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Solutions to Exercises

t
R ∞ R R ∞
Exercise 156. u(x, t) = (S(x−y, t)−S(x+y, t))φ(y)dy+ (S(x−y, t−
0 0 0
s) − S(x + y, t − s))f(y, s)dyds.

∞ 1 3π(2k − 1)t π(2k − 1)x
Exercise 165. a) u = 32h P cos sin ;
π 3 k=1 (2k − 1) 3 l l
∞
8h P 1 πk πkx πkt
b) u = sin sin cos .
π 2 k=1 k 2 2 l l

2
2
Exercise 167. λ = β , where tan βl = k/βc and X(x) = sin βx.

Exercise 169. ω = Nπc/l, where N is any positive integer, provided that g(x)
and sin Nπx/l are not orthogonal.

Exercise 170. No resonance for any ω.

P ∞ cπnt πnx 2 R L πnx
Exercise 174. u(x, t) = n=1 (A cos L +B sin L ), A = L 0 f(x) sin L dx,
n
n
n
2
B = cπn R 0 L g(x) sin πnx dx, n ≥ 1.
n
L

Exercise 176. u(x, t) = 1 − cos 4πx cos 4πt.

∞ 1 π (2k−1) t π(2k − 1)x
2
2
Exercise 183. a) u = 4l P (−1) n−1 e − l 2 sin ;
π 2 k=1 2k − 1 l
∞ 1
2
2
8 P −π (2k−1) t
b) u = e sin π(2k − 1)x.
π 3 k=1 (2k − 1) 3

2 2
Exercise 185. 4 P 1 −n π kt/l 2 sin πnx .
e
π n is odd n l

2 2
P ∞ πnx −iπ n t 2
Exercise 186. n=1 A sin l e l
n

n
P ∞ 1 πn 2
Exercise 189. u(x, t) = frac4π (cos − (−1) )e −17n t sin nx.
n=1 n 2

Exercise 190. A solution of the heat equation has the form
2 2
P ∞ −kπ n t/L 2 πnx 2 R L πnx
(a) u(x, t) = A 0 + A e cos , where A = f(x) cos dx,
2 n=1 n L n L 0 L
n ≥ 0.
(c) The obtained function is a classical solution of the equation for all t > 0,
since if f is continuous the exponential decay implies that for every ε > 0 the

series and all its derivatives converge uniformly for all t > ε > 0. For the same
reason, the series (without A /2) converges uniformly to zero (as a function
0

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