Course Overview

Math 442 covers basic theory of partial differential equations, with a particular emphasize on the wave, diffusion, Laplace and Schrödinger

equations. Topics include classification of PDEs in terms of order, linearity and homogeneity, finding the solutions of the PDEs using methods such as geometric, operator, Fourier, separation of variables and spherical means.

Course Goals

The course covers topics from the textbook Partial Differential Equations by Walter A Strauss.  Students would learn different methods for

solving the PDEs along with some of the applications in the physical systems such as harmonic oscillator, special relativity and hydrogen atom. The main focus will be on understanding the physical meaning and mathematical properties of solutions of partial differential equations. It is

important to recognize that reading ahead in the textbook before viewing class videos will make the lectures more comprehensible and writing your homework solutions in your own words improves your understanding.

General Information

This is a 3 credit hour course. The course is 16 weeks long and consists of 10 Units. You should dedicate approximately 9 hours per week to work on the course itself, but actual time commitments will vary depending on your input, needs, and personal study habits. It is recommended   that you log on to the course website and check your email frequently for updates, news and announcements.

Required and Recommended Texts

Required

Walter A. Strauss. (2007). Partial Differential Equations. (2nd Edition). John Wiley & Sons, Inc.

Course Components

This course will consist of the following components:

Pre-Assignment

The pre-assignment should be completed first. It is used to ensure that you understand the policies, expectations and resources provided in the course. The pre-assignment is worth half the value of one homework assignment.

Units

Each unit begins with an overview and the learning goals you are expected to achieve. These goals should guide your study through the unit.   Every unit consists of a homework assignment, lectures, readings and additional exercises to support these goals. They are designed with the same structure and components unless otherwise specified. The module activities are explained in greater detail below.

Homework Assignments

Each unit contains a homework assignment consisting of several exercises. After clicking the assignment you will see assignment instructions

and a link to a PDF file containing these exercises. View the PDF file and complete the exercises. When you are finished, scan or take a picture of your work and submit the file via the assignment linkin Moodle.

Lectures

Each unit contains a list of recorded lectures. Clicking a link to a lecture allows you to view the lecture in its entirety. You will also find links to view specific topics in each lecture.

As the lecture is playing, note that there are PDF slides of what is written on the chalkboard. These can be displayed side by side with the

lecture. There are controls in the upper right corner of the video player that allow for viewing of just the lecture, just the slide, or to view both, side by side. You can download these slides and use them to take notes as the lecture plays.

Captions for the video can be displayed by clicking the CC button in the lower right corner of the player. You will find these helpful to clarify any portions of the video where descriptions or audio is not clear. Note that in most cases, subscripts are identified by a preceding underscore character and superscripts are identified by a preceding ^ character.

Readings

Each unit contains assigned readings. You are responsible to complete these items. Lectures cover major topics from the readings but do not necessarily include all important information from the readings.

Discussion Forum

Each unit contains a discussion forum. This forum should be used if you have a question about the assignment or content of the unit. Posting questions here allows everyone to benefit from the answer. Most likely, other students also have the same question.

By default, you will receive an e-mail each time a new post is created. To unsubscribe from the forum, go to Forum administration --> Unsubscribe from this forum under the blue Administration block located to the left of the forum.

Exams

This course includes three midterm exams and a final exam. Midterm Exam #1, Midterm Exam #3 and the Final Exam are proctored. See the Exam tab for details.

Accommodations

To obtain disability-related academic adjustments and/or auxiliary aids, students should contact both the instructor and the Disability Resources and Educational Services (DRES) as soon as possible. You can contact DRES at 1207 S. Oak Street, Champaign, (217) 333-1970, or via e-mail at [email protected].

Course Content

1. First order linear partial differential equations, initial and boundary conditions, well-posed problems, geometric method and coordinate method for solving first order PDE.

2. Second order PDEs, types of second order PDEs, different methods for solving second order PDEs including operator method, Fourier coefficient method, separation or variables method, spherical means method, Greenʼs function formula, Duhamelʼs principle.

3. Wave equation, principle of causality, diffusion equation, Maximum principle, reflection of waves, waves equation with a source, diffusion equation with a source, well posedness of the wave equation. Separation of variables with Dirichlet, Neumann and Robin boundary

conditions. Heat and wave equation in higher dimensions, radial functions.

4. Fourier coefficient method for homogeneous and inhomogeneous wave, diffusion based on even, odd, periodic and complex functions, Orthogonality and convergence. Method of shifting data for inhomogeneous diffusion equation.

5. Laplace Equation with Robin BC, Energy problem, Eigen functions, homogeneous and inhomogeneous periodic differential equation, harmonic boundary problem.

6. PDE based modelling of simplest atom (Schrodinger equation), Harmonic oscillator, hydrogen atom, vibrations of drumhead, solid vibration in a ball.

7. Inner product spaces, Cauchy Schwarz inequality, Green Identity, Hilbert spaces, convergence and orthonormal systems, different types of convergence, harmonic function, Besselʼs inequality, Theorem on Continuous derivatives and pointwise convergence, divergence

theorem, Weierstrass approximation theorem.

Grading

Grading Distribution

When calculating your final grade, your instructor will look at an average of all your exams. Your final grade cannot exceed this average by more than 10%. For example, if the average grade on all of your exams (weighted equally) is 70%, your final grade cannot exceed 80%.

Grades are essentially straight scale:

Grades are not curved. You have an opportunity to earn 1% extra credit by completing the survey at the end of the course. After completing the final, you will receive an email with instructions to complete the survey.