Fourier Series (for PDEs) w/ Fourier Polynomials (Orthogonal Projections in Inner Product Spaces)

Fourier Series (for Partial Differential Equations) are Constructed with Fourier Polynomials, which are Orthogonal Projections in Inner Product Spaces (in this case, the Function Space of Real-Valued Continuous Functions C[-pi,pi] with the inner product of f and g defined to be the integral from -pi to pi of f(t) times g(t)), Introduction to Partial Differential Equations (PDEs): the Heat Equation, the Wave Equation, and Laplace's Equation.

The Orthogonal Decomposition Theorem and Best Approximation Theorem justify our approach to approximating functions with Fourier polynomials based on an inner product defined with an integral on C[-pi,pi], which ultimately lead to Fourier series. The set {1,cos(t),cos(2t),... cos(nt) ,sin(t),sin(2t),...,sin(nt)} forms an orthogonal basis for the finite-dimensional subspace ov C[-pi,pi] we are interested in. Fourier series (which are infinite sums of Fourier polynomials) in turn, have application to Partial Differential Equations (PDEs). We will focus on the Heat Equation, the Wave Equation, and Laplace's Equation.

Linear Algebra and Its Application, 5th Edition (David Lay, Steven Lay, Judi McDonald): https://amzn.to/35qHKc4. Amazon Prime Student 6-Month Trial: https://amzn.to/3iUKwdP.

Bill Kinney's Differential Equations and Linear Algebra Course, Lecture 34B.

(a.k.a. Differential Equations with Linear Algebra, Lecture 34B, a.k.a. Continuous and Discrete Dynamical Systems, Lecture 34B).

#fourierseries #fourierpolynomial #partialdifferentialequation

Google drive link for Differential Equations and Linear Algebra course lecture documents: https://drive.google.com/drive/folders/1kM9ko27wA5NvmZ8CxHIKygdi-klzW0UZ?usp=sharing

Using Mathematica for ODEs (Ordinary Differential Equations) Playlist: https://www.youtube.com/playlist?list=PLmU0FIlJY-MlfXdrINHtYvBJINVIfQu7z

Visual Linear Algebra Online (at infinityisreallybig.com): https://infinityisreallybig.com/category/linear-algebra/

Infinite Powers, How Calculus Reveals the Secrets of the Universe (by Steven Strogatz): https://amzn.to/2XXRCF6

Another Differential Equations lecture I made: Differential Equations: As Much As You Can Possibly Learn About in 50 Minutes, especially Population Models: https://www.youtube.com/watch?v=thX3bgmRbkc

Check out my blog: https://infinityisreallybig.com/

Bethel University is a Christian liberal arts university in St. Paul, Minnesota with strong science, engineering, mathematics and computer science departments. You can also get to know your professors personally. https://www.bethel.edu/

(0:00) We are about to start the basics of partial differential equations
(0:48) Fourier polynomials on [-π,π]
(4:47) An orthogonal set in C[-π,π]
(10:52) Apply the Orthogonal Decomposition Theorem
(12:47) nth degree Fourier polynomial
(13:41) Fourier coefficients
(16:30) Fourier polynomial graphs to approximate f(t) = t and f(t) = t^2
(18:48) What are Fourier series?
(22:16) Fourier series and PDEs (Partial Differential Equations)
(23:32) Heat Equation, Wave Equation, Laplace’s Equation
(25:28) Initial data is often represented by an arbitrary continuous function f(x)
(26:39) Fourier coefficients on Mathematica

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Видео Fourier Series (for PDEs) w/ Fourier Polynomials (Orthogonal Projections in Inner Product Spaces) канала Bill Kinney