Math 404 - Introduction to Partial Differential Equations
Location: Math/Psych 104
Time: TTh 1:00-2:15pm
Course Information
Instructor: Dr. Bradford E. Peercy
Office: Math/Psych 436
Email: bpeercy@umbc.edu
Phone: 410-455-2436
Office Hours: T/Th after class 2:15-3:00 and by appointment
Prerequisite: MATH 251 and MATH 225 with a C or better.
Course Textbook: The course will be taught from notes created by Dr. Jonathan Bell.
These notes are available at http://www.math.umbc.edu/~jbell/pde_notes/. An additional Intro PDE text by John M. Davis Introduction to Partial Differential Equations may be useful for the second part of the course. The textbook by Peter J. Olver, Introduction to Partial Differential Equations may also be a useful support text.
Course Description
In an Introduction to Partial Differential Equations (PDEs) we appreciate the nature of multiple independent variables and how various quantities depend on them and their rates of change. We will define, classify and derive PDEs in various contexts, but focus on the classical transport, heat/diffusion, wave, and Laplace's equations. We will develop an understanding of solutions and the methods to obtain them on an unbounded domain and on bounded domains. We will utilize software to plot solutions.
Focused Objectives:
- Be able to confirm a solution of a PDE
- Be able to characterize a PDE as
- linear, semi-linear, quasi-linear, or nonlinear
- homogeneous or nonhomogeneous
- order
- parabolic, hyperbolic, or elliptic for 2nd order PDEs
- Understand the derivation of several classic PDEs such as wave, transport, and diffusion equations.
- Be able to solve 1st order, linear or semi-linear PDEs using method of characteristics
- Be able to use the heat kernel to solve PDE on the infinite domain
- Be able to derive the Heat kernel using separation of variables and transforms (Boltzman, Fourier, Laplace)
- Be able to apply Duhamel's principle to change non-homogenous equations.
- Be able to solve the wave equation using d'Alembert's solution.
- Understand the Domain of Dependence.
- Understand Dispersion and Dissipation in the telegrapher's equation.
- Be able to apply Fourier Transforms to reduce PDEs.
- Be able to apply Laplace transform to reduce PDEs.
- Understand interpretation of Neumann, Dirichlet, and Robin boundary conditions.
- Be able to apply separation of variables to PDEs and solve the associated eigenvalue problems.
- Be able to calculate Fourier series solutions and their coefficients.
- Be able to calculate solutions to the Sturm-Liouville eigenvalue problem.
- Understand what a Green's function is and how to calculate it.
Grading Policy
There will be several components to your grade.
- Weekly homework (40%).
- Three Exams (36%)
- Final exam (24%)
You will be allowed to drop two homework scores. No late homework will be accepted. I expect your homework to be legible and well-organized.
Blackboard will be used to post grades (Login to https://my.UMBC.edu).
Academic Integrity Statement
By enrolling in this course, each student assumes the responsibilities of an active participant in UMBC's scholarly community in which everyone's academic work and behavior are held to the highest standards of honesty. Cheating, fabrication, plagiarism, and helping others to commit these acts are all forms of academic dishonesty, and they are wrong. Academic misconduct could result in disciplinary action that may include, but is not limited to, suspension or dismissal. To read the full Student Academic Conduct Policy, consult the UMBC Student Handbook, the Faculty Handbook, the UMBC Integrity webpage www.umbc.edu/integrity, or the Graduate School website www.umbc.edu/gradschool.
Tentative Course Schedule
| Thurs Aug 27 | Introduction: What is a PDE? |
| Tues Sept 1 | Useful background from ODE, Lin Alg, and Multivariable Calculus |
| Thurs Sept 3 | Classification/Derivations of Some Classical PDEs. | Due HW1: |
| Tues Sept 8 | First Order PDEs |
| Thurs Sept 10 | First Order PDEs continued | Due HW2: |
| Tues Sept 15 | Wave equation Derivation |
| Thurs Sept 17 | D'Alembert's Solution | Due HW3: |
| Tues Sept 22 | Domain of Dependence/Influence, Well Posedness, Nonhomogeneous Wave Equation |
| Thurs Sept 24 | Diffusion/Heat Equation | Due HW4: |
| Tues Sept 29 | Exam #1 |
| Thurs Oct 1 | Nonhomogeneous Diffusion Equation |
| Tues Oct 6 | Telegraph equation/Dispersion Relation | Due HW5: |
| Thurs Oct 8 | Integral Transforms: Fourier |
| Tues Oct 13 | Integral Transforms: Laplace | Due HW6: |
| Thurs Oct 15 | Boundary Value Problems (BVPs) |
| Tues Oct 20 | Separation of variables/Eigenvalue problems | Due HW7: |
| Thurs Oct 22 | More Problems on the Finite Domain |
| Tues Oct 27 | |
| Thurs Oct 29 | Convergence results, Gibb's Phenomena | Due HW8: |
| Tues Nov 3 | Exam #2 |
| Thurs Nov 5 | Source problems for BVPs | Due HW9: |
| Tues Nov 10 | Sturm-Liouville Theory |
| Thurs Nov 12 | Eigenfunction Theory | Due HW10: |
| Tues Nov 17 | Laplace equation/Poisson's Formula |
| Thurs Nov 19 | Laplace Equation Properties | Due HW11: |
| Tues Nov 24 | |
| Thurs Nov 26 | Thanksgiving Break |
| Tues Dec 1 | Self-Adjoint Form, Green's Functions for ODEs |
| Thurs Dec 3 | Green's Functions for PDEs | Due HW12: |
| Tues Dec 8 | Exam #3 - Last day of classes |
| Thurs Dec 10 | Reading Day |
| Tues Dec 15 | FINAL EXAM 1:00-3:00 MP 104 |
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