Freedom’s Battle: The Origins of Humanitarian Intervention

Write a summary for each book and each article.

the book: each chapter must be in different part(each chapter must be under its own title,DO NOT MIX CHAPTER TOGETHER. Also, you must quote form each chapter something from that chapter

The articles: each article under its own name.

At the end: write 3 questions about the reading that would like to discus more in class.

The first reading: • “Freedom’s Battle: The Origins of Humanitarian Intervention (Hardcover or paperback),” Gary J. Bass, Vintage Press 2009, ISBN 0307279871

this from page 1 to 156.

Here is a link for the book:
https://books.google.com/books?id=70g_Vd8AmO8C&printsec=frontcover&hl=ar#v=onepage&q&f=false

The artcles:

1-
https://www.democracynow.org/2019/1/1/four_days_in_occupied_western_sahara

2-
https://www.theguardian.com/us-news/2019/jan/07/trump-robert-mueller-adam-schiff-investigations?CMP=share_btn_link

2. MATLAB code

Your report should not be longer than 4 pages in total. Please use at least a font size of 11pt and 2cm margins on all sides. All MATLAB code should be attached to the report as an appendix. The appendix does not count towards the 4 page limit. All figures in the report must have captions, properly labelled axes, legends where necessary and must be described and analysed in the text.
Problem. In this exercise, we will study a combination of the transport and heat equation called the advection-diffusion equation ˙ u(x,t) + v0ux(x,t) = νuxx(x,t). (1) For all numerical examples, please consider 0 ≤ x ≤ 2π, a final time of tend = 2.0 and a value of v0 = 1.5.
Task 1. (25 marks) 1. (3×5 = 15 marks) Derive the initial value problem in spectral space that arises when solving (1) using the pseudo-spectral method. Proceed along the following steps: a) How does the equation for the residual R(u(x,t)) look like that comes out of plugging the truncated Fourier series
u(x,t) =
1 N
N−1 X k=0
ˆ uk(t)eikx
into (1)? b) What are the N equations that result from enforcing R(u(xn,t)) = 0 at N equidistant mesh points xn = 2πn N , n = 0,…,N −1? (2)
c) How can those N equations be written compactly using the spectral differentiation matrix D? You do not need to comment on the “trick” we used the rewrite the second part of D and may simply ignore the issue. However, in your MATLAB code, be sure to use the matrix D that is defined in the provided examples.
2. (5 marks MATLAB plus 5 marks explanation) Write a MATLAB script that solves the initial value problem coming out of Task 1.1 for ν = 0.1 using ode45 for u0(x) = cos(3x) and then plots the solution at the end of the simulation in both physical and spectral space using N = 64. How does the solution change for an initial value u0(x) = cos(7x)? Describe how the solution behaves. You do not need to explain the results here, this will be done in Task 2.5.
1
Task 2. (30 marks)
1. (5 marks) Apply the continuous Fourier transform in x to Equation (1). Write down the resulting equation in spectral space. Make sure to clearly explain what identities from Fourier analysis you have used to obtain your result.
2. (5 marks) Write down the general solution of the differential equation arising from Task 1.1.
3. (5 marks) Find a text book that tells you what the Fourier transform of cos(3x) is. Write down the result and cite your source.
4. (5 marks) Compare the numerical solutions from Task 1.2 to your mathematically derived solutions.
5. (10 marks) Based on your results from 2.4, explain your observations in Task 1.2.
Task 3. (30 marks)
1. (5 marks) Write a MATLAB function that solves Equation (1) using the pseudo-spectral method and explicit Euler and returns the approximate solution at the end of the simulation in physical space. The function should have input variables N (number of Fourier modes), tend (final time), v0 (transport velocity), ν (viscosity) and u0 (initial value in physical space) . You can copy the code to generate the needed spectral differentiation matrix from the examples provided in Minerva.
2. (5 marks) Write a MATLAB function that solves Equation (1) using centred finite differences for both ux and uxx and explicit Euler and returns the approximate solution at the end of the simulation. The function should have input variables Nx (number of finite difference points), ∆x (mesh spacing), tend (final time), v0 (transport velocity), ν (viscosity) and u0 (initial value in physical space). You can copy the code to generate the needed finite difference matrices from the examples provided in Minerva.
3. (5 marks) Write a MATLAB script that calls your functions from 3.1 and 3.2 for initial values
u(1) 0 (x) = exp−(x−π)2 0.52 
and
u(2) 0 (x) = heaviside(x−
2 3
π)−heaviside(x−
4 3
π).
Have the script generate four figures, each containing the solution from the pseudo-spectral as well as from the finite difference method. The four figures should show the resulting solutions (in physical space) for u(1) 0 and u(2) 0 with ν = 1.0 and ν = 0.005. Use N = Nx = 64 modes/nodes in all cases. For explicit Euler, use 4×N = 256 time steps. 4. (15 marks) Describe the results in Task 3.3.
Task 4. (15 marks)
1. Explain your observations in Task 3.3. Where necessary, you can generate new figures to help your explanations. For example, it could be instructive to replace the centred finite difference approximation of ux with an upwind approximation and see how the results change. Make sure that all figures clearly explain what they show and are properly discussed in the text.