Syllabus (updated 19 Jan 2010)
Projected schedule
Wed 2/3 HW1 posted
Wed 2/17 HW1 due, HW2 posted
Wed 3/3 HW2 due, HW3 posted
Wed 3/10 Midterm, open book, no laptop (calculator OK)
--- 3/15 - 3/20 spring break ---
Mon 3/22 HW3 due, HW4 posted
Mon 4/5 HW4 due, HW5 posted
Mon 4/19 HW5 due, HW6 posted
Wed 4/28 HW6 due (Notice short turn-around for HW6)
Mon 5/3 Last lecture before final
No lecture on Wednesday 5/5. We will have open office hours
all afternoon that day.
Final exam: 5/12, 4:50-6:40 PM
Six homework assignments = 50% of total score
One midterm = 20%
Final exam = 30%
==========================
We have finished:
Ch. 1, Whole chapter
Ch. 2, Whole chapter
Ch. 3, Whole chapter (will finish on 3/1)
Ch. 4, Sec 4.1-4.4 (will finish on 3/3)
Ch. 8, Sec. 8.2 (pp. 347-349 only, on how to deal with nonhomogeneous b.c.)
= = = = = = Midterm = = = = = =
Anticipated scope of the final exam:
* Sections 4.5 and 7.3 were discussed before midterm but were not included
in that exam. Although they will be included in the final, expect no more
than a minor question from those sections.
Progress after midterm (key sections):
Ch. 10, Sec. 10.1-10.5 (you might skip Appendix to 10.3 and Sec. 10.4.3)
Ch. 5, Sec. 5.3 (you might skip p. 167)
[Sec. 5.5 provides useful background for the main theorem in Sec. 5.3]
Ch. 8, Sec. 8.1-8.3
[Sec. 8.4 provides useful background for understanding Sec. 8.3]
Ch. 9, Sec. 9.1-9.2
Ch. 12, Sec. 12.1-12.3 (you might skip pp. 545-549),
Sec. 12.6.1, 12.6.3, 12.6.5
Additional topics (to be discussed on 4/26, 4/28, and 5/3)
* In the above list, we don't expect a question to come directly from
Sec. 5.5 or 8.4, but reading them will help you better understand
other relevant sections.
*Ch. 6, Sec. 6.1-6.2, 6.6 (used for HW6 Prob 1) will be excluded from
the final.
An example of distribution of questions (assume 5 questions total):
Three "major" questions, one each from Ch. 8, Ch. 10, and Ch. 12
Two "minor" questions from Ch. 5, Ch. 9, Sec. 4.5, Sec. 7.3, or any
combinations of the post-midterm material
(The actual distribution may vary.)
* Exam is open book, open note
Homework
Homework #1 (2 pages)
HW1 Solutions
Homework #2 (3 pages)
HW2 Solutions
(Minor correction: The "thermal conductivity" in Prob 3 should really be "thermal
diffusivity". We had artificially set the density and heat capacity of the rod to 1
in all of our discussions, thus the thermal conductivity and diffusivity have the
same value. Nevertheless, it's thermal diffusivity that has the correct unit
of m^2/s. This does not affect your answer for the problem.)
Homework #3 (2 pages)
HW3 Solutions
Homework #4
HW4 Solutions
Homework #5
HW5 Solutions
Homework #6
HW6 Solutions
Slides
Slides #1 - First-day introduction, 1/20
Slides #2 - General remarks, 1/20
Slides #3 - ASU Matlab setup guide for beginners
Slides #4 - Qualitative behavior of Heat equation, 1/25
Slides #5 - Placeholder only
Slides #6 - Separation of variables + Eigenvalue problem (Heat equation), 1/27-2/1
Slides #6A - Eigenvalue problem (supplement of Slides #6), 1/27-2/1
Slides #6B - Orthogonality relation (supplement of Slides #6), 2/1-2/3
Slides #7 - Placeholder only
Slides #8 - Placeholder only
Slides #9 - Equilibrium solution of Heat Equation; Laplace equation, 2/10
Slides #10 - Laplace equation, 2/15
Slides #11 - Useful math formulas
Slides #12 - Placeholder only
Slides #13 - Fourier Sine/Cosine series, 2/24
Slides #14 - Placeholder only
Slides #15 - Fourier Sine transform, 3/22
Slides #16 - Fourier transform, 3/22
Slides #17 - Sturm-Liouville Problem (Ch.5)
Slides #18 - Nonhomogeneous PDE NEW
Slides #19 - Numerical solution for Laplace's Eq. NEW
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 solution to the Laplace equation in Slides #10.
Discretization interval (resolution of plot) is 0.01 for both x and y.
Contour interval is 0.05 (contour levels are 0.05, 0.1, 0.15, ..., 0.45).
The infinite series in the solution is truncated at n = 20 for this plot.
Example_4 - codes
Example_4 - 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_5A - 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_5B - codes
Same as Example 5A but with the 3rd b.c. changed to u(x,0) = sin(2πx)+0.5sin(10πx)
Examples 5A and 5B are the cases with fixed boundary conditions in x.
Example_5C - codes
Same as Example 5A 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 5C and 5D are the cases with periodic boundary conditions in x.
Example_5D - codes
Same as Example 5A 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_6 - codes
Example_6 - 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_7 - codes
Example_7 - 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.
Examples below this line will not be used this semester
Example_6 - codes
Example_6 - result
This program demonstrates how to make a contour/color map for a
function defined in polar coordinate. The result is a map for
u(r,θ) = r*θ, where (r,θ) are the usual polar coordinate
and u is defined within a 60-degree wedge with radius = 1. More
precisely, the domain is 0 ≤ r ≤ 1, 0 ≤ θ ≤ π/3.
Example_7 - codes
Example_7 - result
Just another example for plotting a function in polar coordinate.
The result is a contour/color map of u(r,θ) = r*cos(3*θ) for
the annular domain, 1/2 ≤ r ≤ 1. This program is not needed for
homework but might help you understand Matlab Example 6.
Example_8 - codes
Example_8 - result
This code plots Bessel function of 1st kind and of 0th, 1st, and
2nd order, J0(z), J1(z), and J2(z), and Bessel function of 2nd kind and
of 0th, 1st, and 2nd order, Y0(z), Y1(z), and Y2(z), for 0 ≤ z ≤ 20.