Math 211: Applied Partial Differential Equations and Complex Variables (Fall 2009)

Mathematical methods for solving problems in linear partial differential equations: linear operators and adjoint problems, eigenfunction expansions, Fourier series, Sturm-Liouville problems, orthogonal functions and generalized Fourier series. Solutions via Green's functions. Complex variables for contour integrals and solutions via integral representations. Integral transforms: Fourier and Laplace transforms.

Textbook: Applied Partial Differential Equations (4th ed), by Richard Haberman, Prentice Hall (2003)

Prerequisites

Schedule

MWF 3:05-3:55 PM, Room 259 Physics Building

Instructor

Office hours

Problem Sheets

• Problem set 1
• Problem set 2
• Problem set 3
• Problem set 4
• Problem set 5
• Problem set 6
• Problem set 7
• Problem set 8
• Problem set 9
• Problem set i
• Problem set 10

Course materials

• Course outline/syllabus

• Review sheets
  • Math Basics Summary
  • 1st order ODE and complex numbers
  • Integration Techniques
  • Trigonometric Fourier Series
  • Undetermined Coefficients
  • Vector Calculus
  • Infinite Series
• Lecture notes
  • Lecture 1: Linear algebra review
  • Lecture 2: Key elements of Linear algebra (typo fixed: eigenvalues of adjoint, final version 9/1/09: 8:22pm)
    • Note on "adjoint eigenvalues"
  • Lecture 3: Key elements of L2 inner products
  • Lecture 4: Properties of Fourier Series
  • Lecture 6: Eigen-expansions for solving inhomogeneous ODE BVP
  • Lecture 7: Sturm-Liouville theory
  • Lecture 8: The Fredholm Alternative Theorem
  • Lecture 10: Fredholm Integral Equations
  • Lecture 11a: Motivation for Variation of Parameters
  • Lecture 11: Variation of Parameters for 2nd order ODE BVP
  • Lecture 12: The Heaviside Step Fcn and Dirac Delta Fcn
  • Lecture 14: Green's functions -- further notes
  • Lecture 14: Intro to Separation of Variables for PDEs
  • Lecture 15: Separation of Variables for inhomogeneous PDEs (1.1 sign error fixed)
  • Lecture 24: Basics of Bessel functions
  • Lecture 26: Laplacian in polar coords + Classic ODE Eigenproblems from separation of variables
  • Lecture 28: Intro to Complex Variables
  • Lecture 28: Branch cuts and Multivalued functions
    • Branch cut on Wikipedia
    • Branch cut on Mathworld
  • Lecture 30: Review of line integrals and intro to contour integrals
  • Lecture 32: The complex Fourier series
  • Lecture 34: Complex Contour Integrals
    • Contour integration: Wikipedia
    • Contour integration: MathWorld

• Test 1: Weds Sep 30, 2009, Solution of inhomogeneous ODE BVP via eigenfunction expansions. Fourier series. Adjoint eigenvalue problems. Sturm-Liouville problems. Integral equations: eigenvalue and inhomogeneous problems. The Fredholm alternative theorem for existence/uniqueness/non-existence.
  Optional review session: Sunday Sep 27, 2009, 3:00-5:00pm, Room 259 Physics. No new material will be covered. Questions will be answered and examples worked out. Feel free to come to all/any-part-of the review sesssion block time.
  • Math 211 Fall 2008 Test 1
  • Math 211 Fall 2009 Test 1
  • High 88, Low 30, Avg 67
• Test 2: Mon Nov 9, 3:05(OASAEGH)-4:00pm. Green's functions for ODE BVPs (9.3), separation of variables and eigenfunction expansions for PDEs (2.3, 2.4, 8.4, 8.6), multi-dimensional problems (2.5, 7.2--7.7), the material covered on Homeworks 5-8.
  Optional review session: Friday, Nov 6, 4:05-6:00 pm, Room 259 Physics. No new material will be covered. Questions will be answered and examples worked out. Feel free to come to all/any-part-of the review sesssion block time.
  • Math 211 Fall 2008 Test 2
  • Math 211 Fall 2009 Test 2
  • High 82, Low 35, Avg 62, Distribution: 80's: 1, 70's: 5, 60's: 4, 50's: 6, less: 2
• Test 3: Mon Nov 23, 3:05(OASAEGH)-4:00pm. Complex Contour integration, parametrization, complex analytic functions, path independence, Cauchy's theorem, Cauchy's integral formula, residues, log branch cuts, evaluation of real integrals, Jordan curve lemma. Problems from Homework 10, examples from lectures 30 (Nov 2) through lecture 37 (Nov 18).
  Optional review session: Saturday, Nov 21, 10:00-11:59 am, Room 259 Physics. No new material will be covered. Questions will be answered and examples worked out. Feel free to come to all/any-part-of the review sesssion block time.
  • Math 211 Fall 2008 Test 3
• FINAL EXAM: Thurs, December 10, 2009, 2:00-5:00pm