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.
Unit 5 Time Advisory
This unit should take you approximately 20.25 hours to complete.
☐ Subunit 5.1: 8.5 hours ☐ Subunit 5.1.1: 2 hours
☐ Subunit 5.1.2: 0.5 hours
☐ Subunit 5.1.3: 1 hour
☐ Subunit 5.1.4: 5 hours
☐ Subunit 5.2: 11.75 hours ☐ Subunit 5.2.1: 1 hour
☐ Subunit 5.2.2: 0.75 hours
☐ Subunit 5.2.3: 10 hours
Unit5 Learning Outcomes
Upon successful completion of this unit, the student will be able to:
- 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;
- relate Fourier solutions to Green’s functions; and
- use the Fourier transform to solve the heat and wave equations on
unbounded domains.
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:
[“](http://www.math.umn.edu/~olver/pdn.html)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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displayed on the webpages above.
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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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 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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displayed on the webpages above.
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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displayed on the webpages above.
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.
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 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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
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.