EG25H4 – CA2 – Solution of PDEs
Students are expected to independently prepare solutions to the assigned
problems.
Submissions, accompanied by a plagiarism coversheet, should be uploaded to
MyAberdeen by 5pm (BST) on Friday, the 19th of April 2024. Please note that
unauthorised submissions received after the deadline will incur a late penalty as
per the University’s Policy on the Penalty for Unauthorised Late Submission of
Coursework. Solutions submitted without a plagiarism coversheet will not be
marked and will be subject to a late penalty until a plagiarism coversheet is
submitted.
Your submission should be compiled into a single ZIP file containing the
following:
1. Your plagiarism coversheet
2. Octave scripts and functions
3. A MS Word file with the figures you are asked to produce and the
appropriate analysis
Each student’s mark (out of 22), will be directly translated to the corresponding
Common Grading Scale alphanumeric. The marks for each question are shown
below. This assessment constitutes 50% of your overall course mark for EG25H4.
The presentation of your solution is crucial: the solution must be clearly set out
and explained in order to achieve a high mark. Marks will be deducted if the
working is untidy or unclear. Simply obtaining the “correct answer” is not
sufficient for achieving an excellent mark. The clarity and quality of your
explanation are equally important.
Q1. Consider a wall composed of two layers of bricks with a layer of insulation sandwiched in
between. The temperature variation, T(x, t), at a given position x and time t in a one-dimensional
cross-section through the wall can be modelled using a specific partial differential equation.
This equation will be investigated using the provided Octave functions and script.
The thermal conductivities of the different materials, which vary, are accounted for using a
spatially dependent coefficient for the temperature gradient, ∂x/∂T.
Your task is to analyse this model and interpret the results in the context of the physical system.
function Temp0 = icfun(x)
% Icfun – Initial conditions
Temp0 = 273; % unit: K
end
function [c, f, s] = pdfun(x, t, T, dTdx)
% pdfun – Define partial differential equation
c = 1;
f = (2 – 0.8*(heaviside(x-2) – heaviside(x - 3))) * dTdx;
s = 0;
end
function [pl, ql, pr, qr] = bcfun(xl, Tl, xr, Tr, t)
% bcfun – Boundary conditions
pl = 3 ;
ql = 1 ;
pr = 0.1*(Tr -273) ;
qr = 1 ;
end
% Solve the partial differential equation using pde1dm
x = linspace(0, 5, 101);
t = 0:0.1:8;
sol = pde1dm(0, @pdfun, @icfun, @bcfun, x, t);
Temp = sol(:,:,1);
%Plot temperature profiles at different times
plot(x, Temp(1,:), x, Temp(21,:), x, Temp(41,:), x,
Temp(61,:))
xlabel('Length, x');
ylabel('Temperature, T');
(a) Identify the ranges of values of x that correspond to the brick and the insulation layers.
[2 marks]
(b) What is the partial differential equation (PDE) that is being solved in this context?
[3 marks]
(c) What is the boundary condition at x=0? Please simplify your answer as much as
possible.
[2 marks]
(d) What is the boundary condition at x=5? Please simplify your answer as much as
possible.
[3 marks]
(e) At what specific times t are the temperature profiles plotted by the script?
[2 marks]
Q2. Propose a PDE of your choice, distinct from the examples provided in the lecture and
tutorial notes. This PDE should have appropriate initial and mixed Dirichlet / Neumann
boundary conditions. Solve this PDE using the PDE1DM solver. Create a figure that illustrates
the solution with respect to time t and position x.
[6 marks]
Q3. Suggest a PDE of your choice, different from the examples provided in the lecture notes,
that cannot be solved using the PDE1DM solver. Provide a detailed explanation as to why the
solver cannot be used in this case.
[4 marks]
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