M.I.T., Department of Mathematics

18.303 Linear Partial Differential Equations, Fall 2002

 Where and when: 2-142, MWF 1-2

Introduction: The main purpose of this course is to introduce the basic linear PDE's, the heat equation, the wave equation, Laplace's equation and some generalizations which describe a broad range of scientific phenomenon. We will study the basic techniques for finding exact, approximate and numerical solutions.

Textbook: Elementary Applied Partial Differential Equations, by Haberman, 3rd Edition. We will cover Chapters 1--6.

Instructor: Plamen Koev, office: 2-334, phone: 253-7775, e-mail: plamen (at) math (dot) mit (dot) edu, NEW: Office hours: Monday 2-4 or by appointment

Exam Dates: (two midterms during class hours) Wednesday, October 16 and Wednesday, November 13

Grading: Midterm 1 -- 25%, Midterm 2 -- 30%, Final Project -- 20%, Homework -- 25%.

Final Project: For a Final Project you will have to compute the numerical solution of a given PDE. It will require minimal programming knowledge and the emphasis will be on the mathematics and not on your programming skills. You can use any programming language you want, although I would suggest that you use Matlab. I plan to assign the Final Project around the middle of November.

Lectures:

09/04 Introduction
09/06 Heat Equation
09/09 Bounary Conditions for the Heat Equation
09/11 Introduction to Matlab I
09/13 Introduction to Matlab II
09/16 Separation of Variables
09/18 Boundary Value Problems
09/20 Product Solutions, worked example
09/23 Holiday. No class.
09/25 Heat Conduction in a Rod with Insulated Ends
09/27 Laplace's equation inside a rectangle
09/30 Qualitative Properties of Laplace's Equation
10/02 Fourier Series
10/04 Fourier Sine and Cosine Series
10/07 Differentiation of Fourier Series
10/09 Integration of Fourier Series
10/11 Assignment 2 due. Review for Midterm 1
10/16 Midterm 1
10/18 Review of Midterm 1.
10/21 Vibrating String with Fixed Ends
10/23 Vibrating Membrane
10/25 Sturm-Liouville Eigenvalue Problems
10/28 Self Adjoint Operators
10/30 Rayleigh Quotient
11/01 Matrix Eigenvalue Problem
11/04 Boundary Conditions of the Third Kind
11/06 Approximation Properties
11/08 Review for Midterm 2
11/13 Midterm 2
11/15 Review of Midterm 2
11/18 Numerical PDEs -- Finite Differences
11/20 Heat Equation -- Partial Difference Equation
11/22 Von Neumann Stability Analysis
11/25 Separation of variables
11/27 Matrix Notation

12/02 Nonhomogenous Problems/Other schemes and boundary conditions
12/05 Heat Equation 2D, Wave Equation
12/07 Laplace Equation

12/09 Jacobi, Gauss-Seidel and S-O-R
12/11 Final Project Presentations


Homework assignments:

Assignment 1. (due 9/27/02) Modify the following code so that it computes the steady state heat distribution on a 20x20 point grid with the bottom 5x5 right corner cut out (25 grid points cut out). Boundary conditions: 5 degrees, except on the border of the cut-out part, where the temperature is 20 degrees. Inital conditions: 0 degrees everywhere. Turn in: a printout of the code and a legible plot of the final temperature distribution.

Assignment 2. (due 10/11/02)
1. Read sections 2.4.2, 2.4.3., 2.5.2. and 2.5.3.
2. From Haberman's textbook. Each problem is worth 10 points unless indicated differently in parentheses. You only need to turn in problems worth 80 points. You may turn in solutions to problems worth more than that to ensure you get full credit. 1) 1.4.1.[b,g,h];
2) 1.4.7.[b,c];
3) 2.2.5 (5);
4) 2.3.3.[a,b,c];
5) 2.4.3.;
6) 2.4.4.(5);
7) 2.5.1.[d,e,f];
8) 2.5.5.[a,b];
9) 2.5.10.;
10) 2.5.15.[c](5)
11) 3.2.3.(5);
12) 3.3.15.(15);
13) 3.4.8.;
14) 3.4.12 (20);
15) 3.5.7.(20). Assignment 3. (due 10/25/02) Turn in problems worth at least 25 points.
  1. 4.4.3.(15)
  2. 4.4.4.(15)
  3. 4.4.6.(10)
  4. 4.4.7.(10)
Assignment 4. (due 11/08/02) Turn in problems worth 80 points.
  1. Reading assignment: Read sections 5.4. and 5.7.
  2. 5.3.3. (5)
  3. 5.3.5. (5)
  4. 5.3.8. (10)
  5. 5.5.8. (10)
  6. 5.5.14. (10)
  7. 5.5.A.3. (5)
  8. 5.5.A.5. (5)
  9. 5.6.2. (10)
  10. 5.8.8. (15)
  11. 5.8.13. (5)
  12. 5.10.2.c (5)
  13. 5.10.5. (10)
  14. (25 points -- this is an open research problem and is posted only as a challenge)
    Consider a real tridiagonal matrix A, whose entries on the main diagonal are a1,...,an, the entries on the first superdiagonal are b1,...,bn-1, the entries on the first subdiagonal are c1,...,cn-1 and all other entries are 0.
    It is also given that |ci|=|bi|>0, that is, the matrix is "balanced" and "unreduced", i.e. entries on the opposite side of the main diagonal are of the same magnitude and the off diagonal entries on the first sub/super diagonal are non zero.
    Prove or disprove the following fact:
    The Jordan normal form of A can contain a Jordan block of size at least 2 corresponding to a complex eigenvalue with nontrivial (i.e. nonzero) imaginary part.
    Hints:
    1. If a is an eigenvalue then the complex conjugate of a is also an eigenvalue (since the matrix is real).
    2. Since the matrix is unreduced, rank(A-xI)=n-1 for any eigenvalue x. I.e. there is only one eigenvector corresponding to x. This means that there is exactly one Jordan block corresponding to an eigenvalue. Of course the size of this Jordan block may be 1 or higher.
    3. If the claim is true, one way to prove it is to exhibit a real unreduced tridiagonal 4x4 matrix with 2 double eigenvalues: i and -i, each corresponding to a 2x2 Jordan block.
    4. I don't know whether the claim is true.


Assignment 5. (due 12/06/02) Turn in problems worth 80 points.
  1. 6.2.5.
  2. 6.2.6.
  3. 6.4.2.
  4. 6.3.9. (a,b,c)
  5. 6.3.12. (15)
  6. 6.3.14.
  7. 6.5.3. (5)
  8. 6.5.5.
  9. 6.6.1.
  10. 6.6.2.
The Final Project ... is here and is due 12/11/2002. The solution is here.