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.

Textbook:

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

Exams:

• Midterm Exam 1:  Feb 5 (Wed),  11:10am–12:00pm  @ 807 DENN [Chp.1]
• Midterm Exam 2:  Mar 24 (Mon),  11:10am–12:00pm  @ 807 DENN [Chp.2]
• Final Exam:  Apr 30 (Wed),  4:00pm–6:00pm  @ 807 DENN [cumulative]

• In the midterm exams you will be allowed to bring in one side of a US Letter size (8.5''x11.0'') paper with notes on it. For the final you will be allowed both sides of a US Letter size paper.
• Calculators and other devices are not allowed in exams.
• Exam dates are absolutely firm. All students enrolled must plan to take exams at their scheduled times.

Grading Policy:

• Homework 10% : Homework will be assigned on Friday and due the following Friday at the beginning of the class. Late homework will not be accepted. Lowest two homework grades will be dropped. You are encouraged to discuss the course material and the assigned homework problems with your colleagues, but are responsible for writing up your own solutions.
• Quiz 10% : Quiz will be given every Friday. No makeup quiz will be given but lowest two grades will be dropped.
• Midterm Exam 1, 20%; Midterm Exam 2, 20%; Final Exam, 40%.

Student Data Form:  Form  FAQ

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Lecture Notes:

• Chapter 1 — Introduction and preliminaries (18pp)
• Chapter 2 — PDEs in rectangular coordinates (24pp)
• Chapter 3 — PDEs in cylindrical coordinates (9pp)
• Chapter 4 — PDEs in spherical coordinates (8pp)
• Chapter 5 — Fourier transforms (9pp)
• Chapter 6 — Method of characteristics (3pp)
• Chapter 7 — Green's functions (9pp)

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Syllabus:

1:	Wed,	Jan	8	Partial Differential Equations
				first day handout
2:	Fri,	Jan	10	Review of ODEs
3:	Mon,	Jan	13	Separation of Variables
4:	Wed,	Jan	15	Fourier Series
5:	Fri,	Jan	17	Even and Odd Functions
				Homework Set 1 Due  (problems)  Quiz 1  (solution)
	Mon,	Jan	20	Martin Luther King, Jr. Day
6:	Wed,	Jan	22	Fourier Sine and Cosine Series
7:	Fri,	Jan	24	Convergence, Complex Form of Fourier Series
				Homework Set 2 Due  (problems)  Quiz 2  (solution)
8:	Mon,	Jan	27	Parseval's Theorem and Mean Square Error
9:	Wed,	Jan	29	The Heat Equation
10:	Fri,	Jan	31	Steady State
				Homework Set 3 Due  (problems)  Quiz 3  (solution)
11:	Mon,	Feb	3	Homogeneous Boundary Conditions on a Slab
	Wed,	Feb	5	Midterm Exam 1  (more hw solutions, formulae, solutions)
12:	Fri,	Feb	7	Homogeneous Boundary Conditions on a Slab
13:	Mon,	Feb	10	Orthogonal Functions
14:	Wed,	Feb	12	Sturm-Liouville Eigenvalue Problems
15:	Fri,	Feb	14	Sturm-Liouville Eigenvalue Problems
				Homework Set 4 Due  (problems)  Quiz 4  (solution)
16:	Mon,	Feb	17	Nonhomogeneous Boundary Conditions
17:	Wed,	Feb	19	Asymptotic Behavior
18:	Fri,	Feb	21	Time-Dependent Sources and Boundaries
				Homework Set 5 Due  (problems)  Quiz 5  (solution)
19:	Mon,	Feb	24	The Wave Equation
20:	Wed,	Feb	26	The Vibrating String
21:	Fri,	Feb	28	Applications of Multiple Fourier Series
				Homework Set 6 Due  (problems)  Quiz 6  (solution)
	Mon,	Mar	3	↑
	Wed,	Mar	5	Spring Vacation
	Fri,	Mar	7	↓
22:	Mon,	Mar	10	Laplace's Equation and Applications
23:	Wed,	Mar	12	Bessel Functions
24:	Fri,	Mar	14	Fourier-Bessel Series
				Homework Set 7 Due  (problems)  Quiz 7  (solution)
25:	Mon,	Mar	17	The Vibrating Drumhead, Heat Flow in the Infinite Cylinder
26:	Wed,	Mar	19	Spherically Symmetric Solutions
27:	Fri,	Mar	21	Legendre Polynomials
	Mon,	Mar	24	Midterm Exam 2  (more hw solutions, formulae, solutions)
28:	Wed,	Mar	26	Laplace's Equation in Spherical Coordinates
29:	Fri,	Mar	28	Basic Properties of the Fourier Transform
				Homework Set 8 Due  (problems)  Quiz 8  (solution, alt-solution, supplement)
30:	Mon,	Mar	31	The Heat Equation for an Infinite Rod
31:	Wed,	Apr	2	The Method of Images
32:	Fri,	Apr	4	d'Alembert's Formula
				Homework Set 9 Due  (problems)  Quiz 9  (solution)
33:	Mon,	Apr	7	Heat Equation and Wave Equation
34:	Wed,	Apr	9	Method of Characteristics
35:	Fri,	Apr	11	Green's Functions
				Homework Set 10 Due  (problems)  Quiz 10  (solution)
36:	Mon,	Apr	14	The Green's Function for the Heat Equation
37:	Wed,	Apr	16	One Dimension
38:	Fri,	Apr	18	Properties of the Green's Function
				Homework Set 11 Due  (problems)  Quiz 11  (solution)
39:	Mon,	Apr	21	Review
	Wed,	Apr	30	Final Exam  (more hw solutions, formulae, solutions)