# MA222: Introduction to Partial Differential Equations

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.

This unit should take you approximately 55.75 hours to complete.

☐    Subunit 4.1: 18.75 hours ☐    Subunit 4.1.1: 0.5 hours

☐    Subunit 4.1.2: 3 hours

☐    Subunit 4.1.3: 8.25 hours

☐    Subunit 4.1.4: 6 hours

☐    Subunit 4.1.5: 1 hour

☐    Subunit 4.2: 22.75 hours ☐    Subunit 4.2.1: 8 hours

☐    Subunit 4.2.2: 3 hours

☐    Subunit 4.2.3: 9 hours

☐    Subunit 4.2.4: 2 hours

☐    Subunit 4.2.5: 0.25 hours

☐    Subunit 4.2.6: 0.5 hours

☐    Subunit 4.3: 14.25 hours ☐    Subunit 4.3.1: 5.25 hours

☐    Subunit 4.3.2: 1 hour

☐    Subunit 4.3.3: 8 hours

Unit4 Learning Outcomes
Upon successful completion of this unit, the student will be able to:
- solve the linear transport equation using the method of characteristics; - solve the nonlinear transport equation in cases where the solutions are rarefaction waves and in cases where there are shock dynamics; - define conservation law; - state the Rankine-Hugoniot shock speed condition; - solve the one-dimensional wave equation using d’Alembert’s formula; - define domain of influence and explain its relationship to the 1-D wave equation; - find similarity solutions to traveling wave problems; - solve the heat and wave equations using separation of variables and apply boundary conditions; - solve the Laplace and Poisson equations in polar coordinates and apply boundary conditions; - solve the Laplace equation in spherical coordinates; - define the delta function and explain the concept of generalized functions; - apply ideas from calculus and Fourier series to generalized functions; - relate Green’s functions to the delta function and the superposition principle; - derive Green’s representation formula; and - use Green’s functions to solve the Poisson equation on the unit disk.

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.

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.

• 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.

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.

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.

• 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.

• 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)

You should dedicate approximately 4 hours to complete this assessment.

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.

• 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)

You should dedicate approximately 4 hours to complete this assessment.

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)

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.

• 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)

You should dedicate approximately 3 hours to complete this assessment.

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.

• 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)

You should dedicate approximately 2 hours to complete this assessment.

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.

• 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)

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.

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.

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.

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.

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.

• 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.

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)

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)

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.