MATH 110 (Spring quarter 2009). Introduction to Partial Differential Equations

Instructor: David A. Meyer
Office hours: AP&M 7256, MTh 1:00pm-2:00pm, or by appointment
Lecture: Warren Lecture Hall, room 2111, MWF 9:00am-9:50am
Email: dmeyer "at" math "dot" ucsd "dot" edu

Teaching assistant: Gordon Honerkamp-Smith
Office hours: AP&M 6414, TW 11:00am-12:00pm, or by appointment
Section A01: AP&M, room B412, Th 5:00pm-5:50pm
Section A02: AP&M, room B412, Th 4:00pm-4:50pm
Email: ghonerka "at" math "dot" ucsd "dot" edu

Course description

This course is an introduction to partial differential equations (PDEs). PDEs originated as the mathematical description of various physical systems, e.g., heat diffusion, vibrations of a string or membrane, fluid flow, the motion of an electron, etc. In this course we will concentrate on the heat equation, the wave equation, and the Laplace equation, studying their solutions, and also qualitative properties of their solutions.

The official prerequisites for this course are ordinary differential equations (MATH 20D) and linear algebra (MATH 20F), but a thorough understanding of (multivariable) calculus (MATH 20ABCE) is also necessary. Everything in Appendices A1-A4 of the textbook, W. A. Strauss, Partial Differential Equations: An Introduction, 2nd ed. (New York: Wiley 2008), should be familiar. We will cover most of chapters 1, 2, 4, 5 and 6.

There will be weekly homework assignments, due in section on Thursdays, or before then in the drop box on the sixth floor of AP&M. Students are allowed to discuss the homework assignments among themselves, but are expected to turn in their own work — copying someone else's is not acceptable. Homework scores will contribute 15% to the final grade.

There will be two midterms, approximately in the fourth and eighth weeks of the quarter. The final is scheduled for 8:00am Wednesday 10 June 2009. Scores on the two midterms and final will contribute 25%, 25% and 35% to the final grade, respectively. There will be no makeup tests.

Related events

7 May 09 panel discussion on applying to (math) graduate schools
AP&M 6402, 12:00pm-1:30pm
14 Apr 09 Leonard Sander will speak on "A generalized Cahn-Hilliard equation for biological applications"
AP&M 5829, 2:00pm-2:50pm

Syllabus (subject to modification)

30 Mar 09 §1.1. What is a PDE?
         general first order PDE; examples
         general second order PDE; examples
         differential operators
                 linearity; Superposition Principle
§1.2. First order linear equations
         with constant coefficients
                 directional derivative solution
HWK (for W 1 Apr 09).
         Read §1.1-1.3.
1 Apr 09                  change of variables solution
                 characteristic curves
                 initial conditions
         with variable coefficients
                 directional derivative solution
                 characteristic curves
§1.3. Flow and diffusion
3 Apr 09          derivation of the transport equation
                 review of differentiatlon of an integral w.r.t. endpoint
         experimental demonstration of diffusion
         derivation of the diffusion equation
HWK (due Th 9 Apr 09).
         §1.1: 3, 11, 12
         §1.2: 4, 5, 8, 9
         §1.3: 5, 6, 8

6 Apr 09 §1.3. Vibration
         derivation of the wave equation
         the Laplace equation
§1.4. Initial and boundary conditions
         Dirichlet conditions
         Neumann conditions
derivation of the minimal surface equation
         review of the surface area integral
8 Apr 09          variational principle
the Schrödinger equation
§1.5. Well-posed problems
         existence, uniqueness and stability
         diffusion and anti-diffusion as examples
10 Apr 09 §2.1. The wave equation in 1+1 dimensions
         solution by change of variables
         interpretation of solution in terms of characteristics
         solution of initial value problem
HWK (due Th 16 Apr 09).
         §1.4: 1, 3
         §1.5: 1, 4, 5
         §2.1: 2, 5, 8, 10

13 Apr 09          examples
§2.2. Causality and energy
         domains of dependence and influence
         conservation of energy
15 Apr 09 §2.3. Diffusion on an interval
         Maximum (and minimum) Principle
                 physical intuition
                 idea of proof
         the Maximum Principle implies uniqueness
17 Apr 09          the Maximum Principle implies stability
§2.4. Diffusion on the real line
         invariance properties
HWK (due Th 23 Apr 09).
         §2.2: 3, 5
         §2.3: 2, 5, 6
         §2.4: 4, 9, 10, 16, 18

20 Apr 09 Midterm 1 in Pepper Canyon Hall, room 109
22 Apr 09          Green's function solution of initial value problem
24 Apr 09          properties of the Green's function
         integration of the Gaussian
§2.5. Comparison of the wave and diffusion equations and their solutions
§4.1. Separation of variables with Dirichlet boundary conditions
         the wave equation on the interval [0,l]
HWK (due Th 30 Apr 09).
         §2.5: 2, 3
         §4.1: 1, 2, 3, 4

27 Apr 09          the diffusion equation on the interval [0,l]
         the analogy with eigenfunction problems in linear algebra
         proof that the eigenvalues of the Laplacian on the interval are positive
29 Apr 09 §4.2. Neumann boundary conditions
         proof that the eigenvalues of the Laplacian on the interval are nonnegative
         computation of the eigenvalues and eigenvectors
         solutions of the wave and diffusion equations
§5.1. Coefficients in Fourier series
         analogy with finite dimensional linear algebra
                 eigenvectors for distinct eigenvalues are orthogonal
                 expansion of arbitrary vector as linear combination of projections
         integration as inner product
         sin(nπx/l) is orthogonal to sin(mπx/l) for mn
         Fourier sine coefficients
1 May 09                  calculation of Fourier sine series for 1 and x on the interval [0,l]
         cos(nπx/l) is orthogonal to cos(mπx/l) for mn
         Fourier cosine coefficients
                 calculation of Fourier cosine series for 1 and x on the interval [0,l]
         extension of Fourier sine and cosine series beyond the interval [0,l]
HWK (due Th 7 May 09).
         §4.2: 1, 4
         §5.1: 2, 5, 8
         §5.2: 2, 4, 8, 11

4 May 09          the full Fourier series on the interval [-l,l]
§5.2. Even, odd, periodic and complex functions
         definition and properties of even and odd functions
         definition and properties of periodic functions
         boundary conditions for periodic functions
         inner product for complex functions
         the complex form of the full Fourier series on the interval [-l,l]
6 May 09 §5.3. Orthogonality and general Fourier series
         proof that eigenfunctions of -X'' = λX are orthogonal for Dirichlet, Neumann, and periodic BCs
         definition of symmetric (Hermitian) BCs
         proof that for convergent series f(x) = &Sigma AnXn(x), the coefficients are An = ⟨Xn, f⟩/⟨Xn, Xn
         orthogonal eigenfunctions in two dimensional eigenspaces
         proof that for symmetric BCs, the eigenvalues are real and the eigenfunctions can be chosen to be real
8 May 09 §5.4. Completeness
         -X'' = λX has an infinite number of real eigenvalues, increasing to infinity
         definition of Fourier coefficients and Fourier series for a function
         definition of pointwise, uniform, and mean-square convergence
         Uniform Convergence Theorem
         Mean-square Convergence Theorem
         definition of jump discontinuity
HWK (due Th 14 May 09).
         §5.3: 1, 2, 3, 6, 9
         §5.4: 2, 4, 7, 8, 12, 16

11 May 09          definition of piecewise continuity
         Pointwise Convergence Theorem
         examples of various kinds of convergence
                 some infinite series for π
                 the failure of term-by-term differentiation
         definition of the L2 norm
         proof of Least-Square Approximation of functions with finite norm
13 May 09          Bessel's inequality
         Parseval's equality
§5.5. Completeness and the Gibbs phenomenon
         derivation of the Dirichlet kernel
15 May 09          proof of pointwise convergence for continuous functions with continuous derivative
         modification of pointwise convergence proof for piecewise continuous functions
         description of Gibbs phenomenon
HWK (due Th 21 May 09).
         §5.5: 2, 4, 12, 13, 14

18 May 09 Midterm 2 in Center Hall, room 105
20 May 09 Midterm 2 solutions
22 May 09 §6.1. Laplace's equation
         definition of harmonic function
         Poisson's equation
                 differentiable functions of a complex variable
                 Brownian motion
                 Extra Credit problem on random walks
         Maximum Principle
                 idea of proof
                 partial proof
         uniqueness of the Dirichlet problem
HWK (due Th 28 May 09).
         §6.1: 1, 10, 11, 12, 13

25 May 09 no lecture; Memorial Day holiday
27 May 09 no lecture
29 May 09          translation invariance
         rotation invariance
         Laplacian in polar coordinates
HWK (due Th 28 May 09).
         §6.1: 2, 3, 5, 9
         §6.2: 1, 2, 4, 7
         §6.3: 1, 2, 4

1 Jun 09                  circularly symmetric harmonic functions
         Laplacian in spherical coordinates
                 spherically symmetric harmonic functions
§6.2. Solving the Laplace equation in rectangular domains
         example of a rectangular domain with non-homogeneous boundary conditions
3 Jun 09 §6.3. Poisson's formula
         solving the Laplace equation in a disk with inhomogeneous boundary conditions
         derivation of the Poisson formula
5 Jun 09          interpretation of the Poisson formula
         the mean value property
         the maximum principle
         differentiability of harmonic functions
10 Jun 09 Final exam: 8:00-11:00am WLH 2111

Last modified: 9 June 09.