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
This unit should take you approximately 11 hours to complete.
☐ Subunit 2.1: 2 hours
☐ Subunit 2.2: 9 hours
☐ Subunit 2.2.1: 2.25 hours
☐ Subunit 2.2.2: 0.25 hours
☐ Subunit 2.2.3: 2 hours
☐ Subunit 2.2.4: 2 hours
☐ Subunit 2.2.5: 1.5 hours
☐ Subunit 2.2.6: 1 hour
Unit2 Learning Outcomes
Upon successful completion of this unit, the student will be able to:
- define linear operators;
- identify linear PDEs and classify them as elliptic, parabolic, or
hyperbolic;
- identify linear operations;
- identify homogeneous PDEs;
- relate solving homogeneous linear PDEs to finding kernels of linear
operators;
- identify evolution and nonevolution equations;
- define initial-value problem;
- define boundary-value problem and identify boundary conditions as
periodic, Dirichlet, Neumann, or Robin (mixed);
- explain the physical significance of boundary conditions;
- show uniqueness of solutions to the Laplace and Poisson equations
with various boundary conditions;
- show uniqueness of solutions to the heat equation with various
boundary conditions;
- show uniqueness of solutions to the wave equation with various
boundary conditions; and
- define well-posedness.
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: Find the links under "MA3132 Lecture Notes and Solution Manual" and download the solution manual. Scroll down to page 1 of the document and attempt problems 1-5. When finished, go to page 2 to find the solutions. You will use this document again in the course, so you may wish to keep your copy available.
You should dedicate approximately 2 hours to completing this assessment.Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
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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displayed on the webpages above.
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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displayed on the webpages above.
- 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.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.
- 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: You should already have this text downloaded to you computer, but if not, click on the link above to download it. Go 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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
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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displayed on the webpages above.