 # MA222: Introduction to Partial Differential Equations

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
This unit should take you approximately 8.25 hours to complete.

☐    Subunit 1.1: 3 hours

☐    Subunit 1.2: 5.25 hours ☐    Subunit 1.2.1: 1.5 hours

☐    Subunit 1.2.2: 2 hours

☐    Subunit 1.2.3: 1.75 hours

Unit1 Learning Outcomes
Upon successful completion of this unit, the student will be able to:

• recall notation for partial derivatives;
• recall how to manipulate complex numbers;
• recall how to use Cartesian, polar, and spherical coordinates;
• recall important concepts from Vector Calculus;
• 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; and
• state the mean value property for harmonic functions.

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.

• 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.

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.

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.

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.

1.2.2.4 Properties of Harmonic Functions   - Reading: Mathnotes.me: “Laplace Equation” The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.

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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.

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.