Math 1470: Review Questions for Final Exam
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Math 1470: Review Questions for Final Exam
1. Find the real and complex Fourier series of the following functions (on [−π, π]). For what values of x ∈ R does each series converge? Graph the function that it converges to. Do you get uniform convergence? Explain your answers.
(1) sin2 x.
(2) |x| .
(3) | sin x| .
(4) e-x.
2. Construct and graph the even and odd 2π periodic extensions of the function
A(f)
:(1) 
)xE(.)ven(Wh)e(a)xten(t are)si(t)on(he):(i)r1
i
c(?)
x(o);(n)co(ver)n(g)verge(ence)s(o)u(f)nifor(each.)mly to
2π periodic extension of the
unction f (x) = 1
|x| .
(2) Odd extension: 4 
2π 
si
i(2)i
1
)x + 
sini2ix ; converges uniformly to
2π periodic extension of the function f (x) = sgnx(1 − |x|).
3. Find the Fourier sine and cosine series of the following functions. Graph the function to which the series converges.
(a) 1; (b) cos x; (c) x(π − x).
Answer: (a) Sine: π(4) 
sin 
i
11)x; cosine: 1.
(b) Sine: 
44i2(i si)
1(i)x ; cosine: cos x.
(c) Sine: π(8) 
si
i(2)
3)x ; cosine: 6(π)2 − 
coi(s)2(2)ix.
4. Suppose f (x) is periodic with period P > 0 and integrable. Prove that, for any a ∈ R,
(1) la a+P f (x) dx = l0 P f (x) dx; (2) l0 P f (x + a) dx = l0 P f (x) dx.
Answer: (1) 1aa+P f (x) dx =10P f (x) dx −10a f (x) dx +1P(a)+P f (x) dx
5. Consider the heat equation ut = uxx on 0 < x < 1 with initial temperature u(0, x) = f (x). Find the series solution to the initial-boundary value problem when
(1) the left end of the bar is held at 0 degree and the other end is insulated.
Answer: u(t, x) = 
dn exp l − (n + 
2 π 2t] sin (n + 
πx, where dn = 2 l0 1 f (x) sin (n + 
πxdx.
(2) both ends are insulated.
Answer: u(t, x) = 
+ 
an exp (−n2 π 2t,cos nπx,
where an = 2 l0 1 f (x) cos nπxdx.
6. Write down the solutions to the following initial-boundary value problems for the wave equation on [0, π], in the form of a Fourier series:
(1) utt = uxx , u(t, 0) = u(t, π) = 0, u(0, x) = 0, ut (0, x) = 1.
Answer: u(t, x) = 
.
(2) utt = 3uxx , u(t, 0) = u(t, π) = 0, u(0, x) = sin3 x, ut (0, x) = 0. Answer: u(t, x) = 
cos √tsin x 
cos 3 √3tsin 3x.
Answer: u(t, x) = 
( − 1)n+1 (cos 2nt − 
.
7. Solve the following boundary value problems for Laplaces equation on the square [0, π]2 .
(1) u(x, 0) = sin3 x, u(x, π) = u(0, y) = u(π, y) = 0.
Answer: u(x, y) = 3 sin 4(x)s(s)i(i)n(n)h(h)(π(π) − y) − sin 3x4(s)
π(π)− 3y) .
(2) u(x, 0) = u(0, y) = u(π, y) = 0, u(x, π) = 1.
Answer: u(x, y) = π(4) 
si
j++1
n(s)h(i)
y .
y (x, π) = u(0, y) = ux (π, y) = 0.
Answer: u(x, y) = sin 2(x) cc(o)o(s)s(h)
π − y) .
8. Use separation of variables to solve the following boundary value problem in the unit square [0, 1]2 .
∆u + 2ux = 0, u(x, 0) = f (x), u(x, 1) = u(0, y) = u(1, y) = 0.
o
Answer: u(x, y) = 
bne-x sinh √ 1 + n2 π 2 x sinnπy ,
where bn = 
1 f (y) sinnπydy.
9. Solve the following boundary value problems for Laplace’s equation in the unit disk x2 + y2 < 1.
(1) u = x3 on x2 + y2 + 1.
3 
3 
2 
.
(2) 
= x on x2 + y2 = 1.
Answer: u(x, y) = x.
10. Let u be harmonic in the unit disk and u = x2 on the boundary. Find u(0, 0). Answer: 1/2.
11. Solve the mixed boundary value problem for Laplace equation on the pie wedge 0 ≤ θ ≤ π/4, 0 ≤ r ≤ 1, with Dirichlet boundary data u(1, θ) = cos4 (2θ) on the curved portion of its boundary and homogeneous Neumann condition on the two straight wedges.
Answer: u(r, θ) = 
+ 
r4 cos(4θ) + 
r8 cos(8θ).
12. Write out the series solution to the boundary value problem u(1, θ) = 0, u(2, θ) =
h(θ) for Laplace equation on an annulus 1 < r < 2.
Answer:
u(r, θ) = 2(A)0 l(l)o(o)g(g) 2(r) + 
2n(rn) 
2(r)-(-)n(n) (An cos nθ + Bn sinnθ) ,
where An , Bn are the usual Fourier coefficients of h(θ).
the constant Dirichlet boundary conditions u = a on r 
. Answer: u should be radially symmetric and hence a linear combination of log r and 1. A short computation shows that
u = l(b)
2(a) log r + b = 2(b)l
2(a) log(x2 + y2 ) + b.
2023-06-24