Syllabus (updated 5 Jan 2012)
Projected schedule (Not final; further adjustments are possible)
Homework is due before class on the due date.
Wed 1/18 HW1 posted
Wed 2/1 HW1 due, HW2 posted
Wed 2/15 HW2 due, HW3 posted
Wed 2/22 Midterm, open book, no laptop
Wed 2/29 HW3 due, HW4 posted
Wed 3/14 HW4 due, HW5 posted
--- 3/19 - 3/23 spring break ---
Wed 4/4 HW5 due, HW6 posted
Wed 4/18 HW6 due
Mon 4/23 Last lecture before final
==========================
We have finished:
Chapter 1, Entire chapter
Chapter 2, Entire chapter except Sec 2.5.2 and 2.5.3 (pp.76-83)
Note: The discussion on the similarity between eigenvector and
and eigenfunction expansion can be found in "Appendix to 2.3",
pp. 58-59. We have also used some material from "Appendix to 5.5",
pp. 184-189.
Chapter 3, Entire chapter except Sec 3.6
Chapter 4, Sec 4.1-4.4
Chapter 8, Sec 8.2
The above will be the scope for midterm.
See here for examples of calculators that are allowed in the exams
---------
Material covered after midterm - this will be the scope for final exam
Chapter 4, Sec 4.5
Chapter 5, Sec 5.1-5.5 except pp. 167, 172, and 180; Slides #10
(For busy people, a summary of the key idea is in our Slides #10)
Chapter 7, Sec 7.2-7.3
Chapter 10, Sec 10.1-10.5 except Sec 10.3.3, Sec 10.4.3, and pp. 453-455, 460-463
Slides #11, #12 (Also, read Sec 3.6 on the complex form of Fourier series.)
Chapter 8, Sec 8.3, Slides #13, #14
Chapter 9, Sec 9.2
Chapter 12, Sec 12.2, Sec 12.6 (skip Sec 12.6.2 and pp. 569-579 but study the
two examples in the bottom of p. 570 and in pp. 573-574)
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For self study (relevant to HW6A Prob 1, but will be excluded from final exam):
Chapter 6, Sec 6.2 and 6.6
Homework
Homework #1
Typos for Prob 1 were corrected; Corrections are marked in red.
Solutions
Homework #2
Solutions
Homework #3
Solutions
Homework #4
Solutions
Homework #5
Note a slight modification of Prob 1 as marked in red. (An update
will be sent to class on April 2)
Solutions
Homework #6A
Homework #6B
Solutions
Slides
Slides #1 - First-day introduction, 1/9
Slides #2 - Place holder only. Please see slides under "Matlab"
Slides #3 - General remarks, 1/9
Slides #4 - Separation of variables, heat equation, 1/11, 1/18
Slides #5 - Orthogonality relation, 1/25
Slides #6 - Eigenvalue problem (for self study)
Slides #7 - Qualitative discussion of heat equation, 1/23
Slides #8 - Laplace equation
Slides #9 - Example of Fourier Sine/Cosine series
Slides #10 - Sturm-Liouville Problem
Slides #11 - Fourier Sine transform
Slides #12 - Fourier transform
Slides #13 - Fourier transform for nonhomogeneous PDE
Slides #14 - Additional example for nonhomogeneous PDE
Slides #15 - Example for the method of characteristics
Typos have been corrected; Corrections are in red.
Matlab
ASU Matlab setup guide for beginners
Basic examples for beginners
Additional examples
Example_1 - codes
Example_1 - result
This program plots the two functions, u(x) = sin(6πx)*exp(-x)
and v(x) = cos(6πx)exp(-x), for x ∈ [0, 1]. The discretization
interval (resolution of plot) is 0.01.
Example_2 - codes
Example_2 - result
This program generates the color+contour map for the function,
u(x,y) = sin(2πx)sin(2πy)exp(-(x2+y2)), for x ∈ [0,1], y ∈ [0,1].
The discretization interval (resolution of plot) is 0.01 for both
x and y. Contour interval is 0.1 (contour levels are -0.9, -0.8, ...,
-0.2, -0.1, 0.1, 0.2, ... , 0.8, 0.9). Contours for negative
values are dashed.
Note: With a given 2-D array, u(q,p), Matlab plots the contours
of u as a "map" of the matrix u(q,p), i.e., q goes up and down
and p goes left and right. The index q would then correspond to our y,
and p to x. This is somewhat counterintuitive so beware.
Example_3 - codes
Example_3 - result
This program plots the Fourier Sine series representation of the
function defined on x ∈ [0,1]: F(x) = 1 for 0 ≤ x ≤ 1/2,
F(x) = 0 for 1/2 < x ≤ 1. Black is the original function. Magenta,
red, and blue are the Fourier series representation truncated at n = 5,
25, and 100.
Example_4A - codes
This is a Matlab code for animation of the solution of a 1-D wave equation
that we discussed in class. The boundary conditions are u(0,t) = 0, u(1,t) = 0,
u(x,0) = sin(πx)+sin(2πx), and ut(x,0) = 0
Example_4B - codes
Same as Example 4A but with the 3rd b.c. changed to u(x,0) = sin(2πx)+0.5sin(10πx)
Examples 4A and 4B are the cases with fixed boundary conditions in x.
Example_4C - codes
Same as Example 4A but with the b.c.'s changed to u(0,t) = u(1,t),
ux(0,t) = ux(1,t), u(x,0) = 0.6cos(6πx)+0.8sin(6πx), and ut(x,0) = 0
Examples 4C and 4D are the cases with periodic boundary conditions in x.
Example_4D - codes
Same as Example 4A but with the b.c.'s changed to u(0,t) = u(1,t),
ux(0,t) = ux(1,t), ut(x,0) = 0, and
u(x,0) = summation of five cosine and five sine modes (see code for detail)
Example_5 - codes
Example_5 - result
This program plots the solution to a 2-D wave equation to be
discussed in class. The expression of the solution is
u(x,y,t) = sin(2πx)sin(3πy)cos(sqrt(13)πt)
The 4 panels, top-left, top-right, bottom-left, and bottom-right,
correspond to the solutions at t = 0, 0.45/sqrt(13), 0.55/sqrt(13),
and 1/sqrt(13).
Example_6 - codes
Example_6 - result
This code solves the nonlinear (quasilinear) PDE, the "shock wave equation",
∂u/∂t + u∂u/∂x = 0, with the b.c., u(x,0) = sin(x).
Shown are the solution u(x,t) at t = 0.3 and 0.7, and the initial state.
The solution is periodic in x. Only one full period (x = 0 to 2π) is shown.
The method of characteristics is used to obtain an analytic expression of
the solution. (This is done by hand, not part of the code.) Numerical evaluation
of u(x,t) requires root-finding of a nonlinear equation, which is dealt with
by using the bisection method. This code will work only up to a certain value
of t (what is it?), beyond which the solution becomes a multiple-valued
function of x.