Welcome!

You have reached the course website for

18.085 Spring 2009

Class stuff:

MWF 12-1pm
4-370

Lecturer: Hans Christianson

Office: 2-304 Evans
Office hours: MF2-3

Teaching Assistants:



NEW

18.085 from the Fall semester is now completely online at OCW-18.085



[announcements] [homework assignments] [quizzes and solutions] [links] [resources and old exams]

Course Topics

  • Applied Linear Algebra
  • Applied Differential Equations
  • Fourier Methods
  • Algorithms
  • Here is a tentative lecture outline (updated regularly)



  • Goals for the Course: See applications of calculus, ODE, linear algebra, and discrete methods without going into too much proof.
  • Textbook: Computational Science and Engineering (Wellesley-Cambridge, 2007).
  • Grades: Homework 70%, 3 evening quizzes 30%, no final.
  • Homework: Due Wednesdays. Please use MATLAB notation to describe algorithms. Use of MATLAB for tedious calculations is encouraged, however you need to know how to do the basic algorithms taught in the course by hand (at least for small matrices) for the quizzes.
  • Evening Quizzes: March 2, April 6, May 4. 7:30-9pm, Walker Memorial 50-340. Quizzes are always open book and open notes. No calculator or computer. You will need to know how to do some basic matrix operations by hand.

    Announcements


    5-Mar-09: Quizzes will hopefully be returned on Monday. Homework #5 is now posted, and is a shorter assignment than usual since I posted it later.
    17-Mar-09: Quiz solutions posted (written by me, so the graders' scoring rubric is not included).
    20-Mar-09: Homework posted due after spring break. Don't forget there is a quiz 6 April, covering through 3.1. Happy Spring Break!
    29-Mar-09: Proffessor Strang received this
    job announcement if any of you are interested.
    1-Apr-09: 085 from the Fall semester is now completely online at OCW-18.085
    8-Apr-09: Quiz 2 solutions now online.
    24-Apr-09: Don't forget Quiz 3 (covers 3.2-3.6, 4.1, 4.3) is on 4 May!

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    Homework


    #1 Due Feb 11 (not graded closely). 1.1: 1, 5, 12; 1.2: 1, 7, 8 (work it out yourself), 14.
    Solutions (Thanks to Chan-Hoo Jeon)
    #2 Due Feb 18. 1.3: 2,8,11; 1.4: 1,3,8; 1.5: 2, 10, 17, 23.
    Solutions (Thanks to Chan-Hoo Jeon)
    #3 Due Feb 25. 1.6: 2,8,12,23,27; 1.7: 7,8,15,18; 2.1: 4,5,7.
    Solutions (Thanks to Travis Wolf)
    Solutions (Thanks to Amanda Gaudreau)
    Solutions (Thanks to Chan-Hoo Jeon)
    #4 Due Mar 4. 2.2: 4,6,7,8,11.
    Solutions (Thanks to Chan-Hoo Jeon)
    #5 Due Mar 11. 2.3: 1, 4, 7, 8; 2.4: 1, 5, 9, 18.
    Solutions (Thanks to Amanda Gaudreau)
    #6 Due Mar 18. 2.7: 1, 2, 5, 7, 8; 2.8: 1, 3(typo, should say \sigma^2 = \sum(n-\bar{n})^2 p_n), 10
    Solutions (Thanks to Chan-Hoo Jeon)
    #7 Due Apr 1. 3.1: 2, 3, 8, 11, 14, 15, 17.
    Solution 1 (ungraded) (Thanks to Chan-Hoo Jeon)
    Solution 2, p.1 (ungraded) (Thanks to Amanda Gaudreau)
    Solution 2, p.2 (ungraded) (Thanks to Amanda Gaudreau)
    #8 Due Apr 8. 3.2: 5, 12; 3.3: 2, 6, 8, 9.
    Solutions (Thanks to Yuta Kuboyama)
    #9 Due Apr 15. 3.3: 12, 18; 3.4: 3, 4, 17, 18; 3.5: 1, 3; and this extra problem(updated 4/10/09)
    Solutions (Thanks to Amanda Gaudreau)
    #10 Due Apr 22. 3.6: 3, 4, 5, 8(a), 13, 14; and this extra problem
    Solutions (Thanks to Frank Fan)
    #11 Due Apr 29. 4.1: 1(a,c,d), 2, 6, 7, 8, 10, 14, 18; 4.3: 12, 13, 14; and this extra problem
    Solutions (not graded yet) (Thanks to Amanda Gaudreau)
    Solutions (not graded yet) (Thanks to Zhenya Gu)
    #12 Due May 13. 4.4: 2, 3, 5; 4.5: 1, 2, 4, 7, 11, 20; and these extra problems

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    Quizzes and Solutions


    Quiz 1 covered 1.1-2.1
    Solutions
    Quiz 2 Solutions (covered 2.2-2.4, 2.7-2.8 and 3.1)
    Quiz 3 and Solutions (covered 3.2-3.6, 4.1, 4.3)

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    Links

    18.085 OCW site
    Note: I've had problems getting this to load the first time; if you do too, just hit the refresh button on your browser.


    Email me:   hans@math.mit.edu
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    Previous Course Content


    This section contains information from the previous courses, videos, practice exams, and extra homework and solutions. (Please report broken links so I can fix them).
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    Videos:   The special event for Fall 2008 is that the lectures will be recorded for OpenCourseWare You will already find a partial earlier set on the website but the course has since evolved. The videos of 18.06 Linear Algebra have been successful on OCW (please use them for help!!). The Lord Foundation gave a new grant for 18.085.

    Assignments

    o HOMEWORK 1 FOR MONDAY, SEPTEMBER 8: Any 3 problems from section 1.1 and from 1.2. Please PRINT your name so that we can make a class list. Not graded in detail—you may always discuss with others!

    o Solutions to Homework 1

    Solution 1

    Solution 2

    o Homework 2 for MONDAY September 15 (and please see the notes below):

    o Solutions to Homework 2

    Solution 1

    Solution 2

    Solution 3

    Solution 4

    Solution 5

    o Homework 3 for WED SEPT 24 // There will soon be a separate MATLAB question for that Friday the 26th

    o Solutions to Homework 3

    Solution 1

    Solution 2

    Solution 3

    Solution 4

    o Matlab 1: This MATLAB homework for FRIDAY 9/26 is separate from the text homeworks. It is about the convection-diffusion equation -u'' + Vu' = load A symmetric part K/h^2 comes from diffusion -u'' plus an antisymmetric part if we use centered differences for the convection term Vu'. As V increases this becomes convection dominated and u changes.

    The homework to turn in asks for V=3 and V=12 each with n=5 and n=21.

    1. Solve the equation for u(x) with point load at x=0.5 by hand
    2. Solve by finite differences for two choices of n
      (Include 4 plots of finite difference solution superimposed on exact solution.)
    3. Find the eigenvalues for both choices of V (remarkable)
    4. Your experiment: try a larger V or larger n or more eig (<1 page!!)

    Here is the Matlab code that will help with this.

    o Solutions to Matlab 1

    Solution 1

    Solution 2

    o Homework 4 for WED OCT 8 (after the 7:30 pm exam on TUES OCT 7 in 54-100)
    I will try to move my review session from WED OCT 8 to MON OCT 6

    One more question: Find 4 solutions to the 2 by 2 system Mu''+Ku = 0 The solutions have the form cos(wt)x and sin(wt)x. First find M and K: Fixed-fixed / spring constants c1=1 c2=2 c3=4 / masses m1=1 m2=2 Look for eigenvectors and eigenvalues (by hand) of K x = w^2 M x.

    **The exam will include the start of Section 2.4 and I will give more detail in class**

    o Solutions to Homework 4

    Solution 1

    Solution 2

    Solution 3

    Prof. Strang's Solution to 2.2.6

    o Solutions to Quiz 1

    Quiz 1

    o Homework 5 for MON OCT 20

    o Solutions to Homework 5

    Solution 1

    o Matlab 2: Create the 5 by 8 matrix A for the regular hexagon truss, with the lowest bar removed between the two supports. The angles with the horizontal will be 0, 60 or 120 degrees with good sines and cosines. Number the bars 1 2 3 4 5 and nodes 1 2 3 4 clockwise from the supports! I believe rank(A)=5. Use null(A) to find three mechanisms (solutions to Au=0). Can you see 3 mechanisms that are simple to draw?

    o Solutions to Matlab 2

    Solution 1

    o Homework 6 for MON OCT 27

    o Solutions to Homework 6

    Solution 1

    o Solutions to Matlab 3

    Solution 1

    Solution 2

    o Homework 7 WILL NEVER BE COLLECTED -- it is practice for next week's exam. 3D vectors and splines WILL NOT be on the exam. I will say more in class.
    Suggested: 3.3: 2, 5, 7, 8, 9, 11 and 3.4: 2, 4, 5, 17

    o Solutions to Homework 7

    Solution 1

    Solution 2

    Solution 3

    Solution 4

    o Homework 8 (MATLAB) for FRIDAY Nov 14
    [A couple corrections have been made to this homework assignment. It should be correct now. -- 11/11/08 11:10 AM. Suggestion added 11/13/08 8:21 AM]

    The function u(x,y) = 1 - x2-y2 solves the Dirichlet problem for Poisson's equation inside the unit circle:

    -uxx-uyy=4 with u=0 on the circle

    [Don't forget that the right side of the equation is now 4 and not 1.]

    I am highly interested in the errors in u and its slope when the circle is replaced by a regular polygon with M sides. You can compute one pie-shaped triangle with angle 2π/M at the center (0,0) since all triangles will be the same.

    Take the triangle T with corners (0,0) and (cos π/M, sin π/M) and (cos π/M,-sinπ/M). Start with this triangular mesh on T: Divide the center line of T from 0 to cos π/M into N equal pieces of length h=N-1 cos π/M. In addition to (0,0), we have 3N meshpoints on the center line and the edges of T:

    (h,0),...,(Nh,0) , (h,h tan π/M),...,(Nh,Nh tan π/M) , (h,-h tan π/M),...,(Nh,-Nh tan &pi/M)

    [These coordinates were wrong originally; now they should be right.]

    The boundary condition is U=0 on the short outer edge, and Neumann dU/dh = 0 on the edges in to (0,0). To make triangles, add diagonals from meshpoints (h,0),...,((N-1)h,0) on the center line to the next points (2h,tan 2h),...,(Nh,tan Nh) and (2h,-tan 2h),...,(Nh,-tan Nh) along the edges of T. Draw the mesh by hand to see 4(N-1) triangles + 2 triangles from (0,0). This is my suggested mesh!

    Adjust the code in the book and on math.mit.edu/cse Section 3.6 to assemble K and F and solve KU=F. The known values are 0 at the three outer meshpoints where x=Nh. You should see symmetry across the x axis.

    [Perhaps the easiest thing to do is to replace the "squaregrid" part of femcode.m with your own code to create the p, t, and b lists for this problem. You don't need any tricks for this; just order the points how you want, and be sure to place the point numbers in the t and b lists correctly.]

    Print out and plot the values of U along that center line x=0 and separately along the upper edge of the triangle. On those plots, also show the exact u=1-x2-y2. I expect errors of order 1/N2 (try 3 values of N). I also expect these errors to decay as you go toward (0,0). Plot also the piecewise constant x-derivative of U along the center line y=0 and the true derivative du/dx = -2x. Here I expect errors of order 1/N.

    If possible, increase M beyond 8 (octagon) to see better accuracy as you come closer to the circle.

    o Solutions to Homework 8

    Solution 1

    Solution 2: Part 1, Part 2, Part 3, Part 4, Part 5

    o Homework 9 for MON NOV 24

    o Solutions to Homework 9

    Solution 1

    Solution 2

    o Solutions to Quiz 2

    Problems 1 & 2

    o Homework 10/Preparation for the quiz in Walker DEC 4 7:30

    o Solutions to Homework 10

    Solution 1

    Solution 2


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    Class Resources

    o Videos of Professor Strang's Lectures (Lincoln Lab, Spring 2001)

    o 18.085 Course information page from Fall 2007.

    o Movie of elimination   moe.m   (also need realmmd.m )

    o Code to create K,T,B,C as sparse matrices

    o MATLAB's backslash command to solve Ax = b   (ps, pdf)

    o Getting started with Matlab: http://ocw.mit.edu/OcwWeb/Mathematics/18-06Spring-2005/RelatedResources/

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    Exams and Solutions from previous years

    o Exams and Solutions (Fall 2006)

    o Exams and Solutions (Fall 2005)

    o Exams and Solutions (Fall 2004)

    o Exams and Solutions (Fall 2003)

    o Exams and Solutions (Fall 2002)

    o Exams and Solutions (Fall 2001)

    o More Exams and Matlab Homeworks from previous years

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