Math 211: Applied Partial Differential Equations and Complex Variables

Math 211: Applied Partial Differential Equations and Complex Variables (Fall 2011)

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

Background in linear algebra and ordinary differential equations: [Math 104 and 131], or [Math 107 and 108], or equivalents.

Schedule

MWF 1:30-2:20 PM, Room 119 Physics Building

Instructor

Thomas Witelski, Associate Professor, Dept of Math

Office hours

Tuesdays, 11:00am-2:00pm, Room 295 Physics Building, or by email request for an appointment for other times.

Problem sheets

Problem set 1
Problem set 2
Problem set 3
Problem set 4 (Typo on Q4b4 fixed)
Problem set 5
Problem set 6
Problem set 7
Problem set 8
Problem set i (not required, will not be collected)
Problem set 9
Problem set 10

Course materials

Course outline/syllabus
Review sheets
- Math Basics Summary
- 1st order ODE and complex numbers
- Techniques of integration review
- Trig. Fourier series review
- Undetermined Coefficients: method for solving LCC ODE's
- Multi-variable/vector calculus
- Infinite series
Lecture notes
- Lecture 1: review of essential linear algebra
- Lecture 2: generalizing linear algebra
- Lecture 3: intro to Fourier series
- Lecture 4: convergence of Fourier series
- Lecture 5adjoint: ODE adjoint operators
- Lecture 5: ODE eigevalue problem intro
- Lecture 5lccce: LCC and CE eqns
- Lecture 6: Solving InHom. ODE BVP via eigen-expansions
- Lecture 7: Sturm-Liouville theory
- Lecture 8: The Fredholm Alternative Theorem (FAT)
- Lecture 9: Intro to Integral Equations
- Lecture 10: Fredholm Integral Equations (typos corrected, Thanks to Ezgi and Qing)
- Lecture 11: Review of Variation of Parameters
- Lecture 13: Green's functions
- Lecture 14: Intro to separation of variables for PDE's
- Lecture 21: Linear operator theory for the Laplacian
- Lecture 24: Bessel functions
- Lecture 26: Helmholtz eigenvalue problems: Laplacian/ODEs for different coord systems
- Lecture 27: Green's functions for PDEs
- Lecture 28: Intro to complex variables
  - Multi-valued functions: evaluating functions with branch points and branch cuts
- Lecture 29: Intro to complex contour integrals
- Lecture 33: Uses of infinite series...
- Lecture 35: Summary of contour integration
- Lecture 36: Intro to Fourier transforms
- Lecture 39: Other integral transforms
  - Integral transforms: Fourier and Laplace summary sheet
Tests
- Test 1: Weds, Oct 5, in class -- 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: -- 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
  - Math 211 Fall 2010 Test 1
  - Math 211 Fall 2011 Test 1
  - Results: Avg 62, High 90, Low 40s: 90s:1, 80s:2, 70s:3, 60s:5, 50s:5, 40s:5
- Test 2: Weds, Nov 9 -- Green's functions for ODE BVPs (9.3), separation of variables and eigenfunction expansions for PDEs (2.3, 2.4, 8.2-8.4, 8.6), Helmholtz equations, Bessel functions, multi-dimensional problems (2.5, 7.2--7.10), material covered on Homeworks 5-8, Lectures 11-26.
  Optional review session: Sunday, Nov 6, 4-6pm, Room 119 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 (historical data: 90s: 2, 80s: 3, 70s: 1, less: 2)
  - Math 211 Fall 2009 Test 2 (historical data: High 82, Low 35, Avg 62)
  - Math 211 Fall 2010 Test 2 (historical data: avg 58, 90s:0, 80s:1, 70s:3, 60s:8, 50s5, 40s:4, 30s:3)
  - Math 211 Fall 2011 Test 2 avg 59, 90s:1, 80s:3, 70s:1, 60s:5, 50s:4, 40s:4, less: 3
- Test 3: Weds, Dec 7 -- 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. Material from Homeworks 9, 10 and all lectures on complex variables.
  Optional review session: Sunday Dec 4, 4-6pm room 119 physics
  - Math 211 Fall 2008 Test 3 (historical data: 60s: 3, 90s: 2, 100: 3)
  - Math 211 Fall 2009 Test 3 (historical data: 90s:3, 80s:4, 70s:4, 60s:4, less:3, Average:74)
  - Math 211 Fall 2010 Test 3: avg 73, 90s: 2, 80s: 8, 70s: 6, 60s: 3, 50s: 1, less: 3
  - Math 211 Fall 2011 Test 3: avg 72, 100s:1, 90s: 4, 80s: 4, 70s: 3, 60s: 4, 50s: 2, less: 3
- Final Exam: THURS, Dec, 15, 9:00-Noon
  - 2010 results: avg 71, high 97, low 30s. Distribution: 90s: 5, 80s: 4, 70s: 4, 60s: 5: 50s: 2, less:3
  - 2011 results: avg 63, high 95, low 30s. Distribution: 90s: 1, 80s: 3, 70s: 2, 60s: 7, 50s: 3, less: 5
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