solution u(x, y) of Laplace’s equation in the rectangle 0 < x < a,0 < y < b,

also satisfying the boundary conditions

u(0, y) = 0, u(a, y) = 0, 0 < y < b,

u(x, 0) = 0, u(x, b) = g(x), 0 < x < a.

(b) Find the solution if

, , \x, 0 < x < a/2,

g(x) =1 n T Z

ya — x, a/2 < x < a.

(c) For a = 3 and b = 1 plot u versus x for several values of y and also plot u versus y for several values of x.

(d) Plot u versus both x and y in three dimensions. Also draw a contour plot showing

several level curves of u(x, y) in the xy-plane.

2. Find the solution u(x, y) of Laplace’s equation in the rectangle 0 < x < a, 0 < y < b, also satisfying the boundary conditions

u(0, y) = 0, u(a, y) = 0, 0 < y < b,

u(x, 0) = h(x), u(x, b) = 0, 0 < x < a.

? 3. (a) Find the solution u(x, y) of Laplace’s equation in the rectangle 0 < x < a,0 < y < b,

also satisfying the boundary conditions

u(0, y) = 0, u(a, y) = f (y), 0 < y < b,

u(x, 0) = h(x), u(x, b) = 0, 0 < x < a.

Hint: Consider the possibility of adding the solutions of two problems, one with homogeneous boundary conditions except for u(a, y) = f (y), and the other with homogeneous boundary conditions except for u(x, 0) = h(x ).

(b) Find the solution if h(x) = (x/a)2 and f (y) = 1 — (y/b).

(c) Let a = 2 and b = 2. Plot the solution in several ways: u versus x, u versus y, u versus

both x and y, and a contour plot.

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Chapter 10. Partial Differential Equations and Fourier Series

4. Show how to find the solution u(x, y) of Laplace’s equation in the rectangle 0 < x < a, 0 < y < b, also satisfying the boundary conditions

u(0, y) = k(y), u(a, y) = f (y), 0 < y < b,

u(x, 0) = h(x), u(x, b) = g(x), 0 < x < a.

Hint: See Problem 3.

5. Find the solution u(r, 0) of Laplace’s equation

urr + (1/r )ur + (1/r 2)u 00 = 0 outside the circle r = a, also satisfying the boundary condition

u(a,0) = f (0), 0 < 0< 2n,

on the circle. Assume that u(r,0) is single-valued and bounded for r > a.

? 6. (a) Find the solution u(r, 0) of Laplace’s equation in the semicircular region r < a,

0 < 0 < n, also satisfying the boundary conditions

u(r, 0) = 0, u(r,n) = 0, 0 < r < a,

u(a,0) = f (0), 0 < 0 < n.

Assume that u is single-valued and bounded in the given region.

(b) Find the solution if f (0) = 0(n — 0).

(c) Let a = 2 and plot the solution in several ways: u versus r, u versus 0, u versus both r and 0 , and a contour plot.

7. Find the solution u(r, 0) ofLaplace’s equation in the circular sector0 < r < a,0 < 0 < a, also satisfying the boundary conditions

u(r, 0) = 0, u(r,a) = 0, 0 < r < a,

u(a, 0) = f (0), 0 < 0 < a.

Assume that u is single-valued and bounded in the sector.

? 8. (a) Find the solution u(x, y) ofLaplace’s equation in the semi-infinite strip 0 < x < a,

y > 0, also satisfying the boundary conditions

u(0, y) = 0, u(a, y) = 0, y > 0,

u(x, 0) = f (x), 0 < x < a

and the additional condition that u(x, y) ^ 0as y ^ to.

(b) Find the solution if f (x) = x(a — x).

(c) Let a = 5. Find the smallest value of y0 for which u(x, y) < 0.1 for all y > y0.

9. Show that Eq. (23) has periodic solutions only if X is real.

Hint: Let X = —/j2 where n = v + ia with v and a real.

10. Consider the problem of finding a solution u(x, y) ofLaplace’s equation in the rectangle 0 < x < a, 0 < y < b, also satisfying the boundary conditions

ux (0, y) = 0, ux (a, y) = f (y), 0 < y < b,

uy (x, 0) = 0, uy (x, b) = 0, 0 < x < a.

This is an example of a Neumann problem.

(a) Show that Laplace’s equation and the homogeneous boundary conditions determine the fundamental set of solutions

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