You CAN Ace Differential Equations  

17calculus > differential equations > fourier series  


Topics You Need To Understand For This Page
precalculus  even and odd functions 
Differential Equations Alpha List
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Fourier Series 

This page covers two areas related to Fourier Series. First, we present an introduction to Fourier Series, then we discuss how to solve differential equations using Fourier Series. If you are just learning about Fourier Series, you can go through the introduction and practice problems and skip the section related to solving differential equations. 
What is a Fourier Series? 
The main idea of Fourier Series is that we want to build an infinite series, using the basic trig functions sine and cosine, that is equivalent to a more complicated function. The series can then be manipulated more easily than the original function.
Here is a great video to get you started. He explains why we need to build these functions, goes through an example and then explains the big picture.
Dr Chris Tisdell  building functions  
How to Calculate Fourier Series 
As you saw in that video, there are some basic equations required to calculate the Fourier Series.
Fourier Series Equations  

original function 
build a Fourier Series for a function \(f(t)\) with period \(2L\) 

requirements of \(f(t)\) 
\(f(t)\) and it's derivative \(f'(t)\) must be piecewise continuous on the interval \([L,L]\) 

Fourier Series 
\(\displaystyle{ f(t) = a_0 + \sum_{n=1}^{\infty}{ \left[ a_n \cos \frac{n \pi t}{L} + b_n \sin \frac{n \pi t}{L} \right] } }\)  
constants 
\(\displaystyle{ a_0 = \frac{1}{2L} \int_{L}^{L}{f(t)~dt} }\) 
\(\displaystyle{ a_n = \frac{1}{L} \int_{L}^{L}{f(t)\cos \frac{n\pi t}{L} ~dt} }\) 
\(\displaystyle{ b_n = \frac{1}{L} \int_{L}^{L}{f(t)\sin \frac{n\pi t}{L} ~dt} }\) 
Knowing if the original \(f(t)\) is either even or odd can help us a lot when finding the Fourier Series. Of course, we do not require that \(f(t)\) be even or odd, but you remember from precalculus that cosine is an even function and sine is odd. So, for even functions \(b_n=0\) and for odd functions \(a_n=0\).
[ more discussion and practice problems on the way ]
Solving Differential Equations 
These videos show how to use Fourier Series to solve differential equations.
MIT OCW Lec 15  MIT 18.03 Differential Equations, Spring 2006  Introduction to Fourier Series; Basic Formulas for Period 2(pi)  
MIT OCW  Lec 16  MIT 18.03 Differential Equations, Spring 2006  Continuation: More General Periods; Even and Odd Functions; Periodic Extension  
MIT OCW  Lec 17  MIT 18.03 Differential Equations, Spring 2006  Finding Particular Solutions via Fourier Series; Resonant Terms; Hearing Musical Sounds  
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Practice Problems 

Instructions   Unless otherwise instructed, find the Fourier Series for these functions.
Level A  Basic 
Practice A01  

\(\displaystyle{ f(t) = \left\{
\begin{array}{rr}
1 & \pi < t < 0 \\
0 & t = 0, \pm \pi \\
1 & 0 < t < \pi
\end{array} \right.
}\)  
solution 
Practice A02  

\(\displaystyle{f(x) = \left\{
\begin{array}{rr}
0 & 1 \leq x \leq 0 \\
1 & 0 < x < 1
\end{array} \right.
}\)  
solution 
Practice A03  

\(\displaystyle{ f(x) = \left\{
\begin{array}{rr}
3 & 1 < x < 0 \\
3 & 0 < x < 1
\end{array} \right.
}\)  
solution 