First Order Second Order Laplace Transforms Additional Topics Applications, Practice
Separation of Variables
Linear
Integrating Factors (Linear)
Substitution
Exact Equations
Integrating Factors (Exact)
Linear
Constant Coefficients
Substitution
Reduction of Order
Undetermined Coefficients
Variation of Parameters
Polynomial Coefficients
Cauchy-Euler Equations
Chebyshev Equations
Laplace Transforms
Unit Step Function
Unit Impulse Function
Square Wave
Shifting Theorems
Solve Initial Value Problems
Classify Differential Equations
Fourier Series
Slope Fields
Wronskian
Existence and Uniqueness
Boundary Value Problems
Euler's Method
Inhomogeneous ODE's
Resonance
Partial Differential Equations
Linear Systems
Exponential Growth/Decay
Population Dynamics
Projectile Motion
Chemical Concentration
Fluids (Mixing)
Practice Problems
Practice Exam List
Exam A1
Exam A3
Exam B2

You CAN Ace Differential Equations

17calculus > differential equations > fourier series

### Topics You Need To Understand For This Page

 precalculus - even and odd functions infinite series basics of differential equations

### Differential Equations Alpha List

 Boundary Value Problems Cauchy-Euler Equations Chebyshev Equations Chemical Concentration Classify Differential Equations Constant Coefficients Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay First Order, Linear Fluids (Mixing) Fourier Series Inhomogeneous ODE's Integrating Factors (Exact) Integrating Factors (Linear) Laplace Transforms Linear Systems Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance Second Order, Linear Separation of Variables Shifting Theorems Slope Fields Solve Initial Value Problems Square Wave Substitution Undetermined Coefficients Unit Impulse Function Unit Step Function Variation of Parameters Wronskian

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### Related Topics and Links

fourier series youtube playlist

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

$$\displaystyle{ f(t) = \left\{ \begin{array}{rr} -1 & -\pi < t < 0 \\ 0 & t = 0, \pm \pi \\ 1 & 0 < t < \pi \end{array} \right. }$$
$$f(t) = f(t+2\pi)$$ for all $$t$$

solution

Practice A02

$$\displaystyle{f(x) = \left\{ \begin{array}{rr} 0 & -1 \leq x \leq 0 \\ 1 & 0 < x < 1 \end{array} \right. }$$
with period $$2$$

solution

Practice A03

$$\displaystyle{ f(x) = \left\{ \begin{array}{rr} -3 & -1 < x < 0 \\ 3 & 0 < x < 1 \end{array} \right. }$$
$$f(x) = f(x+2)$$

solution

3