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IT1005

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IT1005

Lab session on week 11 (7th meeting)

1 or 2 more weeks to go…

- Have you all received my reply for lab 7?
- Of course not. I have not finished grading your submissions yet >.<
- I have important research paper deadline >.<
- http://www.cs.mu.oz.au/cp2008/
- Abstract due: 4 April 2008, Paper due: 8 April 2008.

- My future reply should contains:
- Remarks on your M files (look for SH7 tags again)
- For CSTR_input.m, CSTR_matrix.m, and CSTR_soln.m, etc

- Remarks on your Microsoft Word file
- For the other stuffs

- Your marks is stored in the file “Marks.txt” inside the returned zip file!

- Remarks on your M files (look for SH7 tags again)
- I do not think I will be strict mode , the answers are standard for L7!
- Note that my marking scheme is slightly different from the standard one,as I put emphasize on coding style (indentation, white spaces), efficiency,things like proper plotting, etc…

Q1. A. Transform the non standard set of Linear Equations into standard format:

-(Q+k*V)*CA1 + 0*CA2 + 0*CA3 + 0*CA4 = -Q*CA0

Q*CA1 -(Q+k*V)*CA2 + 0*CA3 + 0*CA4 = 0

0*CA1 + Q*CA2 -(Q+k*V)*CA3 + 0*CA4 = 0

0*CA1 + 0*CA2 + Q*CA3 -(Q+k*V)*CA4 = 0

Q1. B. Convert the standard format into matrix format. Another straightforward task:

[-(Q+k*V)000] *[CA1] = [-Q*CA0]

[Q-(Q+k*V)00][CA2] [0]

[0Q-(Q+k*V)0][CA3] [0]

[00Q-(Q+k*V)][CA4] [0]

Q1. C. You just need to do:

V = 1; Q = 0.01; CA0 = 10; k = 0.01;

A = [-(Q+k*V) 0 0 0; Q -(Q+k*V) 0 0; 0 Q -(Q+k*V) 0; 0 0 Q -(Q+k*V)];

b = [-Q*CA0; 0; 0; 0];

x = A\b % you should get x = [5; 2.5; 1.25; 0.625], x = inv(A) * b is NOT encouraged!

% Using fsolve for this is also not encouraged!

Q2. A. CSTR_input.m

Answer is very generic, similar to read_input.m in L6.Q2 (Car Simulation)

Q2. B. CSTR_matrix.m

% This is my geek version, do NOT USE THIS VERSION (too confusing for novice)!

function [A b] = CSTR_matrix(Q, CA0, V, k, N)

A = diag(repmat(-(Q+k*V),1,N)) + diag(repmat(Q,1,N-1),-1); % This is a bit crazy :$

b = zeros(N,1); b(1) = -Q*CA0;

Q2. C. CSTR_soln.m

clear; clc; clf; % New trick, but important ! Clear everything before starting our program!

[Q CA0 V k N] = CSTR_input();

[A b] = CSTR_matrix(Q, CA0, V, k, N);

plot(A\b,'o'); % I prefer not to connect the plot with line, but it is ok if you do so.

title('CA of each tank'); xlabel('Tank no k'); ylabel('CA_k'); % Good for CSTR_plot.m

axis([0.5 N+0.5 0 CA0]); % Fix y axis so that it is consistent across 4 plots (same CA0!)

Remember:

Plot A against B means thatA is the Y axis, B is the X axis!

Should just stop here (n = 4) for case 1

- Tips for guessing logically:
- Plot the functions with some range, see the region of zero intercepts…
- Guess from the easiest equation!e.g. x + y + z = 5
- log(z)… hm… z should be > 0

Q3. Type these at command window:

syms x y; % no need to say syms f1 f2, the next two lines will create f1 and f2 anyway

f1 = x^2 * y^2 - 2*x - 1;

f2 = x^2 - y^2 - 1;

[a b] = solve(f1,f2); % or >> s = solve(f1,f2); a = s.x; b = s.y;

eval(a), eval(b) % convert the symbolic values to numeric values, these are the roots

Q4. Create this function

function F = lab07d(x)

F(1) = sin(x(1)) + x(2)^2 + log(x(3)) - 7;

F(2) = 3*x(1) + 2^(x(2)) + 1 - x(3)^3;

F(3) = x(1) + x(2) + x(3) - 5;

At command window (wild guesses will likely give you many imaginary numbers):

fsolve(@lab07d, [1 1 1]) x=0.5991, y=2.3959, z=2.0050

fsolve(@lab07d, [5 -1 1]) x = 5.1004, y = -2.6442, z= 2.5438

- Equation
- A statement showing the equality of two expressions usually separated by left and right signs and joined by an equals sign.

- Differential Equation
- A description of how something continuously changes over time.

- Ordinary Differential Equation
- A relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable.

- Initial Value Problem
- An ODE together with specified value, called the initial condition, of the unknown function at a given point in the domain of the solution.

- We have seen several examples of IVP throughout IT1005
- Spidey Fall example (Lecture 2, Lecture ODE 1)
- How velocity v change over time dt? (dv/dt)
- Depends on gravity minus drag!

- How displacement s change over time dt? (ds/dt)
- Depends on the velocity at that time.

- The Initial Values for v and s at time 0 v(0) = 0; s(0) = 0;

- How velocity v change over time dt? (dv/dt)
- Car accelerates up an incline example (Lab 6, Q2)
- How velocity v change over time dt? (dv/dt)
- Depends on engine force on some kg car minus friction and gravity factor!

- How displacement s change over time dt? (ds/dt)
- Depends on the velocity at that time.

- The Initial Values for v and s at time 0 v(0) = 0; s(0) = 0;

- How velocity v change over time dt? (dv/dt)
- And two more for Term Assignment (Q2 and Q3)

- Spidey Fall example (Lecture 2, Lecture ODE 1)

- Solving IVP (either 1 ODE or set of coupled ODEs):
- Hard way/Traditional way/Euler method:
- Time is chopped into delta_time, then starting from the initial values for each variable, simulate its changes over time using the specified differential equation!
- What Colin has shown in Lecture 2 for Spidey Fall is a kind of Euler method.
- What you have written for Lab 6 Question 2 (Car Simulation) is also Euler method.

- Matlab IVP solvers: mostly numerical solutions for ODE.
- Create a derivative function to tell Matlab about how a variable change over time!
function dydt = bla(t,y) % always have at least these two arguments

% explain to Matlab how to derive dy/dt! Can be for coupled ODEs!

dydt = dydt'; % always return a column vector!

- Call one of the ODE solver with certain time span and initial values
[t, y] = ode45(@bla, [tStart tEnd], IVs); % IVs is a column vector for coupled ODEs!

plot(t,y(:,1)); % we can immediately plot the results (also in column vector!)

- Create a derivative function to tell Matlab about how a variable change over time!

- Hard way/Traditional way/Euler method:

- This is 30% of your final IT1005 grade... Be very serious with it.
- No plagiarism please!
- Even though you can ‘cross check’ your answers with your friends (we cannotprevent that), you must give a very strong ‘individual flavor’ in your answers!
- The grader will likely grade number per number, so he will be very curiousif he see similar answers across many students. Do not compromise your 30%!

- Who will grade our term assignment?
- I may not be the one doing the grading! Perhaps all the full time staff… dunno yet.

- Submit your zip file (containing all files that you use to answer the questions)to IVLE “Term Assignment” folder! NOT to my Gmail!
- Your zip file name should be: yyy-uxxxxxx.zip, NOT according to my style!
- Strict deadline, Saturday, 5 April 08, 5pmThat IVLE folder will auto close by Saturday, 5 April 08, 5.01pmBe careful with NETWORK CONGESTION around these final minutes…To avoid that problem, submit early, e.g. Friday, 4 April 08, night.

- No plagiarism please!

- Question 1: Trapezium rule for finding integration
- A. Naïve one. Explain your results!
- B. More accurate one. Explain your results!
- References:
- help quad
- http://en.wikipedia.org/wiki/Numerical_integration
- http://en.wikipedia.org/wiki/Trapezium_rule

- Revision(s) to the question:
- Symbol ‘a’ changed to ‘c’ inside function f(t)!
- In Q1.B, the rows in column ‘c’ are [0.001 0.5 10.0 100.0] not [0.01 0.5 1.0 10.0]!
- In Q1.B, the range of k is changed from k = 2:n-1 to k = 1:n-1,but it is ok if you stick with the old one!

- Question 2: Zebra Population versus Lion Population
- A. IVP, coupled, non-linear ODEs
- B. Explain what you see in the graph of part A above.
- C. Steady state issue.
- D. IVP again, but change the IVs according to part C above. Comment!
- E. IVP with different IVs, and different plotting method. Comment!
- References:
- Google the term ‘predator prey’ as mentioned in the question.
- help odeXX (depends on the chosen solver)
- http://en.wikipedia.org/wiki/Steady_state

- Revision(s) to the question:
- No change so far…

- Question 3: Similar to Q2, Predator-Prey: n = 4 species
- A. IVP again, 4 coupled, non-linear ODEs. Dr Saif said that we must use ode15s! (See ODE 3 & 4 lecture note)
- B. IVP, same IVs, 1.000 years, 3D plot x1-x2-x3 (x4 is not compulsory),and explain.
- C. Explain plot in B as best as you can.
- References:
- http://en.wikipedia.org/wiki/Lotka-Volterra_equation (mentioned in the question).
- Google ‘Matlab 3D plot’
- help odeXX (depends on the chosen solver)

- Revision(s) to the question:
- The ODE equations are updated! Read the newest one!
- The coefficient r(3) is changed from 1.53 to 1.43!

- Now, you are free to explore Matlab to:
- Do your Term Assignment (all q1, q2, and q3)
- You should NOT use me as an oracle, e.g.
- I cannot find the bug in my program, can you help me?
- Are my 2-D/3-D plots correct?
- Are my …. bla bla … correct?