# MA222: Introduction to Partial Differential Equations

Unit 3: Fourier Series on Bounded Domains   As you may have learned in a physics (or music!) class, every sound is made up of sound waves with different frequencies. Thus, for instance, a C-major chord based at middle C is made of up waves at 261.6 Hz(C), 329.6 Hz (E), and 392.0 Hz (G). The combination (or superposition) of these three individual sound waves produces a new and distinct sound. In fact, any signal or waveform (subject to some reasonable conditions) can be written as the combination of some fundamental waves, whether it is a water wave, a light or electromagnetic wave, or a sound wave. Mathematically, a waveform is just a function, and, conversely, every function can be viewed as a waveform. Most (again, subject to some reasonable conditions) can be written as a combination of some fundamental wave-like functions, namely sines and cosines. This was the observation of Fourier, a great scientist of the nineteenth century. It took years for other mathematicians to accept his work and come up with a theoretical framework for his results. The study of such ideas is today called Harmonic Analysis, and it has gone far beyond Fourier’s original ideas to touch almost every part of modern science and technology, from medical imaging to wireless communication.

In this unit, you will learn to apply Fourier’s original insight – that most useful functions can be written as linear combinations of sine and cosine waves. To do this rigorously, some mathematical underpinnings must be established, such as the definition and elaboration of the square-integrable functions, the differences between different modes of convergence, and the orthogonality of the basis of trigonometric functions, some of which you will recall from MA241. After this has been accomplished, you will learn how to calculate Fourier series for suitable functions.

This unit should take you approximately 14.5 hours to complete.

☐    Subunit 3.1: 4.75 hours ☐    Subunit 3.1.1: 0.75 hours

☐    Subunit 3.1.2: 0.25 hours

☐    Subunit 3.1.3: 0.5 hours

☐    Subunit 3.1.4: 3 hours

☐    Subunit 3.1.5: 0.25 hours

☐    Subunit 3.2: 9.75 hours ☐    Subunit 3.2.1: 4 hours

☐    Subunit 3.2.2: 1 hour

☐    Subunit 3.2.3: 0.25 hours

☐    Subunit 3.2.4: 4.5 hours

Unit3 Learning Outcomes
Upon successful completion of this unit, the student will be able to: - define, characterize, and use inner products; - define the space of L2 functions and state its key properties; - show that a function belongs to the space of Lfunctions; - define orthogonality and show the orthogonality of certain trigonometric functions; - define the Haar Basis; - distinguish between pointwise, uniform, and L2 convergence; - show convergence of sequences; - use the Weierstrass M-test to show that Fourier series converge; - define orthogonal and orthonormal basis; - define Fourier series on [0,π] and [0,L] and identify sufficient conditions for their convergence and uniqueness; - define the Gibbs Phenomenon; and - compute Fourier coefficients and construct Fourier series.

3.1 Necessary Functional Analysis   3.1.1 Inner Products   - Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “6A: Some Functional Analysis: Inner Products” Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “6A: Some Functional Analysis: Inner Products” (PDF)

Instructions: Click on the link above and read section 6A (pages 103-105).

Solving linear PDEs using algebraic methods requires an understanding of inner product spaces.

Studying this reading should take approximately 20 minutes.

3.1.2 L2 Space   - Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “6B: Some Functional Analysis: L2 Space” Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “6B: Some Functional Analysis: L2 Space” (PDF)

Instructions: Click on the link above and read section 6B (pages 105-108).

In order to apply ideas from linear algebra to PDEs, one requires a normed linear space with a well-defined norm. Using the inner product ideas from the previous section, the author defines the vector space L2 of square-integrable functions.

Studying this reading should take approximately 15-20 minutes.

3.1.3 Orthogonality   - Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “6D: Some Functional Analysis: Orthogonality” Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “6D: Some Functional Analysis: Orthogonality” (PDF)

Instructions: Click on the link above and read section 6D (pages 112-116).

In this section, the author shows the orthogonality of trigonometric functions, which will be a crucial fact for the development of Fourier series in upcoming units.

Studying this reading should take approximately 30 minutes.

3.1.4 Convergence Concepts   - Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “6E: Some Functional Analysis: Convergence Concepts” Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “6E: Some Functional Analysis: Convergence Concepts” (PDF)

Instructions: Click on the link above and read section 6E (pages 116-131).

In many cases, solutions to PDEs can be constructed using sequences or series of functions. However, as you know from calculus, the construction of a sequence or series does not guarantee that it has a well-defined limit. For this reason, the author defines several modes of convergence, which will be used to establish the validity of these types of solutions. The diagrams used to illustrate these concepts are excellent.

Studying this reading should take approximately 3 hours.

3.1.5 Orthogonal and Orthonormal Bases   - Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “6F: Some Functional Analysis: Orthogonal and Orthonormal Bases” Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “6F: Some Functional Analysis: Orthogonal and Orthonormal Bases (PDF)

Instructions: Click on the link above and read section 6F (pages 131-133).

Studying this reading should take approximately 15-20 minutes.

3.2 Fourier Series   3.2.1 Eigensolutions to Linear Evolution Equations   - Reading: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 3: Fourier Series: 3.1: Eigensolutions to Linear Evolution Equations” Link: University of Minnesota: Professor Peter Olver’s Introduction to Partial Differential Equations: “Chapter 3: Fourier Series: 3.1: Eigensolutions to Linear Evolution Equations (PDF)

In this reading, the author motivates the construction of Fourier series by tying them to eigenvalue problems for linear evolution equations. We will return to these ideas when we discuss the technique of separation of variables.

Studying this reading should take approximately 4 hours.

3.2.2 Fourier Series on [0, ?]   - Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “7A: Fourier Sine Series and Cosine Series: Fourier (Co)sine Series on [0, ?]” Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “7A: Fourier Sine Series and Cosine Series: Fourier (Co)sine Series on [0, π]” (PDF)

`````` Instructions: Click on the link above and read section 7A (pages
137-144).

All of the facts that were established in section 3.1 about inner
products, orthogonality, the L<sup>2</sup> functions, and
orthonormal bases are brought into play in this section. Fourier
series are one of your most powerful tools for solving PDEs. They
are also extremely important for signal and image processing, so
investment in understanding this concept is well-worth your time.

Studying this reading should take approximately 1 hour.

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3.2.3 Fourier Series on [0,L]   - Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “7B: Fourier Sine Series and Cosine Series: Fourier (Co)sine Series on [0, L]” Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “7B: Fourier Sine Series and Cosine Series: Fourier (Co)sine Series on [0, L] (PDF)

Instructions: Click on the link above and read section 7B (pages 144-146).

Studying this reading should take approximately 15-20 minutes.

3.2.4 Computing Fourier Coefficients   - Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “7C: Fourier Sine Series and Cosine Series: Computing Fourier (Co)sine Coefficients” Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “7C: Fourier Sine Series and Cosine Series: Fourier (Co)sine Coefficients (PDF)

Instructions: Click on the link above and read section 7C (pages 147-158).

This section reviews some important methods of computing Fourier coefficients for various functions, including polynomials, step functions, and piecewise linear functions.

Studying this reading should take approximately 1 hour and 30 minutes.