Textbook: Elementary
Applied Partial Differential Equations,
by Richard Haberman, 3rd Edition, Prentice
Hall.
Prerequisites: Multivariable
Calculus and Linear Algebra are
required.
Analysis (211.405) is highly recommended
especially knowledge of
convergence of functions and series of functions. Experience
solving
Ordinary Differential Equations is
also highly recommended.
Grading Policy:
Homework
30%: All homework which is assigned is due the following
Monday .
It is expected that some reading and problems will be done Mondays and
Tuesdays.
More homework will be assigned during the first half of the semester than
the later
half and will be weighted accordingly.
Midterm
20%: This exam will be given in the fifth week
of class. It will consist
of problems similar to homework problems and some multiple choice.
Project
20%: This group project can be worked on with up to
three students,
with the quantity of work reflecting the number of students involved.
It should consist of a numerical solution of a PDE using any computer language
the students choose. It should also include an analysis of
the convergence
of the solution. Students who haven't used computers in the past
should work
with those who have.
Final
30%: This exam will be given during the
finals week. It will consist of
problems similar to homework problems, some multiple choice on general
knowledge of PDEs, and a question or two regarding the project.
Syllabus: (This syllabus
will be updated as the course progresses)
Week 1:
Introduction to PDEs
Derivation of the Heat Equation
(1.2 ,1.5)
Boundary Conditions and Steady State Solutions (1.3,1.4)
HW: Steady
State Solutions, Review Gradients
Handout #1 1-7, 1.2.3, 1.5.8, 1.4.1, 1.4.4, 1.4.6, 1.4.7 (a),
1.5.3, 1.5.8, 1.5.14, 1.5.19, 1.5.20
Week 2:
Real Vector Spaces (Defn, Spanning, Linear Indep, Dimension)
Linear Operators (2.2)
Separation of Variables (2.3.2-2.3.3)
HW: Handout #2, 2.2.1,
2.2.2, 2.2.4, 2.2.5, 2.3.1
Week 3:
Pointwise Convergence of Fourier Series (Handout
#3, 2.3.4, 2.3.5-6)
Integrating Series, L2
Inner Product (2.3.6, appendix to 2.3, Handout #4),
Self Adjoint Operators and Eigenfunctions
(Handout #3, 2.4)
HW: Read all of 2.1-2.4
(covers explicit techniques used to solve problems)
Do: Handout #3, Handout #4, 2.3.3 ab, 2.3.5, 2.3.7, 2.3.10abc,
2.4.1 abcd (verify convergence of these Fourier Series), 2.4.3
Week 4:
President's Day
The Convergence Theorem of
Fourier Series (3.2)
The Gibbs Phenomenon (3.3.1) and Uniform
Conv (Handout #5)
HW: Handout #5 1.2/ 2,5,8, 9ab,10 Read
through 1.3.4
Text read 3.1-3.3, do 3.2.1, 3.2.2, 3.2.4, 3.3.1, 3.3.7
Extra Credit : Graph the
Fourier Series up to 10 terms for various functions
using Mathematica or some other language to illustrate uniform
convergence.
Week 5:
Uniform Conv, (3.3-3.5, Handout #5)
Estimating and Maximizing Solns of the Heat Eqn
(Handout #6)
Differentiation and Integration of Series (3.3.2-3.3.4)
HW: Read 3.3.2-3.3.4 , Examples
Week 6:
Midterm Exam on 1.1-1.5, 2.1-2.4, 3.1-3.5, Handouts
#1-#6
Laplace's Equation and the Maximum
Principle (2.5.1-2)
Laplace's Eqn on the disk and the Mean Value Prop
(2.5.3-2.5.4)
HW: Handout
#7, 1-3, 2.5.2, 2.5.10, 2.5.11, 2.5.12, 2.5.13, 2.5.14
Week 7:
Vibrating Strings and Membranes (4.2, 4.5)
Travelling Waves, Dependence
and D'Alembert Solution (12.3)
Travelling Waves and Boundary Data (12.4-5)
HW : 12.3.1, 12.3.4, 12.3.5, 12.4.1, 12.4.2,12.4.4 , 4.4.9,
4.4.11,
4.4.12, Handout #8 on uniqueness of solutions of
the wave eqn.
Read Chapter 4 on solving the wave eqn with Fourier Series.
Spring Break
Week 8:
Fourier Series and Waves (4.4)
Fourier Series and Mathematica (Solns for Handout #7)
Finite Difference Equations (6.1-6.2)
HW : Handout #9 using
mathematica (may take a lot of time),
12.3.3, 12.3.6 , 6.2.1,
Week
9: Numerically Solving
the Heat Eqn, (6.3.2)
Propogation of Disturbances (6.3.3)
Difference Equations (Handout #10, 6.3.5, 6.3.6)
HW: 6.2.3, 6.2.4, Read
Handout #10 and consult
Sections 6.3.4-6.3.6 (which are a bit confusing)
Do problems 1-10 of the handout.
Choose project partners.
Week
10: Stability Analysis and Lax
Equivalency Theorem (Handout #10, 6.3.4)
Other Numerical Schemes (6.3.7-6.3.9) (Skip Random Walks)
Two Dimensional Heat Equation and Stability (6.4)
HW: Problems 11-13 of Handout #10 , 6.2.6,
6.2.7, 6.3.1, 6.3.3, 6.3.6
6.3.12, 6.3.13 Write up the Project's Aims
Week
11: Neumann Boundary Conditions (6.3.9)
Wave Equation solved Numerically (6.5)
Laplace's Equation with Jacobi and Gauss-Seidel (6.6)
Project HW: Write out the exact PDE including
boundary conditions
for the project. Be specific about all known functions
and unknown functions. Write up a consistent difference
equation (for PDE and boundary data and initial data)
and find the truncation errors.
Verify stability of this equation and discuss appropriate
grid ratios. Check that the truncation errors are the
same size given these grid ratios. If not, make better
approximations where appropriate and repeat the process.
Hand in all work. If your partners are
not available, do this
yourself and hand it in with your own name on it.
HW:
6.5.3, 6.5.4, 6.5.5, 6.5.6, 6.6.3 a,b
Extra Credit:
6.5.2, use any programming language you
wish.
Week 12:
Laplace's Equation with S-O-R iteration(6.6)
Von Neumann Stability with Complex eigenvalues
Hints for various Projects
HW: 6.6.1, 6.6.3c, 6.6.4,
and 6.6.5
Project HW: Write
the computer program for your project and debug it!
Week 13:
Sturm-Liouville and Helmholtz Eigenvalue Problems:
Part I: Green's Formula and Orthogonality (5.2-5.3.3, 7.4)
Part II : Uniqueness and Eigenfunctions (5.5, 7.5)
Part III: Rayleigh Quotient (5.6, 7.6)
HW:
Fill
out the Teaching Evaluations at http://www.jhu.edu/Merlin
Read 7.3 for the Helmholtz Equation on a rectangle
5.3.1, 5.4.2, 5.5.11 (reviews self adjointness Handout #3)
7.4.1, 7.5.1a, 7.5.2a, 7.6.1, 7.6.2a, 7.5.4
Please show me some code for your project during office hours Friday
if not before. I can help catch bugs.
Week 14: Heat Flow in
a Nonuniform Rod and a Survey of PDE (5.4)
Review Session May 7, 6-7pm Room 211 Krieger Hall
Projects are Due May 7, 6pm (Each
student should keep a
copy of the project to study for the final)
Final
Exam: Tuesday May 12, 2-5 pm, Room 304 Krieger
Hall
Covers all sections and Handouts listed above.
Types of questions that might be asked:
Linear Differential Equations, Homogeneous Equations
Identify, verify
Seperation of Variables
find product solutions,
maybe seperate 3 variables)
Fourier Series, Gibbs, (as in exam I)
Uniform Convergence
verify that something converges uniformly
Max principle, Mean Value Property
prove uniqueness,
identify solutions of Laplace's eqn given their graphs
Charateristics, reflection...
solve a given wave eqn using this method
verify that a soln of a wave equation can be found this way
Find a consistant difference equation
verify using Taylor series
Check stability for time dependant difference equation
Numerics for Laplace's equation
discuss relation to mean value prop
Eigenfunctions and solving the heat equation
given eigenfunctions find solution using series,
verify orthogonality
Eigenvalues, Rayleigh quotient
check if positive, give example where evalue isn't positive
find eigenvalue for a given eigenfunction
Self Adjoint Operators
verify a given linear operator is self adjoint
(by integrating by parts or using Green's formula)