Math. 401, Sec. 500 Spring, 2005
Homework 10, due April 8
1. [Schaum’s, p. 46, Ex. 2.46 and 2.48)] Use Parseval’s identity along with Exercise 5
of Homework 8 and Exercise 6 of Homework 9 to show (in any convenient order)
(a)
∞
X
n=1
1
n4=π4
90 .
(b)
∞
X
n=1
1
n6=π6
945 .
(c)
∞
X
n=1
1
(2n−1)4=π4
96 .
(d)
∞
X
n=1
1
(2n−1)6=π6
960 .
2. [Schaum’s, p. 46, Ex. 2.56 and 2.57)]
(a) A square plate of side Lhas one side maintained at temperature f(x)andthe
others at zero. Find the steady-state temperature at any point of the plate (as a
Fourier series of appropriate type).
(b) Explain how to solve the problem if the four sides are maintained at temperatures
f1(x), g1(y), f2(x), and g2(y). (Write out the answer in full for the case f2=0=
g
2.)
3. [Schaum’s, p. 46, Ex. 2.58(a))] An infinitely long plate of width Lhas its two parallel
sides maintained at temperature 0 and its other side at constant temperature T.Find
the steady-state temperature.
4. [Schaum’s, p. 46, Ex. 2.63)] Solve the boundary value problem
∂u
∂t =∂2u
∂x2
−α2u, u(0,t)=u
1,u(L, t)=u
2,u(x, 0) = 0,
where 0 <x<L,0<t,andα,L,u
1,andu
2are constants.
Instructions for Exercises 5 and 6: J. B. Fourier was ridiculed by some of the
mathematicians of his day when he first announced his discovery that an arbitrary
function on the interval 0 <x<π,suchasf(x)=x
2
, can be expanded in a series of
sine functions. Some of the criticisms were like the two statements which follow. In
each case explain in a short essay how the mathematicians were confused (and Fourier
was right).
5. “x2is an even function; but any fool can see that a sum of sines will always be an odd
function.”
6. “x2is not zero at the right endpoint (π); but any fool can see that a sum of the
functions sin nx will always vanish at x=π. The same criticism applies if we consider
the limits of functions as x→πinstead of the values of the functions when x=π.”