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

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.

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 |

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 [solutions] |

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 [solutions] |

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 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 [solutions] |

20 Apr 09 |
Midterm 1 in Pepper Canyon Hall, room 109 [solutions] |

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 [solutions] |

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 m≠nFourier 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 m≠nFourier 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 [solutions] |

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 BCsdefinition of symmetric (Hermitian) BCs proof that for convergent series f(x) = &Sigma A(_{n}X_{n}x), the coefficients are
A = ⟨_{n}X, _{n}f⟩/⟨X, _{n}X⟩_{n}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 infinitydefinition of Fourier coefficients and Fourier series for a function definition of pointwise, uniform, and mean-square convergence examples 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 [solutions] |

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 L norm^{2}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 [solutions] |

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 examples electrostatics 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 [solutions] |

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 [solutions] |

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 review |

10 Jun 09 |
Final exam: 8:00-11:00am WLH 2111 |

Last modified: 9 June 09.