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MATH 323 Mathematical Sciences

创建日期：2024-04-24 15:10:56

所属分类：数学mathematics

标签： MATH 323

Mathematical Sciences

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MATH 323
DEPARTMENT: Mathematical Sciences
FURTHER METHODS OF APPLIED MATHEMATICS
Time allowed: Two and a half hours
INSTRUCTIONS TO CANDIDATES:
This paper contains SEVEN questions, all carrying equal weight. Full marks will
be awarded for complete solutions to FIVE questions. Candidates may attempt
all questions, but only the best FIVE solutions will be taken into account.
Candidates are only permitted to use calculators deemed acceptable, and affixed
with an official holographic sticker, by the Department of Mathematical Sciences.
Paper Code MATH 323 Page 1 of 6 CONTINUED
1. (a) By substituting y = xn find the general solution to the Euler’s differ-
ential equation
x2y′′ + 3xy′ + y = 0, x > 0.
[4 marks]
(b) Use the method of variation of parameters to find the solution to the differ-
ential equation
x2y′′ + 3xy′ + y = 4 ln x, x > 0
satisfying boundary conditions y(1) = 0, y′(2) =
1
4
. [8 marks]
(c) For the differential equation
y′′ − 1
x
y′ − 4x2y = 0
verify that y1(x) = e
−x2 is a solution, find the second solution and write down
the general solution. [8 marks]
2. (a) Write down the Euler–Lagrange equation for a function y(x) which ex-
tremises the functional
J [y] =
∫ b
a
F (x, y, y′)dx, where y′ =
dy
dx
.
Write down the sufficient condition for J [y] to have a minimum at y = y0.
[3 marks]
(b) For the given functional
J [y(x)] =
∫ 2
1
(x2y′2 + 12y2 + 4yx4)dx
write down and solve the differential equation for y(x) which extremises J [y] and
satisfies the boundary conditions y(1) = −4 and y(2) = 127. [8 marks]
(c) Use the sufficient condition to investigate the nature of the extremum of J [y].
[3 marks]
(d) Using the result from (b) write down the solution of the Euler-Lagrange
equation if the functional is now
J [y(x)] =
∫ 2
1
(x2y′2 + 12y2)dx
and y(1) = 1, y(2) = 8. Calculate the extremal value of J [y] corresponding to the
solution y(x) and compare it with the value of J [y] for y(x) = 7x− 6. Comment
on the nature of the extremum. [6 marks]
Paper Code MATH 323 Page 2 of 6 CONTINUED
3. (a) Explain how to find a function y(x) : [a, b]→ R for which the functional
I[y] =
∫ b
a
F (x, y, y′)dx
has an extremum and for which the constraint equation
J [y] =
∫ b
a
G(x, y, y′)dx = C, C ∈ R
and the boundary conditions y(a) = y0 and y(b) = y1 are satisfied.
Explain the origin of the three constants required to fulfill y(a) = y0, y(b) = y1
and J [y] = C. [2 marks]
(b) The shape of a flexible string of length 10 fixed at two points (−4, 0) and
(4, 0) is described by the function y(x). By maximising the functional
I(y) =
∫ 4
−4
y(x)dx for J [y] =
∫ 4
−4
√
1 + y′2dx = 10,
show that the area between the string and the x axis is a maximum if the shape
of the string is an arc of a circle. Find the radius and the centre of the circle.
[13 marks]
[Hint: The numerical solution to the equation
10 = 2
√
16 + c21 tan
−1 4
c1
is c1 = 1.88163.]
(c) Consider now a string fixed at the same two points as in (b) but in the shape
of a semicircle centered on the origin. What is the length of the string and the
area A it encloses with the x axis. Compare A with a rectangular shaped string
of the same length and comment on the result.
[5 marks]
Paper Code MATH 323 Page 3 of 6 CONTINUED
4. (a) The functions u(x, y) and v(x, y) satisfy the following system of partial
differential equations
yux + vy = y sinx
yvx + uy = −y cosx.
Show that this system is hyperbolic with the characteristics
α = x− y
2
2
, β = x+
y2
2
. [4 marks]
(b) Use the characteristics to find the Riemann invariants. [6 marks]
(c) Find the general solution for u(x, y) and v(x, y) which satisfy the boundary
conditions u(x, 0) = sin x and v(x, 0) = cos x. [6 marks]
(d) Check that u(x, y) and v(x, y) found in (c) satisfy the system of differential
equations and given boundary conditions. [4 marks]
5. (a) State well posedness for initial/boundary value problems. [2 marks]
(b) Consider the following initial-boundary value problem:
utt = 4uxx + 3t, 0 < x < 2, t > 0, (1)
ux|x=0 = 0, ux|x=2 = 0, (2)
u|t=0 = cos(2pix), ut|t=0 = 0. (3)
Determine the type of the PDE (1). Explain briefly without proof whether the
problem (1)-(3) is or is not well posed. [3 marks]
(c) Show that for y 6= 0, x 6= 0 the partial differential equation
y2uxx − 2xyuxy + x2uyy = 1
xy
(
x3uy + y
3ux
)
(4)
for the function u(x, y) is parabolic with the family of characteristics
ξ(x, y) = x2 + y2 = const. [3 marks]
(d) Explain briefly when the equations
ξ = x2 + y2, η = η(x, y)
define a functionally independent coordinate transformation. By setting η = x
show that this transformation reduces equation (4) to the canonical form
uηη =
1
η
uη. [7 marks]
(e) Find the solution of equation (4) that satisfies the boundary conditions
u(0, y) = e−y
2
, u(1, y) = 0.
[5 marks]
Paper Code MATH 323 Page 4 of 6 CONTINUED
6. (a) Show that the Joukowski transformation
w = u+ iv =
(
z +
1
z
)
, z 6= 0
is conformal for all z 6= ±1, where u and v are real and z = x + iy is a com-
plex variable. Explain what happens in the w-plane with angles between lines
intersecting at these points in the z plane.
[4 marks]
(b) Find u and v in terms of polar coordinates r and θ where x = r cos θ,
y = r sin θ. Show that the circles |z| = 2 and |z| = 1
2
are mapped to the same
ellipse
E1 :
4u2
25
+
4v2
9
= 1
in the w−plane. Explain what happens with the circle |z| = 1. Hence show that
the exterior of |z| = 2 is mapped to the exterior of E1. [9 marks]
(c) The potential Φ outside the circular region |z| = 2 is given by
Φ = Re(A ln z) = A ln r,
where A is a real constant. Show that Φ satisfies Laplace’s equation in polar
coordinates
∂2Φ
∂r2
+
1
r
∂Φ
∂r
+
1
r2
∂2Φ
∂θ2
= 0.
By clearly giving your reasoning, show that the potential outside the ellipse E1
is given by
Φ = ARe ln
[
w +
√
(w2 − 4)
2
]
.
Determine the potential at the point w = 6i.
[7 marks]
Paper Code MATH 323 Page 5 of 6 CONTINUED
7. (a) The Fourier transform of a suitable function f(x) defined on −∞ < x <∞
is
F{f(x)} = f¯(ω) =
∫ ∞
−∞
f(x)e−iωxdx.
Show that the Fourier transform f¯(ω) of the function f(x) = e−a
2x2 , where a is
a real non-zero constant, satisfies the first order differential equation
f¯ ′(ω) = − ω
2a2
f¯(ω).
[4 marks]
(b) By considering the square of f¯(0) show that for f(x) = e−a
2x2
f¯(0) =
√
pi
a
.
[4 marks]
(c) Hence show that f¯(ω) =
√
pi
a
e−
ω2
4a2 [3 marks]
(d) The function u(x, t) satisfies the partial differential equation
∂u
∂t
= c2
∂2u
∂x2
for −∞ < x <∞ and t ≥ 0. In addition u satisfies the conditions
u(x, 0) = e−a
2x2 , u→ 0 as x→ ±∞.
By using a Fourier transform show that u(x, t) is given by
u(x, t) =
1
2a
√
pi
∫ ∞
−∞
e−
ω2
4a2
−c2ω2t+iωxdω.
Hence by substituting
1
X2
=
1
a2
+ 4c2t or otherwise, show that
u(x, t) =
1√
1 + 4a2c2t
e
− a2x2
1+4a2c2t .

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MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 IE5600 ECOS 3997 EDST2003 FIN 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