CAAM 336 – Differential Equations in Science and Engineering

MWF 1:00-1:50 p.m. in Duncan Hall 1070

Instructor:

Derek Hansen (Derek.J.Hansen |AT| rice |DOT| edu), Duncan Hall 2004, (713)348-2290

Office Hours:

Tues 2:30-4:00 p.m.
Fri 4:30-5:30 p.m.
I will be in my office — Duncan Hall 2004 — at these times to answer any questions you have. No appointment necessary. Feel free to stop by.

Teaching Assistant:

Drew Kouri (Drew.Kouri |AT| rice |DOT| edu), Duncan Hall 3016
Drew's office hours: Tuesday, 10:00–11:30 a.m.

Recitation:

Monday, 7:30–9:00 p.m. in Herzstein 210

Course Information:

• Prerequisites: MATH 212 and CAAM 210. Less formally: you should be comfortable with multivariable calculus and basic matrix operations, and you should be able to write MATLAB programs.
• Course Text: Mark S. Gockenbach, Partial Differential Equations: Analytical and Numerical Methods, SIAM, Philadelphia, 2002.
  • Errata
• Required Work:
  • Reading: Students should carefully read all of the assigned sections from the course text and all supplemental material posted below in the course schedule.
  • Problem Sets: Problem sets will be assigned roughly once a week. Mathematically rigorous solutions are expected; strive for clarity and elegance.
    You may collaborate on the problems, but your write-up must be your own independent work. Transcribed solutions are unacceptable. You may not consult solutions from previous sections of this class.
    Unless it is specified that a particular calculation must be performed by hand, you are encouraged to use MATLAB's Symbolic Math Toolbox (or Mathematica or Maple) to compute and simplify tedious integrals and derivatives on the problem sets. As always, you must document your calculations clearly.
    • Late Policy: No late homework will be accepted (unless you have a good excuse, in which case you may not look at the posted solutions before you hand in the assignment).
  • Exams: Two take-home exams. Each exam must be your individual, unassisted effort. Indicate compliance by writing out in full and signing the traditional pledge.
• MATLAB: Most homework assignments will require a modest amount of MATLAB programming. Your solutions should adhere to good programming standards and must not be copied from another student. For help with MATLAB, check out Getting Started with MATLAB from MathWorks.
• (Tentative) Grading Scheme:

  Problem Sets	40%
  First Exam	30%
  Second Exam	30%

MATLAB:

Course Schedule:

Lecture	Date	Topics/Sections Covered	Homework	Additional Material
1	Mon 8/24	Chapter 1 — Classification of differential equations.	Ch. 1 Exercises 1–10.
2	Wed 8/26	2.1 — Derivation of the heat equation.		bar_bc.m
3	Fri 8/28	2.1.1, 2.1.2 — Heat Equation: forcing terms, boundary conditions, initial conditions, steady state.	Due at 1:00: Ch. 1 Exercises 2,4,6,8,10. (You should do all of them, but only hand in the evens.)	bar_bc_ss.m
4	Mon 8/31	2.3 — Vibrating Strings: the wave equation.		Java applet showing string vibrations by Paul Fastad.
5	Wed 9/2	3.1 — Vector spaces and linear operators.	Due at 1:00: hw2.pdf Solutions: sol2.pdf	vib_string1.m, vib_string2.m
6	Fri 9/4	3.1 — More on Vector spaces and linear operators. Finite difference approximation.		Required reading: finite_difference.pdf
	Mon 9/7	Labor Day
7	Wed 9/9	Finite differences (required reading: finite_difference.pdf). 3.2 — Subspaces, null space, range.	Due at 1:00: hw3.pdf Solutions: sol3.pdf	finite_diff_bessel.m
8	Fri 9/11	3.3, 3.4 — Basis, linear independence, and orthogonality.
9	Mon 9/14	3.4 — Orthogonality and projections.
10	Wed 9/16	3.4 — Best approximation. Begin 3.5 — Eigenvalues and eigenvectors.	Due at 1:00: hw4.pdf Solutions: sol4.pdf	proj_example1.m
11	Fri 9/18	3.5 — The spectral method.		lecture11.pdf
12	Mon 9/21	5.1 — The analogy between BVPs and linear systems; symmetric operators (Definition 5.4 and subsequent examples). 5.2 — The spectral method.
13	Wed 9/23	5.1, 5.2 continued.	Due at 1:00: hw5.pdf Solutions: sol5.pdf
14	Fri 9/25	5.2, 5.3 — The spectral method for solving BVPs.
15	Mon 9/28	5.3 — Convergence of Fourier series. Inhomogeneous boundary conditions.
16	Wed 9/30	5.3 — More examples.	Due at 1:00: hw6.pdf Solutions: sol6.pdf
17	Fri 10/2	Fourier series recap. 5.4 — Weak formulation.
18	Mon 10/5	5.4, 5.5 — Weak formulation and the Galerkin method.
19	Wed 10/7	5.5, 5.6 — The Galerkin method, finite elements.	Due at 1:00: hw7.pdf Solutions: sol7.pdf
20	Fri 10/9	5.6 — Piecewise linear finite elements.		galerkin_example1.m hat.m, fem_demo1.m
	Mon 10/12	Midterm Recess
21	Wed 10/14	5.6 continued.	Due at 1:00: hw8.pdf Solutions: sol8.pdf
22	Fri 10/16	6.1.6 — Homogeneous heat equation; separation of variables.
23	Mon 10/19	More heat equation: 6.1.0, 6.1.1 — Fourier series methods. 6.1.3 — Inhomogeneous heat equation with homogeneous Dirichlet conditions. 6.1.4 — Inhomogeneous Dirichlet conditions. Begin 6.2 — Homogeneous Neumann conditions.
24	Wed 10/21	6.2 — Heat equation with homogeneous Neumann conditions. Fourier cosine series.
25	Fri 10/23	6.3 — Periodic boundary conditions and the full Fourier series.	Due at 5:00: exam1.pdf	exam1_practice_problems.pdf
26	Mon 10/26	6.3 continued.
27	Wed 10/28	6.4.0 — Finite element method for the heat equation.	Due at 1:00: hw9.pdf Solutions: sol9.pdf
28	Fri 10/30	4.2, 4.3 — ODE and linear homogeneous systems (review). 4.4 — Matrix exponentials, Euler's method.
29	Mon 11/2	4.5 — Backward Euler method.
30	Wed 11/4	4.5 continued.		euler_demo.m
31	Fri 11/6	6.4, 6.5 — Finite elements and Neumann conditions.	Due at 1:00: hw10.pdf Solutions: sol10.pdf
32	Mon 11/9	6.5 continued.
33	Wed 11/11	6.5 continued.
34	Fri 11/13	7.1 — Infinite string: d'Alembert's solution.		For a careful derivation of a model of a vibrating string, see: Stuart S. Antman, The equations for large vibrations of strings, American Math. Monthly 87 (1980) 359--370.
35	Mon 11/16	7.1 continued. 7.2 — Homogeneous wave equation: separation of variables.
36	Wed 11/18	7.2 — The Fourier method for the wave equation.	Due at 1:00: hw11.pdf Solutions: sol11.pdf	hw11prob6.m wave_DirichletIBVP.m
37	Fri 11/20	7.3 — FEM for the wave equation. 7.4 (especially 7.4.2) — Resonance.
38	Mon 11/23	Finish ch. 7. 8.1 — 2 and 3 dimensions, the Divergence Theorem.
39	Wed 11/25
	Fri 11/27	Thanksgiving Recess
40	Mon 11/30
41	Wed 12/2		Due at 1:00: hw12.pdf
42	Fri 12/4
			exam2.pdf Exam 2 Due at 5 p.m., Wednesday, Dec 16

Any student with a disability requiring accommodation in this course is encouraged
to contact the instructor during the first week of class, and also to contact
Disability Support Services in the Ley Student Center.