Mid-Year Examinations, 2022

MATH302-22S1 (C)

Mathematics and Statistics

EXAMINATION

Mid-Year Examinations, 2022

MATH302-22S1 (C) Partial Differential Equations

Time allowed:

THREE HOURS (which includes the time to upload your solutions)

Number of questions:

5

Number of pages:

5

Instructions to Students:

â€¢ Answer all questions.

â€¢ All questions have equal weight.

â€¢ Show all working.

â€¢ Your solutions should be written in your own handwriting.

â€¢ Do NOT communicate with anyone while completing this exam.

â€¢ Use black or blue ink (or equivalent) only.

â€¢ Show all working.

â€¢ This exam is open book.

â€¢ Your solutions need to be submitted via Learn WITHIN the THREE hours.

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Mid-Year Examinations, 2022

MATH302-22S1 (C)

Questions Start on Page 3

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Mid-Year Examinations, 2022

1.

MATH302-22S1 (C)

Consider the equation

u u x + uy = 1

with

u(x2 , x) = 0.

At each point (x0 , y0 ) âˆˆ R2 , how many values does u take? Find the explicit solution to this equation.

2.

Consider the equation

uxx + 6uxy âˆ’ 16uyy = 0.

i. What is the type of this equation?

ii. Find the canonical form of this equation.

iii. Find the general solution u(x, y).

iv. Find the solution(s) u(x, y) which satisfies

u( âˆ’ x, 2x) = x

v.

and

u(x, 0) = sin 2x.

and

u(x, 0) = cos 2x?

Is there a solution that satisfies

u( âˆ’ x, 2x) = x

Explain your result.

3.

Consider the following problem on the half line 0 < x < âˆž with a Robin boundary condition at x = 0;
utt = uxx
for 0 < x < âˆž and 0 < t < âˆž
u(x, 0) = sin x
for 0 < x < âˆž
ut (x, 0) = 0
for 0 < x < âˆž
ux (0, t) âˆ’ Î±u(0, t) = 0
for 0 < t < âˆž
where Î± is a constant. Find u(x, t).
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Mid-Year Examinations, 2022
4.
MATH302-22S1 (C)
(a) Use the following form of the expression for a regular Sturm-Liouville eigenvalue problem on the
interval [a, b] to derive an expression for the Rayleigh Quotient:
[p(x)Ï• â€² ] â€² + q(x)Ï• + Î»w(x)Ï• = 0.
(b) Consider the regular Sturm-Liouville problem
Ï• â€²â€² + Î»Ï• = 0,
0 < x < 1,
with boundary conditions
Ï•(0) âˆ’ Ï• â€² (0) = 0
and Ï•(1) + Ï• â€² (1) = 0.
i. Use the Rayleigh Quotient to show that Î» â©¾ 0.
ii. Why is Î» > 0?

iii. Show that

âˆš

âˆš

2 Î»

.

tan Î» =

Î»âˆ’1

iv. Sketch a suitable graph which could be used to estimate the eigenvalues, and use the graph

to estimate the large eigenvalues.

5.

(a) Use Besselâ€™s equation to solve for u(r, t) which satisfies the circularly symmetric heat equation

âˆ‚u

1 âˆ‚

âˆ‚u

=k

r

âˆ‚t

r âˆ‚r

âˆ‚r

subject to the conditions u(a, t) = 0 and |u(0, t)| < âˆž. Note: without an initial condition, your
solution will contain unknown coefficients.
(b) Reduce the following non-homogeneous heat equation with a source term to its corresponding
source-free homogeneous problem, showing all steps. Do NOT solve the resulting problem!
Ï€ âˆ‚2 u
âˆ‚u
= sin
x + 2,
âˆ‚t
2
âˆ‚x
with
u(0, t) = 0,
and
u(x, 0) =
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0 < x < 1, t > 0

âˆ‚u

(1, t) = t,

âˆ‚x

x x2

âˆ’ .

2

3

Mid-Year Examinations, 2022

MATH302-22S1 (C)

6. In this question on Fourier Transforms, you can use without proof any transform and inverse transform

from the tables in class.

(a) Without using the general result from class, use Fourier Transforms to solve the source-free heat

equation on âˆ’âˆž < x < âˆž and t > 0, and subject to u(x, t) â†’ 0 as x â†’ Â±âˆž, with

0 x0.

(b) Consider the steady-state temperature distribution in a quarter plane with one insulated wall and

one wall on which the temperature is held fixed:

âˆ‡2 u =

âˆ‚2 u âˆ‚2 u

+

=0

âˆ‚x2

âˆ‚y2

subject to the conditions u(0, y) = f(y) and âˆ‚u

âˆ‚y (x, 0) = 0. You may assume that f(y) â†’ 0 as

y â†’ âˆž and therefore that u(x, y) â†’ 0 as both x â†’ âˆž and y â†’ âˆž â€” you will need this as one

of your boundary conditions!

i. Explain briefly why you could analyse this problem with either a Fourier Sine Transform or a

Fourier Cosine Transform. Your answer should indicate in which direction these transforms

would be made.

ii. Now use a Fourier Cosine Transform in the appropriate direction to show that the solution

to this problem is

Z

x âˆž

1

1

u(x, y) =

+

dyÌ„.

f(yÌ„) 2

Ï€ 0

x + (y âˆ’ yÌ„)2 x2 + (y + yÌ„)2

(Solutions by Fourier Sine Transforms get 0 marks.)

End of Examination

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