Math 454   Boundary Value Problems for Partial Differential Equations (Section 002)

Instructor Manabu Machida
Office 3836 East Hall
Office hours  MWTh 2pm–3pm (and by appointment)

Meeting times  MWF 11am–12pm at 807 DENN

Prerequisites: Math 216, 256, 286, or 316. (Permission required after credit earned in Math 354 or 450.)

Background and Goals:  This course is devoted to the use of Fourier series and other orthogonal expansions in the solution of boundary-value problems for second-order linear partial differential equations. Emphasis is on concepts and calculation. The official prerequisite is ample preparation.

Content:  Classical representation and convergence theorems for Fourier series; method of separation of variables for the solution of the one-dimensional heat and wave equation; the heat and wave equations in higher dimensions; spherical and cylindrical Bessel functions; Legendre polynomials; methods for evaluating asymptotic integrals (Laplace's method, steepest descent); Fourier and Laplace transforms; and applications to linear input-output systems, analysis of data smoothing and filtering, signal processing, time-series analysis, and spectral analysis.

Alternatives:  Both Math 455 and 554 cover many of the same topics but are very seldom offered.

Subsequent Courses:  Math 454 is prerequisite to Math 571 and 572. Although it is not a formal prerequisite, it is good background for Math 556.

Partial Differential Equations and Boundary-Value Problems with Applications
by Mark A. Pinsky
American Mathematical Society 3rd ed. 2011  (ISBN: 9780821868898)


Grading Policy:

Student Data Form:  FormFAQ

Lecture Notes:


1:  Wed,Jan8   Partial Differential Equations
  first day handout
2:  Fri,Jan10   Review of ODEs
3:  Mon,Jan13   Separation of Variables
4:  Wed,Jan15   Fourier Series
5:  Fri,Jan17   Even and Odd Functions
  Homework Set 1 Due  (problems)  Quiz 1  (solution)
Mon,Jan20   Martin Luther King, Jr. Day
6:  Wed,Jan22   Fourier Sine and Cosine Series
7:  Fri,Jan24   Convergence, Complex Form of Fourier Series
  Homework Set 2 Due  (problems)  Quiz 2  (solution)
8:  Mon,Jan27   Parseval's Theorem and Mean Square Error
9:  Wed,Jan29   The Heat Equation
10:  Fri,Jan31   Steady State
  Homework Set 3 Due  (problems)  Quiz 3  (solution)
11:  Mon,Feb3   Homogeneous Boundary Conditions on a Slab
Wed,Feb5   Midterm Exam 1  (more hw solutions, formulae, solutions)
12:  Fri,Feb7   Homogeneous Boundary Conditions on a Slab
13:  Mon,Feb10   Orthogonal Functions
14:  Wed,Feb12   Sturm-Liouville Eigenvalue Problems
15:  Fri,Feb14   Sturm-Liouville Eigenvalue Problems
  Homework Set 4 Due  (problems)  Quiz 4  (solution)
16:  Mon,Feb17   Nonhomogeneous Boundary Conditions
17:  Wed,Feb19   Asymptotic Behavior
18:  Fri,Feb21   Time-Dependent Sources and Boundaries
  Homework Set 5 Due  (problems)  Quiz 5  (solution)
19:  Mon,Feb24   The Wave Equation
20:  Wed,Feb26   The Vibrating String
21:  Fri,Feb28   Applications of Multiple Fourier Series
  Homework Set 6 Due  (problems)  Quiz 6  (solution)
Wed,Mar5   Spring Vacation
22:  Mon,Mar10   Laplace's Equation and Applications
23:  Wed,Mar12   Bessel Functions
24:  Fri,Mar14   Fourier-Bessel Series
  Homework Set 7 Due  (problems)  Quiz 7  (solution)
25:  Mon,Mar17   The Vibrating Drumhead, Heat Flow in the Infinite Cylinder
26:  Wed,Mar19   Spherically Symmetric Solutions
27:  Fri,Mar21   Legendre Polynomials
Mon,Mar24   Midterm Exam 2  (more hw solutions, formulae, solutions)
28:  Wed,Mar26   Laplace's Equation in Spherical Coordinates
29:  Fri,Mar28   Basic Properties of the Fourier Transform
  Homework Set 8 Due  (problems)  Quiz 8  (solution, alt-solution, supplement)
30:  Mon,Mar31   The Heat Equation for an Infinite Rod
31:  Wed,Apr2   The Method of Images
32:  Fri,Apr4   d'Alembert's Formula
  Homework Set 9 Due  (problems)  Quiz 9  (solution)
33:  Mon,Apr7   Heat Equation and Wave Equation
34:  Wed,Apr9   Method of Characteristics
35:  Fri,Apr11   Green's Functions
  Homework Set 10 Due  (problems)  Quiz 10  (solution)
36:  Mon,Apr14   The Green's Function for the Heat Equation
37:  Wed,Apr16   One Dimension
38:  Fri,Apr18   Properties of the Green's Function
  Homework Set 11 Due  (problems)  Quiz 11  (solution)
39:  Mon,Apr21   Review
Wed,Apr30   Final Exam  (more hw solutions, formulae, solutions)

Valid HTML 4.01! Valid CSS!