Welcome!

18.085 Fall 2014

Professor Gilbert Strang

Office: E17-420

e-mail: gs@math.mit.edu

Office Hours by Appointment

Lecture:

• Tuesday and Thursday, 10-11AM in 1-190
• Tuesday, 3-4PM in 54-100

Teaching Assistants:

• Jason Choi (choij@mit.edu). Office Hours by Appt.
• Jan-Christian Huetter (huetter@mit.edu)
• Florent Bekerman (bekerman@math.mit.edu)

Fall 2014 Exam Dates:

• Tuesday, Sep. 30th (26-100), Solutions. Average: 78, Standard Deviation: 15.
• Thursday, Oct. 30th (Walker 50-340), Solutions. Average: 86, Standard Deviation: 11.
• Thursday, Dec. 4th (Walker 50-340), Solutions. Average: 82, Standard Deviation: 9.

Help Sessions:

• Monday 3-4PM in E17-401Q (Office Hour with Florent Bekerman)
• Wednesday 4-5PM in 66-168 (Help Session with JC Huetter)

Announcements:

• Congratulations on finishing 18.085! The final lecture is Tuesday, Dec. 9, at 10 am. Exam 3 and other graded assignments will be returned then.
• The 3pm lecture on Tuesday, Dec. 9 is CANCELED. Enjoy the break!
• Solutions and grades to exam 3 have been posted. If you have any discrepancies, please contact Jason to make an appointment on Wednesday or see Prof. Strang AFTER looking at the solutions.
• Any graded assignment not picked up in class can be picked up during Florent's office hour or from Prof. Strang's office 17-420.
• The course Piazza is up! Please register by following this link: http://piazza.com/mit/fall2014/18085. You will get more rapid responses from the TAs and your peers through Piazza.

Homework: [Fall 2014]

#9 (Due: Tuesday, Nov. 25 @ 10am), Solutions, courtesy of Shuai Li

A scan of the problems is here: Textbook Scan
• Compute the following convolutions:
• $(1, 2, 3) \ast (3, 2, 1)$
• $(1, 2, 3) \circledast (3, 2, 1)$
• $f(x) \ast \delta(x-a)$
• $\delta(x-a)\ast \delta(x-b)$
• 4.3: 21
• 4.4: 1, 6, 8
• 4.5: 1, 2, 8, 10, 12, 22

#8 (Due: Thursday, Nov. 13), Solutions

A scan of the problems is here: Textbook Scan
• 4.1: 1, 3, 7, 11, 13
• 4.2: 1, 3
• 4.3: 2, 8, 10

#7 (Due: Thursday, Nov. 6), Solutions

Divide the square $-1 \leq x \leq 1$, $-1 \leq y \leq 1$ into 4 unit squares (not triangles!) The values of $u$ are given at the 8 boundary points (call those values $u_E$ and $u_{NE}$ and $u_N$, ....) The only unknown is $U_0$ at the center. The one TEST function is BILINEAR in each square: $V(x,y) = a + bx + cy + dxy$ It is zero around the boundary and 1 at the center point. What are $a, b, c, d$ for its 4 pieces in the 4 squares? The one TRIAL function is also BILINEAR. It matches all 8 boundary values. We only have to find its value $U_0$ at the center, in terms of those 8 boundary values.
• Write the weak form of Laplace's equation
• Substitute the 1 test function $v$ and the trial function with unknown value at the center point.
• Integrate the weak form to find that unknown value in terms of the 8 known boundary values.
This will be the difference equation for the BILINEAR choice of finite elements

#6 (Due: Thursday, Oct. 23), Solutions, courtesy of Boris Lipchin

• $u(x,y) = x + y - x^2 + y^2$ solves Laplace's eqn.
• What is $v(x,y) = \nabla u$ ?
• What is the stream function $s(x,y)$ that satisfies the Cauchy-Riemann eqns $du/dx = ds/dy$ and $du/dy = - ds/dx$?
• The gradient $v(x,y)$ is perpendicular to every equipotential curve $u =$ constant. Find $g = \nabla s$ that is perpendicular to the streamlines $s =$ constant. Check that $g$ is perpendicular to $v$.
• These equipotential curves and streamlines are (ellipses) (parabolas) (hyperbolas). [Choose the correct answer]
• 3.3: 1, 7, 8, 11
• 3.4: 2, 4, 5

#5 (Due: Thursday, Oct. 16), Solutions, courtesy of Rachel Mok

A scan of the textbook problems is here: 3.1/3.2 Problems
• The Matlab problem is to approximate the eqn $+u'' + u = x$ by linear finite elements with boundary conditions $u(0)=0$ and $u(1)=0$.
• First find the solution $u= x + A sin x + B cos x$ (particular + null) with $A,B$ from boundary conditions.
• Then find the stiffness matrix $K$(for $u''$) and mass matrix $M$ for the $u$ term and the load vector $F$ for the right side $f(x)=x$.
• The $N$ equations to solve will be $(K + M)U = F$. This is for $N$ hat functions $\phi(x)$ centered at $x=h, 2h, ...,Nh$ with $(N+1)h = 1$. Solve for $N=4, 8, 16, 32$ and plot $N=32$ against the true $u(x)$.
• The maximum error at meshpoints should decrease like $(1/N)^2$--Does it?
• 3.1: 5, 9, 11, 12, 13, 18
• 3.2: 3, 17 (on cubic splines)

#4 (Due: Thursday, Oct. 9), Solutions, courtesy of Hyunwoo Yuk

• Problem 2.4.17 in the book has a square grid with 9 nodes and 12 edges. Count the zeros in $A'A$, find its main diagonal (degree matrix), and find its middle row with 4 and four -1's.
• Start with 4 nodes and all 6 edges. A is 6 by 4. (note: you have to use MATLAB/Julia/some other programming language for this problem. Attach printouts.)
• Find $A'A$.
• Then ground node 4 by $u_4 = 0$. Remove column 4 so $A$ is 6 by 3 with independent columns: invertible $A'A$. PUT IN RANDOM BATTERIES $b = rand(6,1)$.
• Solve $A'Au = A'b$, and find the voltage at node 1 --- do this 1000 times to find the average voltage $u_1$ at node 1.
• For the same 4-node graph with all 6 edges, fix $u_1 = 1$ as well as $u_4 = 0$. The vectors $b$ and $f$ are zero and $C = I$.
• Solve for $u$ and $w$ and find the total current to node 4.
(hint: WHAT ARE THE EQUATIONS? WHEN YOU SET $u_1 = 1$, you have to bring the KNOWN terms of $A'Au$ to the right hand side of $A'Au = 0$ - leaving 2 eqns for $u_2$ and $u_3$.)
• 2.7: 1, 2, 3, 4, 9 (Truss problems)

#3 (Due: Thursday, Sep. 25), Solutions, courtesy of Hyunwoo Yuk

• 1.6: 27
• 2.1: 3, 7, 8
• 2.3: 7, 8, 24
• 2.4: 1, 2, 7

#2 (Due: Thursday, Sep. 18), Solutions

The following questions come from the course textbook. For this week ONLY, the problems in the book are provided, here.
• 1.4: 7, 9, 11
• 1.5: 9 (and find the eigenvalues by MATLAB/Julia)
• 1.6: 3, 9, 20
• Master Equations. Please see the last page of the above posted link.

#1 (Due: Thursday, Sep. 11), Solutions

• $A$ = [1 3 2] is a 1 by 3 matrix and $A'$ = 3 by 1 transpose
• Find $K = A' * A$ and explain why this $K$is singular (many answers, all OK)
• Find all solutions to $Ku=0$.
• Compute $K^4 = A'AA'AA'AA'A$ by hand in a smart order
• Page 16 of the text multiplies our 2nd difference matrix $K$ (make it infinite to avoid boundary effects) by vectors ones = 1,1,1,.. linear 0,1,2,3,.. squares 0,1,4,9,...
• Do the same for cubes 0,1,8,27,.. and quartic 0,1,16,81,..
• Which ones give the wrong derivative and what is the error
• Multiply these vectors by the FORWARD 1st difference matrix $D_+$ and the CENTERED first difference matrix $D_0$ (infinite) and compare to the derivatives of $x^2, x^3, x^4$.
• Solve $-u'' = \delta(x-a)$ with $u(0)=0$ and $u'(1)=0$ (free). Graph $u(x)$ and explain what is happening at $x=a$ between 0,1
• Problem 1.2.9 on page 24 is about the 4th difference with coefficients 1,-4,6,-4,1. Where did those numbers come from and should you divide by a power of $h=\Delta x$ ?
• Problem 1.2.19 on page 25 is about $-u'' + u' = 1$. Choose a finite difference approximation. Solve for $u$.
• Invert our $T$ matrix with $T_{11}=1$ and diagonals $-1,2,-1$ for $n=3$ and $n=4$. From the pattern guess a formula for any $n$. Can you show that your guess is the correct inverse (any $n$)?

Homework Notes:

• Good to underline your last name.
• The Class List with class numbers will be created from these homeworks.
• Problems will often come from the 18.085 textbook on CSE. (Printouts / Graphs for MATLAB questions.)
• Section 1.1 is available on math.mit.edu/cse
• A printing of 1.1.27 has a typo (my name for a mistake)
• These are the simplest corrections: Verify that
• K comes from A0 A0 '     (prime = transpose)
• T comes from A1 A1 '
• B comes from A2 A2 '
• (Those have 2 then 1 then 0 boundary conditions. They change from positive definite to semidefinite.)
• problem set 1.2 from CSE textbook

Course Topics:

• Applied Linear Algebra
• Applied Differential Equations
• Fourier Methods
• Algorithms
• Course outline