Math 4340 (Advanced Engineering Mathematics), Summer II 2018 (online)

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty." --Bertrand Russell

About the class

This course is an introduction to Fourier Series and Partial Differential Equations. Throughout the country, these topics are taught in a variety of contexts -- from a very theoretical course on PDEs and Applied Analysis for senior math majors, to a more computational course geared torwards engineers, e.g., a "Differential Equations II" class. My goal in this course is to strike a balance between these two extremes. I have included enough basic linear algebra (vector spaces, independence, basis, inner products, self-adjoint operators) so the students can see the mathematical structure behind the scenes. However, I have omitted advanced details such as Hilbert spaces, and different types of norms and convergence (pointwise, uniform, and in norm). My goal is for this to be useful to math, science, and engineering majors alike.

Class essentials

Free books

Peter J. Olver. Introduction to Partial Differential Equations. Springer Undergraduate Texts in Mathematics, 2014.
John Douglas Moore. Introduction to Partial Differential Equations. Kendall Hunt, 2005.
Marcus Pivato. Linear Partial Differential Equations and Fourier Theory. Cambridge University Press, 2010.

More resources

Math 2080: Differential equations, which is a prerequestite for this course
Fourier Series applet
Wave Equation applet

Lectures

Links to the individual lectures are listed below. Or, you can view the full YouTube playlist here. To avoid blurriness, these are best viewed by changing the settings to 720p (High Definition) rather than the default of 240p. This can be easily done by clicking the "wheel" on the lower right corner; right next to the "cc" button.

Editor's note: Any YouTube link with (??:??) means I haven't made the video yet.

Section 1: Some linear algebra. (4 lectures: 2 hr 55 min).

Lecture 1.1: Vector spaces [YouTube (32:49) | Slides]
Lecture 1.2: Linear independence and spanning sets [YouTube (46:33) | Slides]
Lecture 1.3: Linear maps. [YouTube (41:08) | Slides]
Lecture 1.4: Inner products and orthogonality [YouTube (47:46) | Slides]

Section 2: Linear differential equations. (7 lectures: 4 hr 51 min).

Lecture 2.1: The fundamental theorem of linear ODEs [YouTube (45:07) | Slides]
Lecture 2.2: Linear independence and the Wronskian [YouTube (39:35) | Slides]
Lecture 2.3: Inhomogeneous ODEs and affine spaces [YouTube (37:53) | Slides]
Lecture 2.4: Undetermined coefficients [YouTube (45:51) | Slides]
Lecture 2.5: Power series solutions to ODEs [YouTube (44:24) | Slides]
Lecture 2.6: Singular points and the Frobenius method [YouTube (43:21) | Slides]
Lecture 2.7: Bessel's equation [YouTube (36:34) | Slides]

Section 3: Fourier series. (8 lectures: 5 hr 16 min.)

Lecture 3.1: Fourier series and orthogonality. [YouTube (29:29) | Slides]
Lecture 3.2: Computing Fourier series and exploiting symmetry. [YouTube (50:09) | Slides]
Lecture 3.3: Solving ODEs with Fourier series [YouTube (31:48) | Slides]
Lecture 3.4: Fourier sine and cosine series. [YouTube (35:07) | Slides]
Lecture 3.5: Complex inner products and Fourier series. [YouTube (51:14) | Slides]
Lecture 3.6: Real vs. complex Fourier series. [YouTube (37:03) | Slides]
Lecture 3.7: Fourier transforms. [YouTube (49:37) | Slides]
Lecture 3.8: Pythagoras, Parseval, and Plancherel. [YouTube (31:32) | Slides]

Section 4: Boundary value problems and Sturm-Liouville theory. (6 lectures: 4 hr 1 min).

Lecture 4.1: Boundary value problems. [YouTube (56:59) | Slides]
Lecture 4.2: Symmetric and Hermitian matrices. [YouTube (32:46) | Slides]
Lecture 4.3: Self-adjoint linear operators. [YouTube (46:44) | Slides]
Lecture 4.4: Sturm-Liouville theory [YouTube (42:30) | Slides]
Lecture 4.5: Generalized Fourier series [YouTube (26:12) | Slides]
Lecture 4.6: Some special orthogonal functions [YouTube (35:43) | Slides]

Section 5: Partial differential equations (PDEs) on bounded domains. (4 lectures, 2 hrs 47 min)

Lecture 5.1: Fourier's law and the diffusion equation [YouTube (44:28) | Slides]
Lecture 5.2: Boundary conditions for the heat equation [YouTube (51:23) | Slides]
Lecture 5.3: The transport and wave equations [YouTube (40:32) | Slides]
Lecture 5.4: The Schrödinger equation [YouTube (31:04) | Slides]

Section 6: PDEs on unbounded domains.

Lecture 6.1: The heat and wave equations on the real line. [YouTube (48:39) | Slides]
Lecture 6.2: Semi-infinite domains and the reflection method. [YouTube (42:05) | Slides]
Lecture 6.3: Solving PDEs with Laplace transforms. [YouTube (??:??) | Slides]
Lecture 6.4: Solving PDEs with Fourier transforms. [YouTube (??:??) | Slides]

Section 7: Higher-dimensional PDEs. (5 lectures: 3 hr 57 min).

Lecture 7.1: Harmonic functions and Laplace's equation. [YouTube (53:53) | Slides]
Lecture 7.2: Eigenfunctions of the Laplacian. [YouTube (29:06) | Slides]
Lecture 7.3: The heat and wave equations in higher dimensions. [YouTube (54:04) | Slides]
Lecture 7.4: The Laplacian in polar coordinates. [YouTube (51:53) | Slides]
Lecture 7.5: Three PDEs on a disk [YouTube (48:02) | Slides]

Homework

Homework should be written up carefully and concisely. Please write in complete sentences. Part of your grade will be based on the presentation and clarity of your answers.

HW 1: pdf | tex. Topics: Vector spaces, linear independence, and bases. Due Thursday, June 28, 2018.
HW 2: pdf | tex. Topics: Linear maps, inner products, and orthogonality. Due Friday, June 29, 2018.
HW 3: pdf | tex. Topics: The Wronskian, affine spaces, and linear ODEs. Due Tuesday, July 3, 2018.
HW 4: pdf | tex. Topics: Cauchy-Euler equations and power series solutions. Due Friday, July 6, 2018.
HW 5: pdf | tex. Topics: The Frobenius method and Bessel's equation. Due Monday, July 9, 2018.
HW 6: pdf | tex. Topics: Real Fourier series, and Fourier sine & cosine series. Due Thursday, July 12, 2018.
HW 7: pdf | tex. Topics: Complex Fourier series, Fourier transforms, and Parseval's theorem. Due Monday, July 16, 2018.
HW 8: pdf | tex. Topics: Self-adjoint operators and Sturm-Liouville theory. Due Wednesday, July 18, 2018.
HW 9: pdf | tex. Topics: Diffusion and the heat equation. Due Friday, July 20, 2018.
HW 10: pdf | tex. Topics: The transport, wave, and Schrödinger equation. Due Monday, July 23, 2018.
HW 11: pdf | tex. Topics: PDEs on unbounded domains. Laplace & Fourier transforms. Due Thursday, July 26, 2018.
HW 12: pdf | tex. Topics: Harmonic functions and higher dimensional PDEs. Due Monday, July 30, 2018.
HW 13: pdf | tex. Topics: PDEs in other coordinate systems. Due Thursday, August 2, 2018.