## Course Syllabus for "MA222: Introduction to Partial Differential Equations"

Partial differential equations (PDEs) describe the relationships among the derivatives of an unknown function with respect to different independent variables, such as time and position. For example, the heat equation can be used to describe the change in heat distribution along a metal rod over time. PDEs arise as part of the mathematical modeling of problems connected to different branches of science, such as physics, biology, and chemistry. In these fields, experiment and observation provide information about the connections between rates of change of an important quantity, such as heat, with respect to different variables. These connections must be exploited to find an explicit way of calculating the unknown quantity, given the values of the independent variables – that is, to derive certain laws of nature. While we do not know why partial differential equations provide what has been termed the “unreasonable effectiveness of mathematics in the natural sciences” (the title of a 1960 paper by physicist Eugene Wigner[1]), they provide the foundation for a robust and important field concerned with applied mathematics. A very large fraction of solvable PDEs are either linear first- or second-order PDEs, or are related to such PDEs by transformation or perturbation theory. Fortunately, these PDEs also make up the language for much of the mathematical description of nature. Most of this class will concentrate on those equations whose tremendous importance to real-world applications has been established. Because methods for finding exact or approximate solutions to partial differential equations tend to be rather specialized, it is important to be able to classify these equations. Accordingly, notation, specialized terminology, and the classification scheme for partial differential equations will constitute the subject of Units 1 and 2. Subsequent units will examine some major solution methods: Fourier series and the Fourier transform, separation of variables, the method of characteristics, and impulse-response methods. This course is a way of dipping your toes into the vast pool that is analysis and solution of PDEs, a place where some people spend their whole lives.

[1] This paper first appeared in *Communications
in Pure and Applied Mathematics*, v13, pages 1–14.

### Learning Outcomes

Upon successful completion of this course, the student will be able to:

- state the heat, wave, Laplace, and Poisson equations and explain their physical origins;
- define harmonic functions;
- state and justify the maximum principle for harmonic functions;
- state the mean value property for harmonic functions;
- define linear operators and identify linear operations;
- identify and classify linear PDEs;
- identify homogeneous PDEs and evolution equations;
- relate solving homogeneous linear PDEs to finding kernels of linear operators;
- define boundary value problem and identify boundary conditions as periodic, Dirichlet, Neumann, or Robin (mixed);
- explain physical significance of boundary conditions;
- show uniqueness of solutions to the heat, wave, Laplace and Poisson equations with various boundary conditions;
- define well-posedness;
- define, characterize, and use inner products;
- define the space of L
^{2}functions, state its key properties, and identify L^{2}functions; - define orthogonality and orthonormal basis and show the orthogonality of certain trigonometric functions;
- distinguish between pointwise, uniform, and L
^{2}convergence and show convergence of Fourier series; - define Fourier series on [0,π] and [0,L] and identify sufficient conditions for their convergence and uniqueness;
- compute Fourier coefficients and construct Fourier series;
- use the method of characteristics to solve linear and nonlinear first-order wave equations;
- solve the one-dimensional wave equation using d’Alembert’s formula;
- use similarity methods to solve PDEs;
- solve the heat, wave, Laplace, and Poisson equations using separation of variables and apply boundary conditions;
- define the delta function and apply ideas from calculus and Fourier series to generalized functions;
- derive Green’s representation formula;
- use Green’s functions to solve the Poisson equation on the unit disk;
- define the Fourier transform;
- derive basic properties of the Fourier transform of a function, such as its relationship to the Fourier transform of the derivative;
- show that the inverse Fourier transform of a product is a convolution;
- compute Fourier transforms of functions; and
- use the Fourier transform to solve the heat and wave equations on unbounded domains.

### Course Requirements

In order to take this course, you must:

√ have access to a computer;

√ have continuous broadband internet access;

√ have the ability/permission to install plug-ins or software (e.g. Adobe Reader of Flash);

√ have the ability to download and save files and documents to a computer;

√ have the ability to open Microsoft files and documents (e.g. .doc, .ppt, .xls, etc.);

√ be competent in the English language;

√ have read the Saylor Student Handbook; and

√ have completed the following courses: MA103 (Multivariable Calculus), MA221 (Differential Equations), MA211 (Linear Algebra), and MA241 (Real Analysis I). MA243 (Complex Analysis) provides useful background, but is not required.

### Course Information

Welcome to MA222: Introduction to Partial Differential Equations. Below, please find general information on this course and its requirements. This a very challenging course! In addition, while it does have strong theoretical underpinnings, it is much more a survey of methods than a construction of theory from first principles. Think of it as the assembly of a toolkit for the solution of PDEs, rather than a definitive overview of the theory of PDEs, a field far too broad for any course to encompass. For that reason, the same problems are solved multiple times and from a variety of perspectives. This means that if something does not click for you after a great deal of persistence, move on. You can return to any of these concepts/methods at a later point, and you may do better once you have gained a different perspective on the problem (This advice applies most strongly to Units 3-5, but the exception to it is the technique of separation of variables. If you only have one hammer in your toolkit, that is the one you need, so persevere!)

**Course Designer: Clare Wickman**

**Primary Resources: **This course is comprised of a range of different
free, online materials. However, the course makes primary use of the
following materials:

- Cambridge University Press: Professor Marcus Pivato’s
*Linear Partial Differential Equations and Fourier Theory* - University of Minnesota: Professor Peter Olver’s
*Introduction to Partial Differential Equations*

**Requirements for Completion:** In order to complete this course, you
will need to work through each unit and all of its assigned materials.
Pay special attention to Unit 1 as this lays the groundwork for
understanding the more advanced, exploratory material presented in
latter units. You will also need to complete:

- Subunit 2.1.2 Assessment
- Subunit 2.2.1 Assessment
- Subunit 2.2.5 Assessment
- Subunit 3.1.1 Assessment
- Subunit 3.2.4 Assessment
- Subunit 4.1.3 Assessment
- Subunit 4.1.4 Assessment
- Subunit 4.2.1 Assessment
- Subunit 4.2.2 Assessment
- Subunit 4.2.3 Assessment
- Subunit 4.3.1 Assessment
- Subunit 4.3.3 Assessment
- Subunit 5.1.4 Assessment
- Subunit 5.2.3 Assessment
- The Final Exam

Note that you will only receive an official grade on your Final Exam.
However, in order to adequately prepare for this exam, you will need to
work through the assessments listed above.

In order to “pass” this course, you will need to earn a 70% or higher
on the Final Exam. Your score on the exam will be tabulated as soon as
you complete it. If you do not pass the exam, you may take it again.

**Time Commitment:** This course should take you a total of **109.75
hours** to complete. Each unit includes a “time advisory” that lists the
amount of time you are expected to spend on each subunit. These should
help you plan your time accordingly. It may be useful to take a look at
these time advisories and to determine how much time you have over the
next few weeks to complete each unit, and then to set goals for
yourself. For example, Unit 1 should take you 5 hours. Perhaps you can
sit down with your calendar and decide to complete subunit 1.1 (a total
of hours) on Monday night; subunit (a total of hours ) on Tuesday night;
etc.

**Tips/Suggestions:** There is too much in the primary resources for
this course to use in just one semester, so explore the books a bit on
your own to see if you wish to learn more about any particular
subject.

As you read, take careful notes on a separate sheet of paper. Mark down
any important equations, formulas, and definitions that stand out to
you. These notes will be useful as a review when you study for your
Final Exam.

For extra help with this material, see Professor Paul Dawkins’ *Paul’s
Online Math Notes*. He has some good material on PDEs and boundary value
problems.
(Link)