IT1005

1
IT1005

• Lab session on week 11 (7th meeting)
• 1 or 2 more weeks to go

2
Lab 7 Quick Check

• Have you all received my reply for lab 7?
• Of course not. I have not finished grading your
  submissions yet gt.lt
• I have important research paper deadline gt.lt
• 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!
• 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

3
L7.Q1 4 tanks CSTRs

• Q1. A. Transform the non standard set of Linear
  Equations into standard format
• -(QkV) CA1 0 CA2 0 CA3
  0 CA4 -QCA0
• Q CA1 -(QkV) CA2 0 CA3
  0 CA4 0
• 0 CA1 Q CA2 -(QkV) CA3
  0 CA4 0
• 0 CA1 0 CA2 Q CA3
  -(QkV) CA4 0
• Q1. B. Convert the standard format into matrix
  format. Another straightforward task
• -(QkV) 0 0 0 CA1 -QCA0
• Q -(QkV) 0 0 CA2 0
• 0 Q -(QkV) 0 CA3 0
• 0 0 Q -(QkV) CA4 0
• Q1. C. You just need to do
• V 1 Q 0.01 CA0 10 k 0.01
• A -(QkV) 0 0 0 Q -(QkV) 0 0 0 Q -(QkV)
  0 0 0 Q -(QkV)
• b -QCA0 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!

4
L7.Q2 n tanks CSTRs

• 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(-(QkV),1,N))
  diag(repmat(Q,1,N-1),-1) This is a bit crazy
  
• b zeros(N,1) b(1) -QCA0
• 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 N0.5 0 CA0) Fix y axis so that it
  is consistent across 4 plots (same CA0!)

5
L7.Q2 Good Plot
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
6
L7.Q3 and Q4

• 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 x2 y2 - 2x - 1
• f2 x2 - y2 - 1
• a b solve(f1,f2) or gtgt 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) 3x(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(_at_lab07d, 1 1 1) ? x0.5991, y2.3959,
  z2.0050
• fsolve(_at_lab07d, 5 -1 1) ? x 5.1004, y
  -2.6442, z 2.5438

7
Application 5 IVP (revisited)

• 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.

8
Application 5 IVP (revisited)

• 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
• 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
• And two more for Term Assignment (Q2 and Q3) ?

9
Application 5 IVP (revisited)

• 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(_at_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!)

10
Term Assignment Admin

• 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 minutesTo avoid that
  problem, submit early, e.g. Friday, 4 April 08,
  night.

11
Term Assignment Q1

• 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 2n-1
  to k 1n-1,but it is ok if you stick with the
  old one!

12
Term Assignment - Q2

• 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

13
Term Assignment Q3

• 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_equati
  on (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!

14
Free and Easy Time

• 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?