MATH 423: Elementary Partial Differential Equations, Spring 2013
Instructor: Shabnam Beheshti
Office: #214, Hill Centre, Busch Campus
e-mail: beheshti[at]math[dot]rutgers[dot]edu
Lecture: Tue/Thu 13:40-15:00 ARC-205
Office Hours: Mon 9-10, Tue/Thu 15:00-16:00, or by appointment
Course Information
Required Course Textbook: Applied Partial Differential Equations with Fourier Series and
Boundary Value Problems,
by Richard Haberman, 4th edition, Prentice Hall, 2003 (ISBN: 0130652431; ISBN13: 978-0130652430).
Recommended Course Textbook: Introduction to Partial Differential Equations, by Walter Strauss, 2nd edition, John Wiley & Sons, 2008 (ISBN: 0470054565; ISBN-13: 978-0470054567)
A Remark on the Course Textbook(s): I will be blending the approaches of the above two books in my lectures and will put current editions of both on reserve in the Mathematics Library (ground floor of the Hill Centre). PLEASE NOTE: Although a newer edition of the textbook is now available, I will be referring to exercises from the 4th edition. Should you wish to purchase other editions of these texts, please consult the reserves to be sure you are completing the correct assigned exercises.
The prerequisites for 640:423 are Differential & Integral Calculus, Multivariable Calculus, Ordinary Differential Equations. Lectures and coursework will involve computation as well as proofs, so courses such as 640-300 (Math Reasoning), 311/312 (Real Analysis I/II) are also extremely helpful.
Our (rather ambitious) Course Syllabus:
- What are PDEs?
- Heat equation: Derivation, Steady-state, Boundary conditions,
Method of separation of variables, Initial value problems (Dirichlet
and Neumann boundary conditions.) Energy.
- Laplace equations on recatangles, discs. Qualitative properpties
of Harmonic functions (Mean Value Property, Maximum Principle,
Uniquness of the solution). Well-posedness of a PDE.
- Wave equation: Derivation of the PDE. Boundary conditions
(Dirichlet. Neumann and Robin using a spring-mass system) Initial and
boundary value problems. Notions such as n-th modes, natural
frequencies, standing waves, travelling waves, energy.
- Fourier Series: Convergence Theorem, Odd/Even/Periodic extensions.
Fourier Sine/Cosine Series. Continuity and convergence of Fourier
(Sine/Cosine) Series. Term-by-term integration/differentiation of
Fourier Series. Method of eigenfunction expansion (for both
homogeneous and non-homogeneous PDEs.)
- Infinite Domain Problems: Fourier transform (as a limit of
Fourier Series viewed as Riemann sums.) Fourier transforms of
Gaussian, Dirac delta function, etc. Heat equation on R. Duhamel
Principle for nonhomogeneous Heat equation. Wave equation on R,
D'Alembert's formula. Fourier Sine/Cosine transforms. Laplace equation
on Half plane, Poisson's formula, Green's function.
- Sturm-Liouville problems. Robin Boundary Condition, heat flow on
non-uniform rod. Self-adjoint operators, Rayleigh quotient.
- Further Topics: Higher dimensional PDEs (on rectangular regions, on disks)
There will be two examinations during the term, and a cumulative final. All of my quizzes and exams will be written so that you can complete them without a calculator. Please contact me TWO WEEKS prior to any exam if you have an excused absence. Homework and quizzes will typically be due weekly.
While I do not require attendance, you will be responsible for all the
material covered in lecture, including announcements, changes to
homework as well as the corresponding sections assigned from the text.
Grading Scheme: 100 HW/Quizzes, 200 Midterms (100 each), and 200 Final.
It is also important for you to know the Rutgers University Academic Integrity Policy.
Supplementary Reading
1. Vector Calculus, by J. Marsden and A. Tromba. This, or any
other calculus textbook covering Green's, Gauss'/Divergence and Stokes'
Theorem is highly suggested for a review of chain rule, directional
derivatives, gradient, line and surface integrals. Rutgers uses Calculus: Early Transcendentals, by Rogawski (2nd edition).
2. Elementary Differential Equations, by W. Boyce and R. DiPrima.
Often used as a course textbook for ODEs.
3. Introduction to Real Analysis, by R. Bartle and D.
Sherbert. Useful for basic calculus proofs on the real line.
4. Partial Differential Equations, by W. Strauss. A classic
textbook in PDE, on which much of course lectures are based, as well as
the homework sets.
5. The Heat Equation, by D.V. Widder. An advanced
undergraduate/beginning graduate student text assuming no prior knowledge
on heat conduction or PDE. It does require background in complex
varaibles, Lebesgue integration and Laplace transform theory.
6. 124/215 Lecture Notes, by V. Grigoryan. Notes for a PDE-Fourier Series-Numerical Methods course sequence at UCSB, on which most of our handouts will be based.
Suggested Exercises
Changed homework/quiz due dates will be announced in
class. There will be no late homeworks or quizzes without a
documented reason (medical note, varsity sports event, etc.).
- Lecture 1 Course outline, what are PDEs? review of
prerequisites, (non)linear (in)homogeneous operators
- Lecture 2 Review of operators, what are initial &
boundary conditions (IC/BC), well-posedness of PDE
- Problem Set
0
- Homework 0 due Thu 31 Jan: Problem
Set
0, #2b, 4, 8, 9, 11c
- Lecture 3 Strauss' Appendix A, changes of
coordinates/chain rule, harmonic functions, method of characteristics-1
- Lecture 4 Method of characteristics-2, motivation for 1D
diffusion equation
- Problem Set
1
- Homework 1 due Thu 07 Feb: Problem
Set 1, #1, 3, 5, 7, 14, 17, 18d, 19 (12.2.2 only)
- Lecture 5 Fick's Law, Derivation of the 1D/3D Heat
Equations, Fourier's Law
- Lecture 6 ICs/BCs for the Heat Equation, Equilibrium
state, Steady state calculations
- Problem Set
2
- Homework 2 due Thu 14 Feb:
Problem Set 2, #4, 7, 9 (1.2.8), 11 (1.3.2), 12 (1.4.1d,f,h, 1.4.5,
1.4.7a), 14 (1.5.5, 1.5.11)
- Lecture 7 Separation of
variables for 1D Heat Equation with homogeneous BCs
- Lecture 8 Separation of variables for 1D Heat Equation
with homogeneous BCs part II, Fourier sine and cosine series
- Problem Set
3
- Homework 3 due Thu 21 Feb: Problem
Set 3, #3d, 4 (2.3.2d, 2.3.3d, 2.3.8), 6 (2.4.3), 8
(optional: 2.5.1c, 2.5.2)
- Lecture 9 1D Heat Equation with periodic BCs of mixed
type, separation of variables in other domains
- Lecture 10 Laplace's Equation on a rectangle, Laplace's
Equation on a disk
- Problem Set
4
- Homework 4 Problem
Set 4 is Exam practice
Exam #1 will be on Thu 28 Feb (in
class), covering lectures 1-10.
- Lecture 11 Well-posedness of PDE, revisited: example from
ODE
- Lecture 12 Exam 1
- Problem Set
5
- Homework 5 due Tue 12 Mar: Problem
Set 5, #1 (2.5.10, 2.5.14), 3, 4a
- Lecture 13 Well-posedness of 1D Heat Equation I: energy
method (uniqueness, L^2 stability),
- Lecture 14 Well-posedness of 1D Heat Equation
II: maximum principle method (uniqueness, uniform stability)
- Problem Set
6
- Homework 6 due Tue 26 Mar: Problem
Set 6, #2, 4, 6 (3.2.1f, 3.2.3), 7 (3.3.1e, 3.3.5b, 3.3.10,
3.3.13)
- Lecture 15 Invariance properties and Green's function for Heat Equation on the line;
Other Heat Equations (half-line/sources handout)
- Lecture 16 Fourier series (full, cosine, sine),
even/odd extensions, Fourier's Theorem
- Problem Set
7
- Homework 7 due Thu 04 Apr: Problem
Set 7, #1 (3.3.15), 4, 5 (3.4.6, 3.4.12), 7,
(3.5.3), 8 (3.6.2), 9 (|x| only)
- Lecture 17 Continuity of Fourier series, Eigenfunction
expansion method, term-by-term
differentiation
- Lecture 18 Term-by-term integration, Complex Fourier
series, complex orthogonality
- Problem Set
8
- Homework 8 Problem Set 8 is
Exam practice
- Lecture 19 General Fourier Series, Notions of
convergence, Three Convergence Theorems, Bessel's
Inequality
- Lecture 20 Completeness, Parseval's Equality, pointwise
convergence of Fourier Series, Dirichlet kernel
- No office hours on Mon 08
Apr
Exam #2 will be on Tue 09 Apr (in
class), covering lectures 10-20.
- Lecture 21 Exam 2
- Lecture 22 Wave Equation I: Derivation of 1D/3D
Wave Equations, separation of variables, harmonics, standing/traveling
waves
- Problem Set
9
- Homework 9 due Tue 23 Apr:
Problem Set 9, #2, 3 (4.3.2), 4 (4.4.1b, 4.4.7), 5, 9
(12.5.2)
- Lecture 23 1D Wave Equation II: Infinite string &
d'Alembert solution, characteristic coordinates, domains of
dependence/influence
- Lecture 24 Well-posedness of 1D Wave Equation:
Invariance, energy conservation, uniqueness,
- Problem Set
10
- Homework 10 due Tue 30 Apr: Problem
Set 10, #4, 5, 6 (4.4.13), 8
- Lecture 25 Comparison of Wave/Heat Equations
, Wave Equation on the disc & the Bessel function
- Lecture 26 Fourier transforms, distributions, Heisenberg Uncertainty Principle
- Problem Set
11
- Homework 11 Problem
Set 11 is Exam practice
- Problem Set
12
- Homework 12: Problem Set 12 is Exam practice as well as extra topics for SUMMER VACATION!
- Here are some selected solutions for
Problem Sets 9-12
Final Exam is on Mon 13 May in ARC-205, 8:00 AM - 11:00 AM.
Formula Sheet