EG25H4代做、代写Python/c++程序

日期：2024-04-08 07:51

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]