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MA30059 Mathematical Methods
Mathematical Methods
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This is an open book exam. You may refer to your own course and revision notes and look up
information in offline or online resources, for example textbooks or online journals.
This exam is designed to take approximately 3 hours (including up to 1 hour for upload) to complete.
Which questions should be answered: Answer THREE of the four questions. Only the BEST three
answers will contribute to the assessment.
Filenames: If you are required to upload a file as part of your exam attempt, to maintain anonymity
please use the following naming convention for your file:
CandidateNumberUnitCodeQuestionNumber.pdf (e.g. 01234AR10001Q2a.pdf).
If the exam only requires one file to be submitted, you do not need to include the question
number(s).
Additional materials needed to complete the assessment: A calculator is not necessary for this
assessment.
Further instructions: In marking, emphasis will be placed upon clear mathematical argument and
precision. Do NOT submit your rough work.
Academic Integrity for Remote Exams
When you registered as a student you agreed to abide by the University's regulations and rules, and
agreed that you would access and read your programme handbook. These documents contain
references to, and penalties for, unfair practices such as collusion, plagiarism, fabrication or
falsification. The University's Quality Assurance Code of Practice, QA53 Examination and Assessment
Offences , sets
out the consequences of committing an offence and the penalties that might be applied.
By submitting your exam as instructed, you confirm that:
1. You have not impersonated, or allowed yourself to be impersonated by, any person for the
purposes of this assessment.
2. This assessment is your original work and no part of it has been copied from any other
source except where due acknowledgement is made. The University may submit your work
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3. You have not previously submitted this work for any other unit/course.
4. You give permission for your assessment response to be reproduced, communicated,
compared and archived for plagiarism detection, benchmarking or educational purposes.
5. You understand that plagiarism is the presentation of the work, idea or creation of another
person or organisation as though it is your own. It is a form of cheating and is a very serious
academic offence that may lead to disciplinary action.
6. You understand that this assessment is undertaken without invigilation, and that you have
not communicated with and will not communicate with anyone concerning this assessment
before the deadline for submission unless it is expressly permitted by the assessment
instructions.
7. No part of this assessment has been produced for, or communicated to, you by any other
person, unless it is expressly permitted by the assessment instructions.
If you have any questions about the exam you should contact the exams helpline.
Information and contact details can be found on our help and advice webpage
1. (a) Consider the ball [8]
B1(0) = {(x1, x2) ∈ R2 : x21 + x22 < 1},
and the functions
u1(x1, x2) = x1x2, (x1, x2) ∈ B1(0), u2(x1, x2) = x21x2−x1x22, (x1, x2) ∈ B1(0).
(i) Explain why the function u1 is harmonic on B1(0).
(ii) Is the function u2 harmonic in B1(0)? Justify your answer.
(iii) Determine the maximum of |u1| on the circle {(x1, x2) ∈ R2 : x21 + x22 = 1}.
(iv) Show, without invoking the maximum principle or the weak maximum
principle, that the principle of maximum of modulus holds for u1 on B1(0).
(b) The fundamental solution for the Laplace operator in two dimensions is [6]
Φ(x) =
1
2pi
log
1
|x| , x ∈ R
2 \ {0}.
(i) Show that Φ is harmonic in the annulus
A = {(x1, x2) ∈ R2 : 1/2 < |x| < 2}.
(ii) Like Φ, the constant function f(x) = −(log 2)/(2pi), x ∈ A, is harmonic in A
and has the same boundary values on {(x1, x2) ∈ R2 : |x| = 2}. Explain why
this does not contradict the uniqueness theorem for the Dirichlet problem.
(iii) Using the maximum principle for the Laplace operator, determine the
maximum and the minimum of Φ− f on A.
(c) Consider the boundary-value problem [6]
∆u = 0 on A,
u|S1/2(0) = 0, u|S2(0) = 1,
(A)
where SR(0), R > 0, denotes the circle of radius R centred at zero. Using separation
of variables, determine the solution u to (A). You may wish to use the expression
for the Laplace operator in polar coordinates:
∂2
∂r2
+
1
r
∂
∂r
+
1
r2
∂2
∂φ2
,
where r is the distance from the origin, φ is the polar angle.
Page 3 of 6 MA30059/MA40059/MA50059
2. Consider a bounded region Ω ⊂ Rd, d = 2 or d = 3, with C1 boundary.
(a) Suppose that u ∈ C2(Ω) satisfies the Laplace’s equation ∆u = 0 in Ω. [6]
(i) Using the representation theorem for C2 functions in terms of potentials, argue
that the values of u in Ω can be recovered from the values of u on the boundary
∂Ω and the values of the normal derivative ∂u/∂n on ∂Ω.
(ii) Using the uniqueness theorem for a suitable Dirichlet problem, explain why
just the values of u on ∂Ω fully determine u everywhere in Ω.
(b) Consider the Green’s function G = G(x, y) for the Laplace operator on Ω and write
g(x, y) = G(x, y)− Φ(x− y), x ∈ Ω, y ∈ Ω, x 6= y,
where
Φ(x) =
1
2pi
log
1
|x| , x ∈ R
2 \ {0},
1
4pi|x| , x ∈ R
3 \ {0},
is the fundamental solution for the Laplace operator. [14]
(i) For a given y ∈ ∂Ω, express the function g(x, y), x ∈ Ω, in terms of the
fundamental solution Φ.
(ii) For each x ∈ Ω, write, in terms of Φ, the data for the Dirichlet problem solved
by g(x, y), y ∈ Ω.
(iii) Suppose that Ω is the ball of radius 2 centred at zero. Explain why the function
g(0, y), y ∈ ∂Ω, is positive valued for d = 2 and negative valued for d = 3.
(iv) Given that for the case when Ω = BR(0) ⊂ R3 is the ball of radius R centred
at zero, one has
∂G
∂ny
(x, y) = − R
2 − |x|2
4piR|y − x|3 , x ∈ Ω, y ∈ ∂Ω,
use the formula for the solution to the Dirichlet problem on BR(0) to verify the
validity of the mean-value theorem for the value at zero of a harmonic function
in BR(0) in terms of its values on ∂BR(0).
Page 4 of 6 MA30059/MA40059/MA50059
3. Consider the square Ω := {x = (x1, x2) ∈ R2 : 0 < x1 < 1/2, 0 < x2 < 1/2} and a value
T > 0.
For continuous functions f = f(x, t), (x, t) ∈ Ω× (0, T ], ψ = ψ(x, t), (x, t) ∈ ∂Ω× [0, T ],
and ϕ = ϕ(x), x ∈ Ω, consider the initial boundary-value problem for u = u(x, t) :
ut(x, t)−∆xu(x, t) = f(x, t), x ∈ Ω, t ∈ (0, T ], (B)
u(x, 0) = ϕ(x), x ∈ Ω, (C)
u(x, t) = ψ(x, t), x ∈ ∂Ω, t ∈ [0, T ]. (D)
(a) [4]
(i) Suppose that u1, u2 are solutions to the problem (B)–(D). For each T > 0,
show that u1 = u2 on the parabolic boundary of Ω× (0, T ].
(ii) Hence, argue that the problem (B)–(D) cannot have more than one solution.
(b) Consider the initial boundary-value problem (B)–(D) with T = ∞, replacing the
intervals (0, T ], [0, T ] with (0,∞), [0,∞), respectively. [4]
(i) Suppose that the heat-source density f(x, t) and the boundary data ψ(x, t) are
zero for all (x, t). What can be said about the absolute value of the solution
u = u(x, t) to (B)–(D) as t→∞?
(ii) Suppose that f and ψ are independent of time t. What can you say about the
behaviour of the solution u = u(x, t) to (B)–(D) as t→∞?
(c) [12]
(i) Determine all solutions u = u(x, t) of the equation
ut = ∆xu, x = (x1, x2) ∈ Ω, t ∈ (0, T ],
that have the form
u(x, t) = X1(x1)X2(x2)T(t), x ∈ Ω, t ∈ [0, T ], (E)
where X1, X2 have two continuous derivatives and T has one continuous
derivative.
(ii) For u having the form (E) and satisfying the condition u(x, t) = 0, x ∈ ∂Ω,
t ≥ 0, what functions X1, X2 are possible among solutions found in (i)?
(iii) Suppose that f and ψ are zero functions, and ϕ(x1, x2) = 3 sin(4pix1) sin(2pix2),
(x1, x2) ∈ Ω. Write the general solution to (B), subject to the condition (C),
in the form of a series of functions of the form (E), that is
u(x, t) =
∞∑
j=1
X(j)1 (x1)X
(j)
2 (x2)T
(j)(t), x ∈ Ω, t ∈ [0, T ], (F)
where each of the functions T(j) is determined up to the value T(j)(0).
(iv) Using (C), find the values T(j)(0), j = 1, 2, . . . , and hence determine the
solution to (B)–(D) subject to the data in (iii).
Page 5 of 6 MA30059/MA40059/MA50059
4. (a) Consider the integral functional
I(u) := −
∫ 1
−1
log(1 + u2x)dx (G)
on the setA = {u ∈ C1pw([−1, 1]) : u(−1) = 0, u(1) = 1}, where C1pw([−1, 1]) denotes
the set of functions that are continuous and have piecewise continuous derivative on
the interval [−1, 1]. [10]
(i) Explain why all local minimisers u of I in A must be such that ψ′u,φ(0) = 0 for
all φ ∈ C1([−1, 1]) with φ(−1) = φ(1) = 0, where ψu,φ(t) := I(u+ tφ), t ∈ R.
(ii) Show that the necessary condition from (i) implies that all local minimisers u
of I in A satisfy the Euler-Lagrange equation(
fv(ux)
)
x
= 0,
where f(v) := log(1 + v2), v ∈ R.
(iii) Write down the Euler-Lagrange equation for (G) explicitly.
(iv) Determine the solution to the Euler-Lagrange equation for (G) that satisfies
the boundary values for functions in A.
MA30059 Mathematical Methods
This is an open book exam. You may refer to your own course and revision notes and look up
information in offline or online resources, for example textbooks or online journals.
This exam is designed to take approximately 3 hours (including up to 1 hour for upload) to complete.
Which questions should be answered: Answer THREE of the four questions. Only the BEST three
answers will contribute to the assessment.
Filenames: If you are required to upload a file as part of your exam attempt, to maintain anonymity
please use the following naming convention for your file:
CandidateNumberUnitCodeQuestionNumber.pdf (e.g. 01234AR10001Q2a.pdf).
If the exam only requires one file to be submitted, you do not need to include the question
number(s).
Additional materials needed to complete the assessment: A calculator is not necessary for this
assessment.
Further instructions: In marking, emphasis will be placed upon clear mathematical argument and
precision. Do NOT submit your rough work.
Academic Integrity for Remote Exams
When you registered as a student you agreed to abide by the University's regulations and rules, and
agreed that you would access and read your programme handbook. These documents contain
references to, and penalties for, unfair practices such as collusion, plagiarism, fabrication or
falsification. The University's Quality Assurance Code of Practice, QA53 Examination and Assessment
Offences , sets
out the consequences of committing an offence and the penalties that might be applied.
By submitting your exam as instructed, you confirm that:
1. You have not impersonated, or allowed yourself to be impersonated by, any person for the
purposes of this assessment.
2. This assessment is your original work and no part of it has been copied from any other
source except where due acknowledgement is made. The University may submit your work
through a plagiarism detection service.
3. You have not previously submitted this work for any other unit/course.
4. You give permission for your assessment response to be reproduced, communicated,
compared and archived for plagiarism detection, benchmarking or educational purposes.
5. You understand that plagiarism is the presentation of the work, idea or creation of another
person or organisation as though it is your own. It is a form of cheating and is a very serious
academic offence that may lead to disciplinary action.
6. You understand that this assessment is undertaken without invigilation, and that you have
not communicated with and will not communicate with anyone concerning this assessment
before the deadline for submission unless it is expressly permitted by the assessment
instructions.
7. No part of this assessment has been produced for, or communicated to, you by any other
person, unless it is expressly permitted by the assessment instructions.
If you have any questions about the exam you should contact the exams helpline.
Information and contact details can be found on our help and advice webpage
1. (a) Consider the ball [8]
B1(0) = {(x1, x2) ∈ R2 : x21 + x22 < 1},
and the functions
u1(x1, x2) = x1x2, (x1, x2) ∈ B1(0), u2(x1, x2) = x21x2−x1x22, (x1, x2) ∈ B1(0).
(i) Explain why the function u1 is harmonic on B1(0).
(ii) Is the function u2 harmonic in B1(0)? Justify your answer.
(iii) Determine the maximum of |u1| on the circle {(x1, x2) ∈ R2 : x21 + x22 = 1}.
(iv) Show, without invoking the maximum principle or the weak maximum
principle, that the principle of maximum of modulus holds for u1 on B1(0).
(b) The fundamental solution for the Laplace operator in two dimensions is [6]
Φ(x) =
1
2pi
log
1
|x| , x ∈ R
2 \ {0}.
(i) Show that Φ is harmonic in the annulus
A = {(x1, x2) ∈ R2 : 1/2 < |x| < 2}.
(ii) Like Φ, the constant function f(x) = −(log 2)/(2pi), x ∈ A, is harmonic in A
and has the same boundary values on {(x1, x2) ∈ R2 : |x| = 2}. Explain why
this does not contradict the uniqueness theorem for the Dirichlet problem.
(iii) Using the maximum principle for the Laplace operator, determine the
maximum and the minimum of Φ− f on A.
(c) Consider the boundary-value problem [6]
∆u = 0 on A,
u|S1/2(0) = 0, u|S2(0) = 1,
(A)
where SR(0), R > 0, denotes the circle of radius R centred at zero. Using separation
of variables, determine the solution u to (A). You may wish to use the expression
for the Laplace operator in polar coordinates:
∂2
∂r2
+
1
r
∂
∂r
+
1
r2
∂2
∂φ2
,
where r is the distance from the origin, φ is the polar angle.
Page 3 of 6 MA30059/MA40059/MA50059
2. Consider a bounded region Ω ⊂ Rd, d = 2 or d = 3, with C1 boundary.
(a) Suppose that u ∈ C2(Ω) satisfies the Laplace’s equation ∆u = 0 in Ω. [6]
(i) Using the representation theorem for C2 functions in terms of potentials, argue
that the values of u in Ω can be recovered from the values of u on the boundary
∂Ω and the values of the normal derivative ∂u/∂n on ∂Ω.
(ii) Using the uniqueness theorem for a suitable Dirichlet problem, explain why
just the values of u on ∂Ω fully determine u everywhere in Ω.
(b) Consider the Green’s function G = G(x, y) for the Laplace operator on Ω and write
g(x, y) = G(x, y)− Φ(x− y), x ∈ Ω, y ∈ Ω, x 6= y,
where
Φ(x) =
1
2pi
log
1
|x| , x ∈ R
2 \ {0},
1
4pi|x| , x ∈ R
3 \ {0},
is the fundamental solution for the Laplace operator. [14]
(i) For a given y ∈ ∂Ω, express the function g(x, y), x ∈ Ω, in terms of the
fundamental solution Φ.
(ii) For each x ∈ Ω, write, in terms of Φ, the data for the Dirichlet problem solved
by g(x, y), y ∈ Ω.
(iii) Suppose that Ω is the ball of radius 2 centred at zero. Explain why the function
g(0, y), y ∈ ∂Ω, is positive valued for d = 2 and negative valued for d = 3.
(iv) Given that for the case when Ω = BR(0) ⊂ R3 is the ball of radius R centred
at zero, one has
∂G
∂ny
(x, y) = − R
2 − |x|2
4piR|y − x|3 , x ∈ Ω, y ∈ ∂Ω,
use the formula for the solution to the Dirichlet problem on BR(0) to verify the
validity of the mean-value theorem for the value at zero of a harmonic function
in BR(0) in terms of its values on ∂BR(0).
Page 4 of 6 MA30059/MA40059/MA50059
3. Consider the square Ω := {x = (x1, x2) ∈ R2 : 0 < x1 < 1/2, 0 < x2 < 1/2} and a value
T > 0.
For continuous functions f = f(x, t), (x, t) ∈ Ω× (0, T ], ψ = ψ(x, t), (x, t) ∈ ∂Ω× [0, T ],
and ϕ = ϕ(x), x ∈ Ω, consider the initial boundary-value problem for u = u(x, t) :
ut(x, t)−∆xu(x, t) = f(x, t), x ∈ Ω, t ∈ (0, T ], (B)
u(x, 0) = ϕ(x), x ∈ Ω, (C)
u(x, t) = ψ(x, t), x ∈ ∂Ω, t ∈ [0, T ]. (D)
(a) [4]
(i) Suppose that u1, u2 are solutions to the problem (B)–(D). For each T > 0,
show that u1 = u2 on the parabolic boundary of Ω× (0, T ].
(ii) Hence, argue that the problem (B)–(D) cannot have more than one solution.
(b) Consider the initial boundary-value problem (B)–(D) with T = ∞, replacing the
intervals (0, T ], [0, T ] with (0,∞), [0,∞), respectively. [4]
(i) Suppose that the heat-source density f(x, t) and the boundary data ψ(x, t) are
zero for all (x, t). What can be said about the absolute value of the solution
u = u(x, t) to (B)–(D) as t→∞?
(ii) Suppose that f and ψ are independent of time t. What can you say about the
behaviour of the solution u = u(x, t) to (B)–(D) as t→∞?
(c) [12]
(i) Determine all solutions u = u(x, t) of the equation
ut = ∆xu, x = (x1, x2) ∈ Ω, t ∈ (0, T ],
that have the form
u(x, t) = X1(x1)X2(x2)T(t), x ∈ Ω, t ∈ [0, T ], (E)
where X1, X2 have two continuous derivatives and T has one continuous
derivative.
(ii) For u having the form (E) and satisfying the condition u(x, t) = 0, x ∈ ∂Ω,
t ≥ 0, what functions X1, X2 are possible among solutions found in (i)?
(iii) Suppose that f and ψ are zero functions, and ϕ(x1, x2) = 3 sin(4pix1) sin(2pix2),
(x1, x2) ∈ Ω. Write the general solution to (B), subject to the condition (C),
in the form of a series of functions of the form (E), that is
u(x, t) =
∞∑
j=1
X(j)1 (x1)X
(j)
2 (x2)T
(j)(t), x ∈ Ω, t ∈ [0, T ], (F)
where each of the functions T(j) is determined up to the value T(j)(0).
(iv) Using (C), find the values T(j)(0), j = 1, 2, . . . , and hence determine the
solution to (B)–(D) subject to the data in (iii).
Page 5 of 6 MA30059/MA40059/MA50059
4. (a) Consider the integral functional
I(u) := −
∫ 1
−1
log(1 + u2x)dx (G)
on the setA = {u ∈ C1pw([−1, 1]) : u(−1) = 0, u(1) = 1}, where C1pw([−1, 1]) denotes
the set of functions that are continuous and have piecewise continuous derivative on
the interval [−1, 1]. [10]
(i) Explain why all local minimisers u of I in A must be such that ψ′u,φ(0) = 0 for
all φ ∈ C1([−1, 1]) with φ(−1) = φ(1) = 0, where ψu,φ(t) := I(u+ tφ), t ∈ R.
(ii) Show that the necessary condition from (i) implies that all local minimisers u
of I in A satisfy the Euler-Lagrange equation(
fv(ux)
)
x
= 0,
where f(v) := log(1 + v2), v ∈ R.
(iii) Write down the Euler-Lagrange equation for (G) explicitly.
(iv) Determine the solution to the Euler-Lagrange equation for (G) that satisfies
the boundary values for functions in A.
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