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.
Unit 4 Time Advisory
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.
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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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.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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displayed on the webpages above.
- 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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displayed on the webpages above.
- 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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displayed on the webpages above.
- 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
R^{2} 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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