Email me:   hans@math.mit.edu
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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

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!

Solutions to Homework 1

Solution 1

Solution 2

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

• 1.3 2, 7, 11, 16
• 1.4 2, 7, 12
• 1.5 4, 9, 12, 18

Solutions to Homework 2

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

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

• 1.6 15, 20, 24, 27
• 1.7 4, 5, 18
• 2.1 4, 8

Solutions to Homework 3

Solution 1

Solution 2

Solution 3

Solution 4

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.

Solutions to Matlab 1

Solution 1

Solution 2

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

• 2.2 6, 7
• 2.3 7, 18, 24
• 2.4 1, 3, 7, 19

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**

Solutions to Homework 4

Solution 1

Solution 2

Solution 3

Prof. Strang's Solution to 2.2.6

Solutions to Quiz 1

Quiz 1

Homework 5 for MON OCT 20

• 2.7 1, 4, 6, 7
• 3.1 1, 3, 5, 10

Solutions to Homework 5

Solution 1

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?

Solutions to Matlab 2

Solution 1

Homework 6 for MON OCT 27

• 3.1: 11, 14
  
• 3.2: 1, 4, 5, 17 (MATLAB)
  
• MAIN MATLAB: Use linear hat functions with h=Δx=1/8 and also 1/16 to solve the free-fixed problem
  -d/dx(c(x)du/dx)=δ(x-1/3) with u'(0)=0 and u(1)=0
  The stiffness c(x) jumps from c=1 to c=2 at x=2/3.
  Note that the spike in δ and jump in c are not at gridpoints.
  
  • Can you plot and compare the exact u(x) with those approximations u8 and u16?
    
  • At which x is the error a maximum?
    
  • Is the maximum error proportional to h² (error divides by 4 when h divides by 2)?
    
  • What about the error at x=0?
    

Solutions to Homework 6

Solution 1

Solutions to Matlab 3

Solution 1

Solution 2

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

Solutions to Homework 7

Solution 1

Solution 2

Solution 3

Solution 4

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:

-u_xx-u_yy=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.

Solutions to Homework 8

Solution 1

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

Homework 9 for MON NOV 24

• 4.1 3, 7, 10, 13, 18
• 4.3 2, 3, 6, 8, 9, 15

Solutions to Homework 9

Solution 1

Solution 2

Solutions to Quiz 2

Problems 1 & 2

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

• 4.4: 2 (for N=4), 6, 8, 9
• 4.5: 1, 8, 10, 11, 12, 14, 23

Solutions to Homework 10

Solution 1

Solution 2

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

18.085 Course information page from Fall 2007.

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

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

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

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

Exams and Solutions (Fall 2006)

Quiz 1 ( ps | pdf ) and Solutions ( ps | pdf )
Quiz 2 ( ps | pdf ) and Solutions ( ps | pdf )
Quiz 3 ( ps | pdf ) and Solutions ( ps | pdf )

Exams and Solutions (Fall 2005)

Exam 1 (pdf) and solutions (pdf)
Exam 2 (pdf) and solutions (pdf) and supplementary solutions (pdf)
Exam 3 (pdf) and solutions (pdf)

Exams and Solutions (Fall 2004)

Exam 1 (pdf) and solutions (pdf)
Exam 2 (ps) and solutions (pdf)
Exam 3 (ps) and solutions (ps)

Exams and Solutions (Fall 2003)

Exam 1 (pdf) and solutions (pdf)
Exam 2 (pdf) and solutions (pdf)
Exam 3 (pdf) and solutions (pdf)

Exams and Solutions (Fall 2002)

Exam 1 (ps) (pdf) and solutions (ps) (pdf)
Exam 2 (ps) (pdf) and solutions (ps) (pdf)
Exam 3 (ps) (pdf) and solutions (ps) (pdf)

Exams and Solutions (Fall 2001)

Exam 1 (ps) (pdf) and solutions (ps) (pdf)
Exam 2 (pdf) and solutions (pdf)
Exam 3 (ps) (pdf) and solutions (pdf)

More Exams and Matlab Homeworks from previous years

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