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Math 412 Fourier Series and Partial Differential Equations

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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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LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 FINC5090 MATH1052 SciComp Elec4622 FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 MAT 237 BISM2201 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SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum

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