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Partial Differential Equations
Math 4340-1
Louisiana State University
Fall Semester, 2014
Prof. Stephen Shipman
Place: Room 135 of Lockett Hall
Time: Monday, Wednesday, and Friday, from 10:30 to 11:20
Office: Room 314 of Lockett Hall
Telephone: 225/578-1674
Email: shipman@math.lsu.edu
Office Hours: Monday 1:30-4:00 and Thursday 1:30-3:30, or by appointment
For a PDF version of the basic course information on this page,
click here: 4340syl.pdf.
Textbook
Introduction to Partial Differential Equations with MATLAB, by Jeffery M. Cooper.
Prerequisite
The material assumes knowledge of ODEs, linear algebra, and vector calculus.
MATH 2057(vector calc) and 2090(ODE with linear algebra); or MATH 2057(vector calc) and 2065(ODE); or MATH 2070(math methods in engr) and 2085(linear algebra)
Course Content
The course will touch chapters 2 to 9 of the textbook, some in more detail than others. The main topics are
- Chapter 2: First-order equations; method of characteristics.
First-order PDE can be reduced to ODEs, and arise in conservation laws, wave dynamics in many applications, and shock waves.
- Chapter 3: Diffusion and heat (parabolic PDE).
There are lots of ways to solve and analyze the diffusion of heat, chemicals, or other things in different types of domains, depending on the domain and the initial and boundary conditions.
- Chapter 4: Boundary-value problems.
This is where decomposition of solutions into infinitely many "eigenfunctions", or harmonics, becomes prominent.
- Chapter 5: The wave equation (hyperbolic PDE).
The linear wave equation describes waves in acoustics, elasticity, electromagnetics, and many other physical systems. How to solve it depends on the domain and initial and boundary conditions.
- Chapter 6: Fourier Transforms (harmonic analysis).
This chapter discusses the many different ways in which the solution to a PDE can be decomposed into eigenfunctions, or harmonics.
- Chapter 7: Dispersion and the Schrödinger equation.
Quantum mechanics has introduced a huge variety of interesting problems into the realm of PDE.
- Chapter 8: PDE in higher dimension.
We will try to do some of this chapter.
- Chapter 9: Equilibrium (elliptic PDE).
The Laplace equation is the prototypical PDE that governs equilibrium states of heat, displacement of an elastic or acoustic object, and electromagnetism.
Written problems
Students will be expected to present their solutions in readable, logically coherent arguments, with proper use of mathematical symbols.
Due Date |
Section |
Problems |
| Sept. 10
| 2.1
| 3
|
| Sept. 10
| 2.2
| 1, 2, 6
|
| Sept. 10
| 2.3
| 1, 4
|
| Sept. 10
| 2.4
| 3
|
| Oct. 1
| 3.2
| 1, 2, 3
|
| Oct. 1
| 3.3
| 4, 9, 10
|
| Oct. 29
| 5.5
| 1, 2, 4, 10, 11, 12
|
| Nov. 24
| 8.1
| 1, 2, 8
|
| Nov. 24
| 8.2
| 1
|
| Nov. 24
| 8.3
| 5
|
| Nov. 24
| 8.4
| 5, 6
|
| Dec. 5
| 7.3
| 1
|
| Dec. 5
| 7.5
| 4
|
| Dec. 5
| 9.1
| 1, 5, 6 [5 should read vξξ+vηη = ((fξ)2+(fη)2) (uxx+uyy) ]
|
| Dec. 5
| 9.2
| 8, 9
|
MATLAB projects
When learning PDE for applications, being able to compute a numerical solution is at least as important as the study of qualitative behavior and analytic methods of solution. Sometimes, for physical systems with many interacting components, coding the equation on a computer is the only hope for understanding the behavior of the system. Thus MATLAB problems are a major component of this course.
MATLAB projects may be done in groups of up to three students.
| Due Date |
Section |
Project |
| Oct. 8
| 2.8
| 1-3
|
| Nov. 7
| 5.8
| 1, 2, 3
|
| Dec. 5
| 9.2
| 5
|
| Dec. 5
| 9.6
| 1
|
Exam schedule
- Midterm exam: Friday, October 17
- Final exam: Thursday, December 11 from 12:30 to 2:30 in Lockett 135
The midterm and final exams may include an out-of-class part. The final exam will be based on modifications of the written problems in the assignments above.
Evaluation
A subset of the assigned problems will be chosen to be graded. Evaluation of performance in the course is computed as follows:
- Written problems: 25%
- MATLAB projects: 25%
- Midterm exam: 25%
- Final exam: 25%
Grading scale: A---at least 90%; B---at least 80%; C---at least 70%; D---at least 60%.
Ethical Conduct
Students may discuss problems with each other and other people and consult other literature; however, all work that is turned in must ultimately be that of the submitter alone, or submitters in the case of MATLAB projects. If a student receives aid on an assigned problem or project from discussions with people or other sources, he or she must begin from scratch in writing the solution or code so that the result is the product of his or her own understanding alone.
Students must abide by the LSU
Code of Student Conduct.
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