Syllabus (updated 19 Jan 2011)
Projected schedule (Not final; further adjustments are possible)
Mon 1/31 HW1 posted
Mon 2/14 HW1 due, HW2 posted
Mon 2/28 HW2 due, HW3 posted
Mon 3/7 Midterm, open book, no laptop
--- 3/14 - 3/18 spring break ---
Mon 3/21 HW3 due, HW4 posted
Mon 4/4 HW5 posted
Mon 4/11 HW4 due
Mon 4/18 HW5 due, HW6 posted
Wed 4/27 HW6 due (Notice short turn-around for HW6)
Mon 5/2 Last lecture before final
Mon 5/9 4:50-6:50 Final exam, open book, no laptop (calculator OK)
==========================
We have finished:
Ch. 1, whole chapter except pp.27-32
(Also useful: Ch. 8, pp. 347-349)
Ch. 2, whole chapter except Sec 2.5.2 and 2.5.3 (pp.76-82)
(Also useful: Ch. 7, pp. 275-282)
Ch. 3, whole chapter (focus on Sec 3.1-3.4 - might skip detail of 3.4)
Ch. 4, Sec 4.1-4.4
= = = = = = = = midterm = = = = = = =
Material for final exam:
Ch. 4, Sec 4.5, Ch 7, Sec 7.3 (Not included in midterm but useful for HW3)
Ch. 5, Sec 5.3 is the most important
(Sec 5.1-5.2, Sec 5.5, and Appendix to Sec 5.5 are also useful)
Ch. 10, Sec 10.2, 10.3 (you might skip the detail in 10.3.3 and Appendix to 10.3)
Sec 10.4.2 (Sec 10.4.1 is also relevant; we will discuss it at a later time)
Sec 10.5
+ Slides #11 and #12
Ch. 8, Sec 8.1-8.3 (Sec 8.3 is the most important; Sec 8.1-8.2 are old material
that we have already discussed before midterm)
+ Slides #13 (material not in textbook)
Ch. 9, Sec 9.2, 9.3.1, and 9.3.3
Ch. 12, Sec 12.1-12.3, 12.5, 12.6
(You might skip the detail of p.567, pp. 569-570, 572-577)
Introduction to nonlinear PDE (lectures)
Homework
Homework #1
Solutions
Homework #2
Solutions
Homework #3
Solutions
Homework #4
Solutions
Homework #5
Solutions
Homework #6
Solutions
Slides
Slides #1 - First-day introduction, 1/19
Slides #2 - ASU Matlab setup guide for beginners
Slides #3 - General remarks, 1/19
Slides #4 - Separation of variable, heat equation, 1/26
Slides #5 - Orthogonality relation, 1/31
Slides #6 - Eigenvalue problem, 1/31
Slides #7 - Qualitative discussion of heat equation, 1/31
Slides #8 - Laplace equation, 2/7
Slides #9 - Example of Fourier Sine/Cosine series 2/14
Slides #10 - Sturm-Liouville Problem 3/21
Slides #11 - Fourier Sine transform, 3/28
Slides #12 - Fourier transform, 3/30
Slides #13 - Fourier transform for nonhomogeneous PDE, 4/6
Matlab 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.