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Math 412 Fourier Series and Partial Differential Equations

创建日期：2024-05-21 18:06:45

所属分类：数学mathematics

标签： Math 412

Fourier Series and Partial Differential Equations

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Math 412

Fourier Series and Partial Differential Equations
HOMEWORK 4
due March 16, 2023
1. (a) Find the eigenvalues and eigenfunctions of
x2y′′+ xy′+3y=−λy, y(1) = 0, y(2) = 0.
(b) Expand the piecewise smooth function f (x) in terms of the eigenfunctions of (a).
2. (a) Find the eigenvalues and eigenfunctions of
y′′+ y′+ y= λy, 0≤ x≤ 1, with y(0) = 0 and y(1) = 0.
(b) Expand f (x), piecewise smooth for [0,1], in terms of the eigenfunctions of (a).
3. Solve 
ψt −ψxx = h(x, t), 0 < x< pi, t > 0
ψx(0, t) = 0 t > 0
ψ(pi, t) = 0 t > 0
ψ(x,0) = f (x), 0 < x< pi
and interpret physically.
4. (a) Verify the orthogonality of the Legendre polynomials of odd order for the interval [0,1]∫ 1
0
Pn(x)Pm(x)dx=
δnm
2n+1
.
(b) Find the potential ψ(r,ϕ) in the infinite region r > b, 0 < ϕ <
pi
2
, if ψ = 0 on the plane portion of the
boundary
(
ϕ =
pi
2
, r > b
)
, ψ → 0 as r → ∞ and ψ = f (cosϕ) on the hemispherical portion of the boundary(
r = b, 0≤ ϕ < pi
2
)
.
Consider also the special case f (cosϕ) = cos3ϕ .
5. Find the steady-state temperature (potential) distribution inside a hemisphere if the spherical part is maintained at a
temperature f (cosϕ) and the flat part is insulated against the flow of heat.
Hint: 1. Show that the condition of insulation on the flat face, i.e.,
[
∂ψ
∂ z
]
ϕ=pi/2
= 0 implies
[
∂ψ
∂ϕ
]
ϕ=pi/2
= 0.
2. Your solution should involve Legendre polynomials of even order.
6. Find the steady-state temperature ψ(r,z) in the solid cylinder formed by the three surfaces r = 1, z = 0, and z = 1
when ψ = 0 on the side, ψ = 1 on the top, and the bottom is insulated.
Hint: From the recurrence relation
d
dx
[xnJn(x)] = xnJn−1(x) we obtain
∫
xJ0(x)dx= xJ1(x)+C.
7. Obtain the three lowest energy levels for a particle of mass m in a cylindrical box, whose height and radius equal b.
Schro¨dinger’s equation: − }
2m
∇2ψ = Eψ, ψ = 0 on walls, }=
h
2pi
.
8. A cylinder occupies the region r ≤ b, 0 ≤ z ≤ H. It has temperature f (r,z) at time t = 0. For t > 0, its end z = 0 is
insulated, and the remaining two surfaces are held at temperature 0◦C. Find the temperature in the cylinder.
1
9. Show that the solution of the initial-boundary value problem for the wave equation
a2 ∂
2ψ
∂x2 −
∂ 2ψ
∂ t2 = h(x, t), 0 < x< L, t > 0
ψ(0, t) = F(t), t > 0
ψ(L, t) = G(t), t > 0
ψ(x,0) = f (x), 0 < x< L
ψt(x,0) = g(x), 0 < x< L
is unique.
Hint: Let ψ1 and ψ2 be solutions to the same problem. Then apply the energy integral from homework 2 problem 2
to the function U = ψ1−ψ2.
Not for credit
N1 (a) Find the eigenvalues and eigenfunctions of
y′′+ y′ = λy, y(0) = 0, y(L) = 0.
(b) Expand the piecewise smooth function f (x),
f (x) =
{
x2, 0 < x< L/2
x3−3, L/2 < x< L
in terms of the eigenfunctions of (a). In the coefficients of expansion, the numerators may be left in integral form
(but put in the value of f (x)).
N2 The following differential equations have polynomial solutions for integer values of k :
(a) Laguerre: xy′′+(1− x)y′+ ky= 0.
(b) Hermite: y′′−2xy′+2ky= 0.
(c) Chebyshev: (1− x2)y′′− xy′+ k2y= 0.
If yn(x) and ym(x) are two solutions corresponding to the integers k = n and k = m respectively, show for
(a)
∫ ∞
0
ynyme−x dx= 0, n 6= m.
(b)
∫ ∞
0
ynyme−x
2
dx= 0, n 6= m.
(c)
∫ ∞
0
ynym
(1− x2)1/2 dx= 0, n 6= m.
N3 (a) Obtain the potential in the region between two concentric spheres if the inner surface, r = a, is maintained at a
potential f (cosϕ), and its outer surface r = b is at potential g(cosϕ).
(b) Find the potential (steady-state temperature) distribution between two concentric spheres if the inner and outer
surfaces are maintained at
ψ(1,ϕ) = 30+10cosϕ and ψ(2,ϕ) = 50−20cosϕ, respectively.
N4 Explain why:
(a)
∫ 1
−1
Pn(x)dx= 0, n= 1,2,3, . . .
(b)
∫ 1
0
Pn(x)dx= 0, n= 2,4,6, . . .
(c)
∫ 1
−1
(ax+b)Pn(x)dx= 0, n= 2,3,4, . . . , for a and b real constants.
2
N5 Determine the steady-state temperature distribution inside the sphere of radius a if the boundary temperature is
T0
(
1+2sin2ϕ
)
.
N6 The temperature of a very long circular cylinder of unit radius is initially zero. The surface is maintained at a constant
temperature T0. Find the temperature ψ(r, t).
Hint: From the recurrence relation
d
dx
[xnJn(x)] = xnJn−1(x) we obtain
∫
xJ0(x)dx= xJ1(x)+C.
Enrichment
E1 (a) If w(x, t) =
(
1−2xt+ t2)−1/2 is the generating function for the Legendre polynomials, show that (1− 2xt +
t2)
∂w
∂ t
+(t− x)w= 0, and hence
(n+1)Pn+1(x)− (2n+1)xPn(x)+nPn−1(x) = 0
⇒ Pn+1(x) = 2n+1n+1 xPn(x)−
n
n+1
Pn−1(x).
Hence the Legendre polynomials of all orders can be expressed in terms of P0(x) and P1(x).
(b) Show (1−2xt+ t2)∂w
∂x
− tw= 0, and hence that
P′n+1(x)−2xP′n(x)+P′n−1(x)−Pn(x) = 0.
(c) From (a) and (b) deduce
P′n+1(x)−P′n−1(x) = (2n+1)Pn(x).
Hint: Differentiate the result of (a) - call it (d). Then solve for P′n−1(x) and P
′
n+1(x), respectively, in (b). Then
substitute these values into (d).
E2 (a) Show 4x− x3 =−16∑∞k=1
J1(µkx)
µ3k J0(2µk)
where J1(2µ) = 0 give the discrete µk, k = 1,2,3, . . .
(b) Show 1 =−2
3
∑∞n=1
J0(µnx)
(1+µ2n )J0(3µn)
, where the discrete µn values, n= 1,2,3, . . . are determined from J0(3µ)−
d
dx
J0(µx)
∣∣∣∣
x=3
= 0.
(c) Show
∫ b
0 x
(
1− x
2
b2
)
J0(µkx)dx=
4
bµ3k
J1(µkb) where J0(µb) = 0.

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