MA222: Introduction to Partial Differential Equations « Saylor.org

Introduction to Partial Differential Equations

Purpose of Course show close

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.

Course Information show close

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)

Learning Outcomes show close

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 L² functions, state its key properties, and identify L² functions;
define orthogonality and orthonormal basis and show the orthogonality of certain trigonometric functions;
distinguish between pointwise, uniform, and L² 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 show close

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.

Unit Outline show close

1 Introduction to Partial Differential Equations
1.1 Preliminaries
1.1.1 Sets, Functions, and Derivatives
1.1.2 Notation for Derivatives
1.1.3 Complex Numbers
1.1.4 Coordinate Systems
1.1.5 Vector Calculus
1.2 Overview and Motivation
1.2.1 Overview of PDEs
1.2.2 Heat and Diffusion
1.2.3 Waves and Signals

2 Classification of Linear PDEs
2.1 Linear Partial Differential Equations
2.1.1 Functions and Vectors
2.1.2 Linear Operators
2.1.3 Homogeneous and Nonhomogeneous Linear PDEs
2.2 Classification of PDEs
2.2.1 Evolution and Nonevolution Equations
2.2.2 Initial-Value Problems
2.2.3 Boundary Value Problems: Types of Boundary Conditions
2.2.4 Uniqueness of Solutions
2.2.5 Classification of Second Order Linear PDEs
2.2.6 Well-Posedness

3 Fourier Series on Bounded Domains
3.1 Necessary Functional Analysis
3.1.1 Inner Products
3.1.2 L2 Space
3.1.3 Orthogonality
3.1.4 Convergence Concepts
3.1.5 Orthogonal and Orthonormal Bases
3.2 Fourier Series
3.2.1 Eigensolutions to Linear Evolution Equations
3.2.2 Fourier Series on [0, ?]
3.2.3 Fourier Series on [0,L]
3.2.4 Computing Fourier Coefficients

4 Solution Methods
4.1 Linear and Nonlinear Waves
4.1.1 Stationary Waves
4.1.2 Transport and Traveling Waves
4.1.3 Nonlinear Transport and Shocks
4.1.4 D’Alembert’s Solution to the Wave Equation
4.1.5 Symmetry and Similarity
4.2 Separation of Variables
4.2.1 Separation of Variables for the Heat Equation
4.2.2 Separation of Variables for the Wave Equation
4.2.3 Separation of Variables for the Laplace and Poisson Equations
4.2.4 Separation of Variables in Spherical Coordinates
4.2.5 Separated and Quasiseparated Solutions
4.2.6 Differential Operators as Polynomials
4.3 Impulse-Response Methods
4.3.1 Generalized Functions: The Delta Function and Calculus
4.3.2 Green’s Functions for One-Dimensional Boundary Value Problems
4.3.3 The Green’s Function for the Poisson Equation

5 The Fourier Transform
5.1 The Fourier Transform
5.1.1 One-Dimensional Fourier Transforms
5.1.2 Fourier Transforms of Derivatives and Integrals
5.1.3 Green’s Functions and Convolution
5.1.4 The Fourier Transform on Hilbert Space
5.2 Fourier Transform Solutions to PDEs on Unbounded Domains
5.2.1 The Heat Equation
5.2.2 The Wave Equation
5.2.3 Laplace’s Equation

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Unit 1: Introduction to Partial Differential Equations

This unit will review the background mathematics that you will need to use in this course. You should be familiar with the majority of the concepts presented here, but some material may be new to you. This unit will also introduce the notation and terminology that we will use in the remainder of the course.

In addition, several of the most important PDEs – the heat equation, the wave equation, and Schrodinger’s equation – will be derived from physical principles. While some of this material is optional, knowledge of it will help you develop intuition about solutions to these equations and understand why the theory developed as it did.

Unit 1 Time Advisory show close

Unit 1 Learning Outcomes show close

1.1 Preliminaries

1.1.1 Sets, Functions, and Derivatives

Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “Chapter Zero: Appendices”
Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “Chapter Zero: Appendices” (PDF)

Instructions: Click on the link above. This reading covers subunits 1.1.1 through 1.1.5.

Read sections 0A through 0E in the Appendices section (pages 545 through 565). Section 0A will explain vocabulary such as path, mass density, and time-varying scalar field. Section 0B explains the author’s notation for derivatives. Section 0C reviews complex numbers. Section 0D reviews several coordinate systems, including polar and spherical. Section 0E reviews some important concepts from vector calculus, including the gradient, the divergence, the Divergence theorem, and Green’s formulas.

Reading these sections should take approximately 3 hours.

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1.1.2 Notation for Derivatives

Note: This topic is covered by the reading assigned below subunit 1.1.

1.1.3 Complex Numbers

Note: This topic is covered by the reading assigned below subunit 1.1.

1.1.4 Coordinate Systems

Note: This topic is covered by the reading assigned below subunit 1.1.

1.1.5 Vector Calculus

Note: This topic is covered by the reading assigned below subunit 1.1.

1.2 Overview and Motivation

If your background or interest in mathematical physics is small, feel free to skim through much of the material on physical motivation for the development of these PDEs, which is given in the readings for subunits 1.2.2–1.2.3. However, this material is worthwhile, and you should be sure to take the time to understand the mathematical properties of the solutions to the PDEs, which some of these readings begin to discuss. The material on the properties of harmonic functions, for instance, is particularly important.

1.2.1 Overview of PDEs

Web Media: YouTube: commutant’s “PDE Part 1: Introduction”
Link: YouTube: commutant’s “PDE Part 1: Introduction ” (YouTube)

Instructions: Click on the link above, and watch the video. The author briefly explains what PDEs are and what it means to solve them and then works through an example.

Watching this lecture and pausing to take notes should take approximately 25 minutes.

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Reading: MIT: Gigliola Staffilani’s Introduction to Partial Differential Equations: “Lecture 1: “Introduction and Basic Facts about PDEs”
Link: MIT: Gigliola Staffilani’s Introduction to Partial Differential Equations: “Lecture 1: Introduction and Basic Facts about PDEs” (PDF)

Instructions: Click on the link above. The lecture notes will open in PDF form (6 pages).

These notes give an overview of much of the course content and define linearity and homogeneity.

Studying these notes should take approximately 45 minutes.

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Web Media: YouTube: commutant’s “PDE Part 2: Three Fundamental Examples”
Link: YouTube: commutant’s “PDE Part 2: Three Fundamental Examples ” (YouTube)

Instructions: Watch this brief video that briefly explores three fundamental PDEs. The physical motivation behind these PDEs will be explained in the next three sections.

Watching this lecture and pausing to take notes should take approximately 20 minutes.

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1.2.2 Heat and Diffusion

1.2.2.1 Fourier’s Law and the Heat Equation

Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “1A: Heat and Diffusion: Fourier’s Law” and “1B: Heat and Diffusion: The Heat Equation”
Links: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “1A: Heat and Diffusion: Fourier’s Law” (PDF) and “1B: Heat and Diffusion: The Heat Equation” (PDF)

Instructions: Click on the links above and read sections 1A and 1B (pages 3-9).

The heat equation describes the erosion or diffusion of a system, and this reading explains its derivation. The reading also gives several solutions to the heat equation.

Studying this reading should take approximately 45 minutes.

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1.2.2.2 Laplace’s Equation

Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “1C: Heat and Diffusion: Laplace’s Equation”
Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “1C: Heat and Diffusion: Laplace’s Equation” (PDF)

Instructions: Click on the link above and read section 1C (pages 9-12).

This reading explains Laplace’s equation and defines harmonic functions.

Studying this reading should take approximately 30 minutes.

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1.2.2.3 The Poisson Equation

Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “1D: Heat and Diffusion: The Poisson Equation”
Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “1D: Heat and Diffusion: The Poisson Equation” (PDF)

Instructions: Click on the link above and read section 1D (pages 12-16).

The Poisson equation describes the steady state of the generation-diffusion equation.

Studying this reading should take approximately 30 minutes.

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1.2.2.4 Properties of Harmonic Functions

Reading: Mathnotes.me: “Laplace Equation”
Link: Mathnotes.me: “Laplace Equation” (PDF)

Instructions: Read this statement of a particular Laplace equation, carefully digest the Maximum Principle and its proof, and see how it is used to prove uniqueness of the trivial solution to the given PDE.

Reading these notes should take approximately 30 minutes.

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1.2.3 Waves and Signals

1.2.3.1 The Laplacian and Spherical Means

Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “2A: Waves and Signals: The Laplacian and Spherical Means”
Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “2A: Waves and Signals: The Laplacian and Spherical Means” (PDF)

Instructions: Click on the link above and read section 2A (pages 23-27).

This reading further justifies the spherical means used in the exploration of the properties of harmonic functions.

Studying this reading should take approximately 30 minutes.

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1.2.3.2 The Wave Equation

Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “2B: Waves and Signals: The Wave Equation”
Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “2B: Waves and Signals: The Wave Equation” (PDF)

Instructions: Click on the link above and read section 2B (pages 27-34).

This reading derives the wave equation. Again, feel free to skip this material if you find it overwhelming or uninteresting.

Studying this reading should take approximately 1 hour.

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Web Media: YouTube: commutant’s “PDE Part 7: Wave Equation: Intuition”
Link: YouTube: commutant’s “PDE Part 7: Wave Equation: Intuition ” (YouTube)

Instructions: Watch this video that explains the physical origins of the wave equation and what exactly the terms mean. In addition, the author analyzes the dimensions of variables.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Unit 2: Classification of Linear PDEs

There is an incredible amount of variety in the study of PDE problems, in part because of the incredible amount of variety in the physical systems that many of them model. Over the course of the last 300 years, many different mathematicians and physicists have tackled the same problems and come up with diverse approaches to solving them, each adding new richness to the field. While this diversity makes for very interesting problems, it can devolve into a theoretical Gordion knot when considered without a strong and well-defined framework to anchor the novice student. For that reason, we will wait to touch on the most common solution methods until Units 3-5. In this unit, we will attempt to impose some order on our approach so that as you learn different solution methods, you will understand the broad classes of functions to which they apply, as well as the complications that make one method appropriate for one situation, but not for another.

The unit begins with a refresher on the concept of linear operators, essential for the classification of linear PDEs. Equations are classified as homogeneous or nonhomogeneous, and evolution or nonevolution. The readings for the unit detail the different types of boundary conditions and their physical significance. The different types of linear PDEs are explained. Finally, the concepts of uniqueness and well-posedness are elaborated, and uniqueness under suitable conditions is established for some of the most important PDEs.

If your mathematical background does not include a course in real analysis, you will likely find this unit more comprehensible if you first skip ahead to Units 3 and 4 and study some of the more tangible solution techniques. In so doing, do not worry about the technical assumptions required to ensure that the techniques “work.” Rather, assume what you need and focus on the techniques and what they produce. Once you are comfortable with this, then go back to Unit 2 and work through the mathematical underpinnings. As you do so, keep in mind what you learned in Units 3 and 4 as a target to which the theory will inevitably be applied.

Unit 2 Time Advisory show close

Unit 2 Learning Outcomes show close

2.1 Linear Partial Differential Equations

2.1.1 Functions and Vectors

Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “4A: Linear Partial Differential Equations: Functions and Vectors”
Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “4A: Linear Partial Differential Equations: Functions and Vectors” (PDF)

Instructions: Click on the link above and read section 4A (pages 57-59).

This reading explains how different sets of functions can be described as vector spaces, most importantly the C-infinity functions – those that are infinitely differentiable.

Studying this reading should take approximately 15-20 minutes.

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2.1.2 Linear Operators

Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “4B: Linear Partial Differential Equations: Linear Operators”
Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “4B: Linear Partial Differential Equations: Linear Operators” (PDF)

Instructions: Click on the link above. The book will open as a PDF. Read section 4B (pages 59-64).

Linear Algebra is a powerful tool used to investigate broad categories of equations and mathematical phenomena. Differentiation is a linear operation, and therefore many of the now familiar characters from PDE, such as the Laplacian, can be analyzed as linear operators.

Studying this reading should take approximately 30 minutes.

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Assessment: MIT: Professor Matthew Hancock’s “Problem Set 2” and MIT: Professor Steven Johnson’s “Problem Set 1”
Links: MIT: Professor Matthew Hancock’s “Problem Set 2 ” (PDF) and MIT: Professor Steven Johnson’s “Problem Set 1 ” (PDF)

Instructions: Click on the first link above, and complete problems 2 and 3 of “Problem Set 2.” Then, click on the second link above, and complete problem 1 of “Problem Set 1.” To check your solutions, follow this link for the Hancock problems, and this link for the Johnson problems.

You should dedicate approximately 1 hour to completing this assessment.

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2.1.3 Homogeneous and Nonhomogeneous Linear PDEs

Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “4A: Linear Partial Differential Equations: Homogeneous vs. Nonhomogeneous”
Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “4A: Linear Partial Differential Equations: Homogeneous vs. Nonhomogeneous” (PDF)

Instructions: Click on the link above. The book will open as a PDF. Read section 4C (pages 64-66).

This reading defines homogeneous and non-homogeneous equations and explains the superposition principle.

Studying this reading should take approximately 15-20 minutes.

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2.2 Classification of PDEs

2.2.1 Evolution and Nonevolution Equations

Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “5A: Classification of PDEs and Problem Types: Evolution vs. Nonevolution Equations”
Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “5A: Classification of PDEs and Problem Types: Evolution vs. Nonevolution Equations” (PDF)

Instructions: Click on the link above and read section 5A (pages 69 and 70).

This reading defines evolution equations and the order of a PDE.

Studying this reading should take approximately 15-20 minutes.

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Assessment: Naval Postgraduate School: Professor Beny Neta’s Partial Differential Equations MA 3132: Solutions of Problems in Lecture Notes “Chapter 1.1: Introduction and Applications: Basic Concepts and Definitions”
Link: Naval Postgraduate School: Professor Beny Neta’s Partial Differential Equations MA 3132: Solutions of Problems in Lecture Notes “Chapter 1.1: Introduction and Applications: Basic Concepts and Definitions ” (PDF)

Instructions: Click on the link above. Under “MA3132 Lecture Notes and Solution Manual,” find the link “Solution Manual for MA3132 in pdf.” Click on the link to download the document; it contains problems and solutions.

In the PDF, scroll down to page 1 and attempt problems 1-5. When finished, go to page 2 to find the solutions.

You should dedicate approximately 2 hours to completing this assessment.

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2.2.2 Initial-Value Problems

Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “5B: Classification of PDEs and Problem Types: Initial Value Problems”
Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “5B: Classification of PDEs and Problem Types: Initial Value Problems” (PDF)

Instructions: Click on the link above and read section 5B (pages 70 and 71).

This reading defines initial-value or Cauchy problems.

Studying this reading should take approximately 15-20 minutes.

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2.2.3 Boundary Value Problems: Types of Boundary Conditions

Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “5C: Classification of PDEs and Problem Types: Boundary Value Problems”
Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “5C: Classification of PDEs and Problem Types: Boundary Value Problems” (PDF)

Instructions: Click on the link above and read section 5C (pages 71-84).

This reading defines boundary-value problems. While initial value problems apply only to evolution equations and may take place on an infinite domain, boundary-value problems are not restricted to evolution equations and allow more realistic modeling of physical phenomena. There are multiple types of boundary conditions, and this reading will explain their physical significance.

Studying this reading should take approximately 2 hours.

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2.2.4 Uniqueness of Solutions

Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “5D: Classification of PDEs and Problem Types: Uniqueness of Solutions”
Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “5D: Classification of PDEs and Problem Types: Uniqueness of Solutions” (PDF)

Instructions: Click on the link above and read section 5D (pages 84-95).

The ability to establish uniqueness of solutions is very important in analysis of PDEs. This reading describes the uniqueness of solutions to various PDEs with different regularity properties and boundary conditions.

Studying this reading should take approximately 1 hour.

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Assessment: MIT: Professor Matthew Hancock’s “Problem Set 2” (PDF) and Penn State: Professor Kris Wysocki’s “Homework 4” (PDF)
Links: MIT: Professor Matthew Hancock’s “Problem Set 2 ” (PDF) and Penn State: Professor Kris Wysocki’s “Homework 4 ” (PDF)

Instructions: Click on the first link above to access the PDF, and complete problems 2 and 3 of “Problem Set 2.” Then, click on the second link above, select the hyperlink to “Homework 4” to download the PDF, and complete problem 5. To check your solutions for the Hancock problems, click on this link. To check your solutions for the Wysocki problems, follow the top link again and click on the corresponding link for solutions.

In the second assessment, note that “energy methods” are used in the readings from Professor Pivato’s textbook to show uniqueness: namely, integrating the square of some quantity to show that it is zero.

You should dedicate approximately 1 hour to completing this assessment.

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2.2.5 Classification of Second Order Linear PDEs

Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “5E: Classification of PDEs and Problem Types: Classification of Second Order Linear PDEs”
Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “5E: Classification of PDEs and Problem Types: Classification of Second Order Linear PDEs” (PDF)

Instructions: Click on the link above and read section 5E (pages 95-98).

Recall that in many cases, solving a (homogeneous) PDE can be formulated as finding the kernel (nullspace) of a linear operator on a function space. In this reading, three classes of linear PDEs are identified by analyzing these operators; the categories are elliptic, parabolic, and hyperbolic equations.

Studying this reading should take approximately 30 minutes.

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Assessment: Naval Postgraduate School: Professor Beny Neta’s Partial Differential Equations MA 3132: Solutions of Problems in Lecture Notes: “Chapter 2.2: Classification and Characteristics: Classification of Linear Second Order PDEs”
Link: Naval Postgraduate School: Professor Beny Neta’s Partial Differential Equations MA 3132: Solutions of Problems in Lecture Notes: “Chapter 2.2: Classification and Characteristics: Classification of Linear Second Order PDEs ” (PDF)

Instructions: Click on the link above. Under “MA3132 Lecture Notes and Solution Manual,” find the link “Solution Manual for MA3132 in pdf.” Click on the link to download the document; it contains problems and solutions.

In the PDF, scroll down to page 14 and attempt problems 1 and 2. When finished, go to page 15 to find the solutions.

You should dedicate approximately 1 hour to completing this assessment.

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2.2.6 Well-Posedness

Reading: Oakridge National Laboratory Physics Division’s “6 Well-Posed PDE Problems”
Link: Oakridge National Laboratory Physics Division’s “6 Well-Posed PDE Problems ” (HTML)

Instructions: Click on the link above, and read through the linked pages. Make sure to click on the “continued” link at the bottom of each webpage to read the entire 8-page tutorial.

These notes give an overview of the important concept of well-posedness, which links mathematical properties of solutions to PDEs to their usefulness in describing natural phenomena.

Studying this reading should take approximately 1 hour.

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Unit 3: Fourier Series on Bounded Domains

As you may have learned in a physics (or music!) class, every sound is made up of sound waves with different frequencies. Thus, for instance, a C-major chord based at middle C is made of up waves at 261.6 Hz(C), 329.6 Hz (E), and 392.0 Hz (G). The combination (or superposition) of these three individual sound waves produces a new and distinct sound. In fact, any signal or waveform (subject to some reasonable conditions) can be written as the combination of some fundamental waves, whether it is a water wave, a light or electromagnetic wave, or a sound wave. Mathematically, a waveform is just a function, and, conversely, every function can be viewed as a waveform. Most (again, subject to some reasonable conditions) can be written as a combination of some fundamental wave-like functions, namely sines and cosines. This was the observation of Fourier, a great scientist of the nineteenth century. It took years for other mathematicians to accept his work and come up with a theoretical framework for his results. The study of such ideas is today called Harmonic Analysis, and it has gone far beyond Fourier’s original ideas to touch almost every part of modern science and technology, from medical imaging to wireless communication.

In this unit, you will learn to apply Fourier’s original insight – that most useful functions can be written as linear combinations of sine and cosine waves. To do this rigorously, some mathematical underpinnings must be established, such as the definition and elaboration of the square-integrable functions, the differences between different modes of convergence, and the orthogonality of the basis of trigonometric functions, some of which you will recall from MA241. After this has been accomplished, you will learn how to calculate Fourier series for suitable functions.

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3.1 Necessary Functional Analysis

3.1.1 Inner Products

Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “6A: Some Functional Analysis: Inner Products”
Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “6A: Some Functional Analysis: Inner Products” (PDF)

Instructions: Click on the link above and read section 6A (pages 103-105).

Solving linear PDEs using algebraic methods requires an understanding of inner product spaces.

Studying this reading should take approximately 20 minutes.

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Assessment: MIT: Professor Steven Johnson’s “Problem Set 2”
Link: MIT: Professor Steven Johnson’s “Problem Set 2 ” (PDF)

Instructions: Click on the link above to access the PDF file, and complete problem 2 from “Problem Set 2.” To check your solutions, follow this link.

You should dedicate approximately 30 minutes to completing this assessment.

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3.1.2 L2 Space

Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “6B: Some Functional Analysis: L2 Space”
Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “6B: Some Functional Analysis: L2 Space” (PDF)

Instructions: Click on the link above and read section 6B (pages 105-108).

In order to apply ideas from linear algebra to PDEs, one requires a normed linear space with a well-defined norm. Using the inner product ideas from the previous section, the author defines the vector space L² of square-integrable functions.

Studying this reading should take approximately 15-20 minutes.

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3.1.3 Orthogonality

Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “6D: Some Functional Analysis: Orthogonality”
Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “6D: Some Functional Analysis: Orthogonality” (PDF)

Instructions: Click on the link above and read section 6D (pages 112-116).

In this section, the author shows the orthogonality of trigonometric functions, which will be a crucial fact for the development of Fourier series in upcoming units.

Studying this reading should take approximately 30 minutes.

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3.1.4 Convergence Concepts

Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “6E: Some Functional Analysis: Convergence Concepts”
Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “6E: Some Functional Analysis: Convergence Concepts” (PDF)

Instructions: Click on the link above and read section 6E (pages 116-131).

In many cases, solutions to PDEs can be constructed using sequences or series of functions. However, as you know from calculus, the construction of a sequence or series does not guarantee that it has a well-defined limit. For this reason, the author defines several modes of convergence, which will be used to establish the validity of these types of solutions. The diagrams used to illustrate these concepts are excellent.

Studying this reading should take approximately 3 hours.

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3.1.5 Orthogonal and Orthonormal Bases

Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “6F: Some Functional Analysis: Orthogonal and Orthonormal Bases”
Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “6F: Some Functional Analysis: Orthogonal and Orthonormal Bases ” (PDF)

Instructions: Click on the link above and read section 6F (pages 131-133).

Studying this reading should take approximately 15-20 minutes.

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3.2 Fourier Series

3.2.1 Eigensolutions to Linear Evolution Equations

Reading: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 3: Fourier Series: 3.1: Eigensolutions to Linear Evolution Equations”
Link: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 3: Fourier Series: 3.1: Eigensolutions to Linear Evolution Equations ” (PDF)

Instructions: Click on the link above. Scroll down to the Chapter 3, and click on the link to download the PDF. Read pages 55-63.

In this reading, the author motivates the construction of Fourier series by tying them to eigenvalue problems for linear evolution equations. We will return to these ideas when we discuss the technique of separation of variables.

Studying this reading should take approximately 4 hours.

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3.2.2 Fourier Series on [0, ?]

Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “7A: Fourier Sine Series and Cosine Series: Fourier (Co)sine Series on [0, ?]”
Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “7A: Fourier Sine Series and Cosine Series: Fourier (Co)sine Series on [0, π]” (PDF)

Instructions: Click on the link above and read section 7A (pages 137-144).

All of the facts that were established in section 3.1 about inner products, orthogonality, the L² functions, and orthonormal bases are brought into play in this section. Fourier series are one of your most powerful tools for solving PDEs. They are also extremely important for signal and image processing, so investment in understanding this concept is well-worth your time.

Studying this reading should take approximately 1 hour.

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3.2.3 Fourier Series on [0,L]

Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “7B: Fourier Sine Series and Cosine Series: Fourier (Co)sine Series on [0, L]”
Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “7B: Fourier Sine Series and Cosine Series: Fourier (Co)sine Series on [0, L]” (PDF)

Instructions: Click on the link above and read section 7B (pages 144-146).

Studying this reading should take approximately 15-20 minutes.

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3.2.4 Computing Fourier Coefficients

Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “7C: Fourier Sine Series and Cosine Series: Computing Fourier (Co)sine Coefficients”
Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “7C: Fourier Sine Series and Cosine Series: Fourier (Co)sine Coefficients ” (PDF)

Instructions: Click on the link above and read section 7C (pages 147-158).

This section reviews some important methods of computing Fourier coefficients for various functions, including polynomials, step functions, and piecewise linear functions.

Studying this reading should take approximately 1 hour and 30 minutes.

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Assessment: MIT: Professor Matthew Hancock’s “Problem Set 1” and Trinity College, Dublin: Professor Sarah McMurry’s “Problem Sheet 2”
Link: MIT: Professor Matthew Hancock’s “Problem Set 1 ” (PDF) and Trinity College, Dublin: Professor Sarah McMurry’s “Problem Sheet 2 ” (PDF)

Instructions: Click on the first link above to access the PDF, and complete problem 1 from “Problem Set 1.” Then, click on the second link above, scroll down to the section “Problem Sheets,” and select the link to “Problem Sheet 4.” Complete all of the problems on “Problem Sheet 4.” To check your solutions for the Hancock assessments click on this link. To check your solutions for the McMurry assessments follow the second link again and click on the corresponding link (“Answers to Sheet 4”) for solutions.

You should dedicate approximately 3 hours to completing this assessment.

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Unit 4: Solution Methods

Dozens of methods have been developed to solve the most common PDEs over the past 300 years, and many tools have been developed to analyze PDEs that cannot be explicitly solved. Many approaches are seemingly ad hoc, although in many cases deeper reasons emerge for their efficacy. In this unit, we deal with three main classes of solution methods, some of which are more broadly applicable than others.

First, we deal with the use of characteristics to solve traveling-wave problems (both linear and nonlinear). The method of characteristics is used for evolution equations in which, in some way, the initial information is conserved. This subject can be quite complicated, but is a good preparation for the d’Alembert solution to the wave equation in one space dimension. Finally, the exploration of solutions to wave equations ends with an introduction to similarity methods. The transport equation is used as an example.

Next, the technique of separation of variables is explained. This is, possibly, the most important technique that you will learn in this course, and it relies on the insight that, in many cases, the effects of changes in the independent variables on the solution to a PDE are independent – they can be isolated. Your readings begin with a recapitulation of the exponential ansatz for solutions to the Heat equation, which is used to motivate separation in the cases of the wave equation and, most powerfully, the Laplace and Poisson equations. Many examples with different boundary conditions are explored, and the technique is linked back to the Fourier series described in unit 3.

The last component to this unit is the use of impulse-response methods. This may be the most challenging material in this course; it relies on the calculus of generalized functions, which is explained in detail before it is used. Full justification of the use of such methods is reserved for a graduate course in functional analysis, but these methods are too powerful to neglect, even without such rigorous defense. In fact, choosing to peruse these topics would provide sufficient exposure at this point of the study of PDEs. Bear in mind that a Dirac delta “function” is used to model a sharp blow to a system, as you did in the ODEs course. Such sharp blows occur for systems whose mathematical description requires PDEs as well. In turn, Dirac delta-type functions naturally arise in the PDEs themselves.

The last of the main solution methods suitable for an undergraduate introduction to PDEs is the Fourier transform, for which the last unit of this course has been reserved.

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4.1 Linear and Nonlinear Waves

4.1.1 Stationary Waves

Reading: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 2: Linear and Nonlinear Waves: Introduction and Stationary Waves”
Link: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 2: Linear and Nonlinear Waves: Introduction and Stationary Waves ” (PDF)

Instructions: Click on the link above. Scroll down to the Chapter 2, and click on the link to download the PDF. Read pages 12-15: the introductory material (about one page) and section 2.1, which will introduce you to wave phenomena with a seemingly simple example.

Studying this reading should take approximately 30 minutes.

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4.1.2 Transport and Traveling Waves

Web Media: YouTube: commutant’s “PDE Part 3: Transport Equation: Derivation” and “PDE Part 4: Transport Equation: General Solution”
Links: YouTube: commutant’s “PDE Part 3: Transport Equation: Derivation ” and “PDE Part 4: Transport Equation: General Solution ” (YouTube)

Instructions: Click on the links above, and watch the two videos. The author briefly explains the transport equation and then derives its general solution.

Watching these lectures and pausing to take notes should take approximately 30 minutes.

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Reading: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 2: Linear and Nonlinear Waves: 2.2: Transport and Traveling Waves”
Link: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 2: Linear and Nonlinear Waves: 2.2: Transport and Traveling Waves ” (PDF)

Instructions: Click on the link above. Scroll down to the Chapter 2, and click on the link to download the PDF. Read section 2.2 on pages 15–25. This section introduces you to the use of characteristic curves, the lines and curves in the plane along which a signal propagates, with or without decay of information. In these examples, the ODEs that the characteristic curves solve are autonomous, and therefore the characteristics do not cross.

Studying this reading should take approximately 2 hours.

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Web Media: YouTube: commutant’s “PDE Part 5: Method of Characteristics”
Link: YouTube: commutant’s “PDE Part 5: Method of Characteristics ” (YouTube)

Instructions: Click on the link above, and watch the YouTube clip. The lecturer gives an introduction to the method of characteristics explained in the reading above.

Watching this video and pausing to take notes should take approximately 20 minutes.

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4.1.3 Nonlinear Transport and Shocks

Reading: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 2: Linear and Nonlinear Waves: 2.3: Nonlinear Transport and Shocks”
Link: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 2: Linear and Nonlinear Waves: 2.3: Nonlinear Transport and Shocks ” (PDF)

Instructions: Click on the link above. Scroll down to the Chapter 2, and click on the link to download the PDF. Read section 2.3 on pages 25-40. For these nonlinear PDEs, the method of characteristics is again employed, but now the characteristics may cross. The author details the possibilities – rarefaction waves and shock dynamics – in this situation. He explains the concept of a conservation law and derives the Rankine-Hugoniot shock speed condition.

Studying this reading should take approximately 4 hours.

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Web Media: YouTube: commutant’s “PDE Part 6: Transport with Decay and Nonlinear Transport”
Link: YouTube: commutant’s “PDE Part 6: Transport with Decay and Nonlinear Transport ” (YouTube)

Instructions: Watch the video and pay close attention as the lecturer gives an introduction to the method of characteristics explained in the reading above.

Watching this lecture and pausing to take notes should take approximately 15-20 minutes.

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Assessment: MIT: Professor Matthew Hancock’s “Problem Set 4” (PDF) and Penn State: Professor Kris Wysocki’s “Homework 1” (PDF) and “Homework 2” (PDF)
Links: MIT: Professor Matthew Hancock’s “Problem Set 4 ” (PDF) and Penn State: Professor Kris Wysocki’s “Homework 1 ” (PDF) and “Homework 2 ” (PDF)

Instructions: Click on the first link above to download the PDF, and complete problems 1, 2 (a, b, and c), and 3 of “Problem Set 4.” Then, click on the second link above, select the “Homework 1” link, and complete problems 2 and 3. Finally, click on the last link above, select the “Homework 2” link, and complete problem 2. To check your solutions for the Hancock problems, click on this link. To check your solutions for the Wysocki problems, follow the second and third links again, and click on the corresponding link for solutions to “Homework 1” and “Homework 2.”

You should dedicate approximately 4 hours to complete this assessment.

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4.1.4 D’Alembert’s Solution to the Wave Equation

Reading: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 2: Linear and Nonlinear Waves: 2.4: The Wave Equation—d’Alembert’s Solution”
Link: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 2: Linear and Nonlinear Waves: 2.4: The Wave Equation—d’Alembert’s Solution ” (PDF)

Instructions: Click on the link above. Scroll down to the Chapter 2, and click on the link to download the PDF. Read Section 2.4 on pages 40-51. D’Alembert’s Solution to the wave equation is one of the most important examples you will learn in this course. Having learned to use characteristics in the previous subunits, the approach should be familiar. Note the definition of the domain of influence (this is also called the domain of dependence).

Studying this reading should take approximately 2 hours.

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Assessment: MIT: Professor Matthew Hancock’s “Problem Set 3” (PDF) and Penn State: Professor Kris Wysocki’s “Homework 3” (PDF)
Links: MIT: Professor Matthew Hancock’s “Problem Set 3 ” (PDF) and Penn State: Professor Kris Wysocki’s “Homework 3” (PDF)

Instructions: Click on the first link above to access the PDF, and complete problems 1-3 for “Problem Set 3.” Then, click on the second link above, select the “Homework 3” link to download the PDF file, and complete all of the problems. To check your solutions for the Hancock problems, click on this link. To check your solutions for the Wysocki problems, follow the top link again and click on the corresponding link for solutions.

You should dedicate approximately 4 hours to complete this assessment.

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4.1.5 Symmetry and Similarity

Reading: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 9: Linear and Nonlinear Evolution Equations: 9.2: Symmetry and Similarity”
Link: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 9: Linear and Nonlinear Evolution Equations: 9.2: Symmetry and Similarity ” (PDF)

Instructions: Click on the link above. Scroll down to Chapter 9, and click on the link to download the PDF. Read section 9.2 on pages 314-320.

The Fourier transform is used here to derive the fundamental solution to the heat equation, which was already seen in the section on Green’s functions. The author also tackles the heat equation with forcing.

Studying this reading should take approximately 1 hour.

The Fourier transform is used here to derive the fundamental solution to the heat equation, which was already seen in the section on Green’s functions. The author also tackles the heat equation with forcing.

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4.2 Separation of Variables

Notice that the technique of Separation of Variables brings us naturally to eigenvalue problems and thence to the construction of Fourier series solutions.

4.2.1 Separation of Variables for the Heat Equation

Reading: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 4: Separation of Variables: Introduction and 4.1: The Diffusion and Heat Equations”
Link: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 4: Separation of Variables: Introduction and 4.1: The Diffusion and Heat Equations ” (PDF)

Instructions: Click on the link above. Scroll down to Chapter 4, and click on the link to download the PDF. Read pages 102-117. This reading recalls the exponential ansatz about solutions to the heat equation, which can be considered a type of separation-of-variables technique. It then discusses smoothing and the long-time behavior of the heat equation (the way solutions decay as time approaches infinity), which are important when discussing Fourier series solutions. Three boundary value problems are then discussed: the heated ring, inhomogeneous boundary conditions, and the heat equation on a semi-infinite interval.

Studying this reading should take approximately 5 hours.

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Assessment: MIT: Professor Matthew Hancock’s “Problem Set 1” and “Problem Set 2” and Penn State: Professor Kris Wysocki’s “Homework 4”
Links: MIT: Professor Matthew Hancock’s “Problem Set 1 ” (PDF) and “Problem Set 2 ” (PDF) and Penn State: Professor Kris Wysocki’s “Homework 4 ” (PDF)

Instructions: Click on the first link above to download the PDF, and complete problems 3 and 5 of “Problem Set 1.” Next, click on the second link above to access the PDF, and complete problems 4 and 7 of “Problem Set 2.” Finally, click on the last link above, select the link to “Homework 4,” and complete problems 3 and 4. To check your solutions for the Hancock problems, click on these links, Problem Set 1 and Problem Set 2. To check your solutions for the Wysocki problems, follow the third link again and click on the corresponding link for solutions to “Homework 4.”

You should dedicate approximately 3 hours to complete this assessment.

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4.2.2 Separation of Variables for the Wave Equation

Reading: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 4: Separation of Variables: 4.2: The Wave Equation”
Link: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 4: Separation of Variables: 4.2: The Wave Equation ” (PDF)

Instructions: Click on the link above. Scroll down to Chapter 4, and click on the link to download the PDF. Read section 4.2 on pages 117-126. This reading works through solutions to several boundary value problems for the wave equation using separation of variables and Fourier series solutions. It then specifically reflects on the d’Alembert formula on a bounded interval, which is the sum of one such series.

Studying this reading should take approximately 1 hour.

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Assessment: MIT: Professor Matthew Hancock’s “Problem Set 3,” Penn State: Professor Kris Wysocki’s “Homework 6,” and Trinity College, Dublin: Professor Sarah McMurry’s “Problem Sheet 7”
Links: MIT: Professor Matthew Hancock’s “Problem Set 3 ” (PDF), State: Professor Kris Wysocki’s “Homework 6 ” (PDF), and Trinity College, Dublin: Professor Sarah McMurry’s “Problem Sheet 7 ” (PDF)

Instructions: Click on the first link above to access the PDF, and complete problem 4 of “Problem Set 3.” Then, click on the second link above, select the link to “Homework 6” to access the PDF, and complete problem 2. Finally, click on the last link above, scroll down to the “Problem Sheets” section, select the link to “Problem Sheet 7,” and complete problems 1 and 3. To check your solutions for the Hancock problems, click on this link. To check your solutions for the Wysocki and the McMurry problems, follow the links above again and click on the corresponding link for solutions to “Homework 6” and “Problem Sheet 7.”

You should dedicate approximately 2 hours to complete this assessment.

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4.2.3 Separation of Variables for the Laplace and Poisson Equations

Reading: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 4: Separation of Variables: 4.3: The Planar Laplace and Poisson Equations”
Link: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 4: Separation of Variables: 4.3: The Planar Laplace and Poisson Equations ” (PDF)

Instructions: Click on the link above. Scroll down to Chapter 4, and click on the link to download the PDF. Read section 4.3 on pages 126-142. In this reading, the Laplace and Poisson equations are solved in both the Cartesian and the more natural polar-coordinate setting. The Poisson integral formula is derived. In the last two pages, the author reflects on the maximum principle, the mean value property, and the analyticity of harmonic functions, concepts to which you were first introduced in Unit 1.

Studying this reading should take approximately 6 hours.

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Assessment: Penn State: Professor Kris Wysocki’s “Homework 5” and “Homework 6;” and Trinity College, Dublin: Professor Sarah McMurry’s “Problem Sheet 7”
Links: Penn State: Professor Kris Wysocki’s “Homework 5 ” (PDF) and “Homework 6 ;” (PDF) and Trinity College, Dublin: Professor Sarah McMurry’s “Problem Sheet 7 ” (PDF)

Instructions: Click on the first link above, select the “Homework 5” link, and complete problems 2 and 3. Then, click on the second link above, select the “Homework 6 link to download the PDF, and complete problem 4. Finally, click on the last link above, scroll down to the “Problem Sheets” section, select the link to “Problem Sheet 7,” and complete problem 2. To check your solutions, follow the links above again, and click on the corresponding link for solutions to “Homework 5,” “Homework 6,” and “Problem Sheet 7.”

Note that problems 2 and 3 in Homework 5 are not about the Laplace equation, but rather about orthogonality and the telegraph equation.

You should dedicate approximately 3 hours to complete this assessment.

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4.2.4 Separation of Variables in Spherical Coordinates

Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “16D: Separation of Variables: Separation in Spherical Coordinates”
Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “16D: Separation of Variables: Separation in Spherical Coordinates ” (PDF)

Instructions: Click on the link above and read section 16D (pages 359-368).

The derivation of Bessel’s equation is a beautiful example of the application of separation of variables. If you have forgotten some of the material you learned about ODEs in MA221, it might help to refresh your knowledge of Cauchy-Euler equations.

Studying this reading should take approximately 2 hours.

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4.2.5 Separated and Quasiseparated Solutions

Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “16E: Separation of Variables: Separated vs. Quasiseparated”
Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “16E: Separation of Variables: Separated vs. Quasiseparated ” (PDF)

Instructions: Click on the link above and read section 16E (page 369).

This short section is really just a remark on why it makes sense to allow complex values for the constants that are introduced in the process of applying separation of variables.

Studying this reading should take approximately 15 minutes.

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4.2.6 Differential Operators as Polynomials

Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “16F: Separation of Variables: The Polynomial Formalism”
Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “16F: Separation of Variables: The Polynomial Formalism ” (PDF)

Instructions: Click on the link above and read section 16F (pages 369-371).

By now you have almost certainly noticed that solving many PDEs, at least using separation of variables, reduces to finding roots of polynomials. There is a deep link between abstract algebra and PDEs, and it is touched upon in this section, which should give you a new perspective on the structure of PDEs.

Studying this reading should take approximately 30 minutes.

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4.3 Impulse-Response Methods

4.3.1 Generalized Functions: The Delta Function and Calculus

Reading: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 6: Generalized Functions and Green’s Functions: Introduction and 6.1: Generalized Functions”
Link: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 6: Generalized Functions and Green’s Functions: Introduction and 6.1: Generalized Functions ” (PDF)

Instructions: Click on the link above. Scroll down to Chapter 6, and click on the link to download the PDF. Read the introduction and Section 6.1 on pages 176-193.

The delta “function” is not a function per se; strictly speaking, it is a distribution. Nevertheless, it is one of the most useful abstractions in analysis, PDEs, and mathematical physics. This reading develops the idea of the delta function and how it can be combined with standard techniques from calculus and Fourier analysis.

As noted earlier, the material in this subunit could technically be skipped without negatively impacting your initial exposure to the study of PDEs.

Reading this chapter should take approximately 5 hours.

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Assessment: MIT: Professor Steven Johnson’s “Problem Set 5”
Link: MIT: Professor Steven Johnson’s “Problem Set 5 ” (PDF)

Instructions: Click on the link above to access the PDF file, and complete problem 3 of “Problem Set 5.” To check your solutions, follow this link.

You should dedicate approximately 15-20 minutes to completing this assessment.

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4.3.2 Green’s Functions for One-Dimensional Boundary Value Problems

Reading: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 6: Generalized Functions and Green’s Functions: 6.2: Green’s Functions for One-Dimensional Boundary Value Problems”
Link: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 6: Generalized Functions and Green’s Functions: 6.2: Green’s Functions for One-Dimensional Boundary Value Problems ” (PDF)

Instructions: Click on the link above and scroll down to Chapter 6, and click on the link to download the PDF. Read section 6.2 on pages 193-199.

Recall Green’s formula, which is practically a restatement of the integration-by-parts formula; it relates integrals of the derivatives of a function on a domain to integrals of the normal derivative of the function around the boundary. Solving boundary value problems involves making this relationship balance out, and one way to conceptualize this balancing act is through the use of Green’s functions. The two most important concepts in the construction of a Green’s function are the delta function introduced in the previous section, and the superposition principle (also called Duhamel’s principle in this setting).

As noted earlier, the material in this subunit could technically be skipped without negatively impact your initial exposure to the study of PDEs.

Studying this reading should take approximately 1 hour.

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4.3.3 The Green’s Function for the Poisson Equation

Reading: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 6: Generalized Functions and Green’s Functions: 6.3: The Green’s Function for the Poisson Equation”
Link: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 6: Generalized Functions and Green’s Functions: 6.3: The Green’s Function for the Poisson Equation ” (PDF)

Instructions: Click on the link above and scroll down to Chapter 6, and click on the link to download the PDF. Read section 6.3 on pages 199-217.

In this reading, the author quickly reviews calculus in R² and then constructs the two-dimensional delta function before constructing the Green’s function for the Poisson equation. To do so, he proves Green’s representation formula, to which you should pay close attention. Be sure to work through the proof yourself!

As noted earlier, the material in this subunit could technically be skipped without negatively impact your initial exposure to the study of PDEs.

Studying this reading should take approximately 6 hours.

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Assessment: Naval Postgraduate School: Professor Beny Neta’s Partial Differential Equations MA 3132: Solutions of Problems in Lecture Notes: “Chapter 10.3: Green's Functions: Green’s Function for Sturm-Liouville Problems”
Link: Naval Postgraduate School: Professor Beny Neta’s Partial Differential Equations MA 3132: Solutions of Problems in Lecture Notes: “Chapter 10.3: Green's Functions: Green’s Function for Sturm-Liouville Problems ” (PDF)

Instructions: Click on the link above. Under “MA3132 Lecture Notes and Solution Manual,” find the link “Solution Manual for MA3132 in pdf.” Click on the link to download the document; it contains problems and solutions.

In the PDF, scroll down to page 320, and attempt problems 1-4. In problem 4, recall that script “L” represents a linear operator. When finished, go to page 326 to find the solutions.

You should dedicate approximately 2 hours to completing this assessment.

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Unit 5: The Fourier Transform

The basic idea behind the Fourier transform is that the frequency information hidden in a waveform or function because of its time-dependence can be extracted by transforming that function into a different space where the time-dimension is replaced with a frequency dimension. In this space, relationships between derivatives of the solution to a PDE are transformed into algebraic constraints that the Fourier transform of the solution must satisfy. While Fourier series are suited for boundary-value problems on bounded intervals, the Fourier transform is used to solve boundary-value problems on infinite domains.

In this unit, proceeding as before, the mathematical foundations for the technique are laid down before the use of the technique is demonstrated. Notice the similarity between the Fourier transform solution to the Heat equation via convolution and the Green’s functions derived earlier.

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5.1 The Fourier Transform

5.1.1 One-Dimensional Fourier Transforms

Reading: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 8: Fourier Transforms: Introduction and the Fourier Transform”
Link: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 8: Fourier Transforms: Introduction and the Fourier Transform ” (PDF)

Instructions: Click on the link above, scroll down to “Chapter 8,” and click on the link to download the PDF. Read the introduction and section 8.1 on pages 283-293.

The Fourier transform is an operator that maps functions into a space where their derivatives obey algebraic relationships. (It is similar to the Laplace transform that you learned about in MA221.)

Studying this reading should take approximately 2 hours.

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5.1.2 Fourier Transforms of Derivatives and Integrals

Reading: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 8: Fourier Transforms: 8.2: Derivatives and Integrals”
Link: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 8: Fourier Transforms: 8.2: Derivatives and Integrals ” (PDF)

Instructions: Click on the link above, scroll down to Chapter 8, and click on the link to download the PDF. Read section 8.2 on pages 293-295.

Be sure that you understand what the identities outlined here are saying and be able to apply them in appropriate situations.

Studying this reading should take approximately 30 minutes.

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5.1.3 Green’s Functions and Convolution

Reading: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 8: Fourier Transforms: 8.3: Green’s Functions and Convolution”
Link: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 8: Fourier Transforms: 8.3: Green’s Functions and Convolution” (PDF)

Instructions: Click on the link above, scroll down to Chapter 8, and click on the link to download the PDF. Read section 8.3 on pages 295-300.

Convolution is a special operation because the Fourier transform of the convolution of two functions is the product of their Fourier transforms. Unsurprisingly, the Green’s function for the heat equation and similar convolution-based formulas can be derived using the Fourier transform.

Studying this reading should take approximately 1 hour.

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5.1.4 The Fourier Transform on Hilbert Space

Reading: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 8: Fourier Transforms: 8.4: The Fourier Transform on Hilbert Space”
Link: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 8: Fourier Transforms: 8.4: The Fourier Transform on Hilbert Space ” (PDF)

Instructions: Click on the link above, scroll down to Chapter 8, and click on the link to download the PDF. Read section 8.4 on pages 300-304.

We have seen the Fourier transform as a mapping from the set of square-integrable functions on the real line to itself; this is a special example of a Hilbert space. Other spaces are possible, but that is outside the scope of this course.

Studying this reading should take approximately 1 hour.

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Assessment: Naval Postgraduate School: Professor Beny Neta’s Partial Differential Equations MA 3132: Solutions of Problems in Lecture Notes: “Chapter 9.2: Fourier Transform Solutions of PDEs: Fourier Transform Pair”
Link: Naval Postgraduate School: Professor Beny Neta’s Partial Differential Equations MA 3132: Solutions of Problems in Lecture Notes: “Chapter 9.2: Fourier Transform Solutions of PDEs: Fourier Transform Pair ” (PDF)

Instructions: Click on the link above. Under “MA3132 Lecture Notes and Solution Manual,” find the link “Solution Manual for MA3132 in pdf.” Click on the link to download the document; it contains problems and solutions.

In the PDF, scroll down to page 284 and attempt problems 1-5. When finished, go to page 285 to find the solutions.

You should dedicate approximately 4 hours to complete this assessment.

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5.2 Fourier Transform Solutions to PDEs on Unbounded Domains

5.2.1 The Heat Equation

Reading: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 9: Linear and Nonlinear Evolution Equations: 9.1: The Fundamental Solution to the Heat Equation”
Link: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 9: Linear and Nonlinear Evolution Equations: 9.1: The Fundamental Solution to the Heat Equation ” (PDF)

Instructions: Click on the link above, scroll down to Chapter 9, and click on the link to download the PDF. Read section 9.1 on pages 307-314 (stop at the section on Black-Scholes).

The Fourier transform is used here to derive the fundamental solution to the heat equation, which you have already seen in the section on Green’s functions. The author also tackles the heat equation with forcing.

Studying this reading should take approximately 1 hour.

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5.2.2 The Wave Equation

Reading: University of British Columbia: Professor Joel Feldman’s “Using the Fourier Transform to Solve PDEs”
Link: University of British Columbia: Professor Joel Feldman’s “Using the Fourier Transform to Solve PDEs ” (PDF)

Instructions: Click on the link above, scroll down to the “Notes” section, and then click on the link titled “Using the Fourier Transform to Solve PDEs.” Read all of the lecture notes (4 pages).

These notes describe how to use the Fourier transform to solve the wave equation and the telegraph equation.

Studying this reading should take approximately 45 minutes.

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5.2.3 Laplace’s Equation

Reading: MIT: Professor Matthew Hancock’s “Infinite Spatial Domains and the Fourier Transform: Fourier Transform Solution to Laplace’s Equation”
Link MIT: Professor Matthew Hancock’s “Infinite Spatial Domains and the Fourier Transform: Fourier Transform Solution to Laplace’s Equation ” (PDF)

Instructions: Click on the link above. These are lecture notes from a PDE course at MIT. The beginning of the notes discusses some of the material covered in the previous sections. The material on Laplace’s equation begins on page 12. Read the entire set of lecture notes (13 pages).

Studying this reading should take approximately 4 hours.

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Assessment: Naval Postgraduate School: Professor Beny Neta’s Partial Differential Equations MA 3132: Solutions of Problems in Lecture Notes: “Chapter 9.4: Fourier Transform Solutions of PDEs: Fourier Transforms of Derivatives”
Link: Naval Postgraduate School: Professor Beny Neta’s Partial Differential Equations MA 3132: Solutions of Problems in Lecture Notes: “Chapter 9.4: Fourier Transform Solutions of PDEs: Fourier Transforms of Derivatives ” (PDF)

Instructions: Click on the link above. Under “MA3132 Lecture Notes and Solution Manual,” find the link “Solution Manual for MA3132 in pdf.” Click on the link to download the document; it contains problems and solutions.

In the PDF, scroll down to page 290 and attempt problems 1-5. When finished, go to page 291 to find the solutions.

You should dedicate approximately 3 hours to complete this assessment.

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Assessment: MIT: Final Exam for Linear Partial Differential Equations (Fall 2006)
Link: MIT: Final Exam (PDF) for Linear Partial Differential Equations (Fall 2006)

Instructions: Click on the link above. In row 3 (Final Exam), go to the third (“tests”) column, and click on the “PDF” link. The final exam will download in PDF. Skip problem 4.

To check your solutions, follow the solutions link on the same page.

You should dedicate approximately 3 hours to complete this assessment.

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Final Exam

Final Exam: The Saylor Founation’s “MA222 Final Exam”
Link: The Saylor Founation’s “MA222 Final Exam”

Instructions: You must be logged into your Saylor Foundation School account in order to access this exam. If you do not yet have an account, you will be able to create one, free of charge, after clicking the link.
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