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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:
• Answer all questions.
• All questions have equal weight.
• Show all working.
• Your solutions should be written in your own handwriting.
• Do NOT communicate with anyone while completing this exam.
• Use black or blue ink (or equivalent) only.
• Show all working.
• This exam is open book.
• 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 ) ∈ R2 , how many values does u take? Find the explicit solution to this equation.
2.
Consider the equation
uxx + 6uxy − 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( − x, 2x) = x
v.
and
u(x, 0) = sin 2x.
and
u(x, 0) = cos 2x?
Is there a solution that satisfies
u( − x, 2x) = x
Explain your result.
3.
Consider the following problem on the half line 0 < x < ∞ with a Robin boundary condition at x = 0; utt = uxx for 0 < x < ∞ and 0 < t < ∞ u(x, 0) = sin x for 0 < x < ∞ ut (x, 0) = 0 for 0 < x < ∞ ux (0, t) − αu(0, t) = 0 for 0 < t < ∞ where α is a constant. Find u(x, t). Page 3 of 5 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)ϕ ′ ] ′ + q(x)ϕ + λw(x)ϕ = 0. (b) Consider the regular Sturm-Liouville problem ϕ ′′ + λϕ = 0, 0 < x < 1, with boundary conditions ϕ(0) − ϕ ′ (0) = 0 and ϕ(1) + ϕ ′ (1) = 0. i. Use the Rayleigh Quotient to show that λ ⩾ 0. ii. Why is λ > 0?
iii. Show that
√
√
2 λ
.
tan λ =
λ−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 Bessel’s equation to solve for u(r, t) which satisfies the circularly symmetric heat equation

∂u
1 ∂
∂u
=k
r
∂t
r ∂r
∂r
subject to the conditions u(a, t) = 0 and |u(0, t)| < ∞. 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! π ∂2 u ∂u = sin x + 2, ∂t 2 ∂x with u(0, t) = 0, and u(x, 0) = Page 4 of 5 0 < x < 1, t > 0
∂u
(1, t) = t,
∂x
x x2
− .
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 −∞ < x < ∞ and t > 0, and subject to u(x, t) → 0 as x → ±∞, 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:
∇2 u =
∂2 u ∂2 u
+
=0
∂x2
∂y2
subject to the conditions u(0, y) = f(y) and ∂u
∂y (x, 0) = 0. You may assume that f(y) → 0 as
y → ∞ and therefore that u(x, y) → 0 as both x → ∞ and y → ∞ — 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 ∞
1
1
u(x, y) =
+
dȳ.
f(ȳ) 2
Ï€ 0
x + (y − ȳ)2 x2 + (y + ȳ)2
(Solutions by Fourier Sine Transforms get 0 marks.)
End of Examination
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