18.4.1 - Fourier Series on [−L,L]
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Introduction to Fourier Series
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Welcome class! Today, we're diving into Fourier series, which are instrumental in expressing periodic functions as infinite sums of sine and cosine functions. Why do you think this is important in engineering?
I think it might help in understanding how different states change over time and space.
Exactly! For example, when we analyze vibrations in structures or heat distribution, we can represent complex behaviors through these series. Does anyone know what the basic formula for a Fourier series looks like?
Isn't it something like f(x) = a0 + sum of a_n cos(nπx/L) + b_n sin(nπx/L)?
Spot on! Now, let’s discuss how we obtain the coefficients a_n and b_n needed for this expansion.
Calculating Fourier Coefficients
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To represent our function using Fourier series, we need to compute the coefficients a0, a_n, and b_n. Who can tell me how we calculate a0?
a0 is the average value of the function over the interval, right?
That's correct! We compute it as a0 = 1/(2L) * integral from -L to L of f(x) dx. Now, what about a_n?
a_n involves integrating the function multiplied by cos, right?
Right again! It's a_n = 1/L * integral from -L to L of f(x) cos(nπx/L) dx. Understanding these formulas is critical because they allow us to break down complex functions into simpler terms!
Application of Fourier Series in PDEs
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Now that we know how to represent a function with a Fourier series, why do you think this method is used in solving PDEs?
Maybe because it simplifies complex problems into manageable parts?
Exactly! Fourier series allow us to represent boundary and initial conditions accurately. For instance, when we have a heat equation, we can use those sine and cosine terms to express temperature distribution over time.
So, by applying different boundary conditions, we can get different solutions?
Correct! Each condition can lead to different sets of coefficients that represent specific physical scenarios. This is foundational in fields like civil engineering where understanding these principles is crucial!
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
The section discusses Fourier series for continuous functions defined on the interval [−L,L]. It explains how to express these functions as infinite sums of sine and cosine terms, providing the necessary formulas for computing the series coefficients. The significance of this representation is emphasized for solving problems in partial differential equations in engineering.
Detailed
In this section, we explore the application of Fourier series for representing piecewise continuous functions defined on the interval [−L,L]. A Fourier series allows us to express a periodic function f(x) as the sum of sinusoidal functions:
$$ f(x) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos\left( \frac{n \pi x}{L} \right) + b_n \sin\left( \frac{n \pi x}{L} \right) \right) $$
The coefficients in this series are calculated using the following formulas:
- $a_0 = \frac{1}{2L} \int_{-L}^{L} f(x) dx$
- $a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos\left( \frac{n \pi x}{L} \right) dx$
- $b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin\left( \frac{n \pi x}{L} \right) dx$
These coefficients allow us to construct the Fourier series, which plays a crucial role in solving partial differential equations by decomposing complex functions into simpler oscillatory components. This is particularly useful in civil engineering applications, facilitating the analysis of structures, heat conduction, and other phenomena governed by differential equations.
Youtube Videos
![Fourier Analysis 4: Example of Fourier Series on [-L, L]](https://img.youtube.com/vi/JgYx5jTYv4s/mqdefault.jpg)
![Fourier Analysis 3: Fourier Series of function on [-L, L]](https://img.youtube.com/vi/M7HLT3cwJPI/mqdefault.jpg)
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Fourier Series Representation
Chapter 1 of 2
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Chapter Content
For a piecewise continuous function f(x) on [−L,L]:
∞
f(x) = a_0 + ∑ (a_n cos(nπx/L) + b_n sin(nπx/L))
Where coefficients are:
Detailed Explanation
In this chunk, we introduce the representation of a piecewise continuous function f(x) defined over the interval [-L, L]. The Fourier series formula represents this function as an infinite sum of cosine and sine terms.
- The term a_0 represents the average value of the function over the interval, and it is the first term of the series.
- The terms a_n and b_n are coefficients that determine how much each sine and cosine function contributes to the overall shape of the function f(x). This infinite sum allows us to build a complex waveform from simple harmonic components.
Examples & Analogies
Think of the Fourier series like a well-orchestrated musical band. Each musician (sine and cosine terms) plays their part to create a beautiful symphony (the complex function f(x)). Just as the main melody can be broken down into harmonics, f(x) can be expressed as a combination of simple waveforms.
Calculating Fourier Coefficients
Chapter 2 of 2
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Chapter Content
• a_0 = (1/2L) ∫ from -L to L f(x) dx
• a_n = (1/L) ∫ from -L to L f(x) cos(nπx/L) dx
• b_n = (1/L) ∫ from -L to L f(x) sin(nπx/L) dx
Detailed Explanation
This chunk explains how to find the coefficients a_0, a_n, and b_n required in the Fourier series representation.
- To calculate a_0, we integrate the function f(x) over the interval from -L to L and take the average.
- The coefficients a_n and b_n are found by multiplying f(x) by the corresponding cosine and sine functions respectively, integrating over the same interval, and then normalizing by dividing by the length of the interval (L). These coefficients are essential as they dictate how each frequency contributes to the overall function.
Examples & Analogies
Consider baking a cake. Just like each ingredient (flour, sugar, eggs) influences the taste and texture of the cake, the coefficients a_0, a_n, and b_n influence the shape and character of the waveform represented by f(x). By adjusting the amounts (coefficients), you create a cake (waveform) that can resemble many different flavors.
Key Concepts
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Fourier Series: A mathematical representation of a periodic function as an infinite series of sine and cosine terms.
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Piecewise Continuous Function: A function with a limited number of discontinuities within a specified interval.
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Coefficients in Fourier Series: These calculate the contribution of each sine and cosine term in accurately expressing a function.
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Boundary Conditions: Conditions that must be satisfied at the boundaries of the domain in PDE problems.
Examples & Applications
A temperature distribution function specified for a material can be expressed as a Fourier series to analyze heat flow.
The vibrations of a beam under periodic loading can be modeled using Fourier series to determine response characteristics.
Memory Aids
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Rhymes
Sines and cosines play their part, building series that seem like art.
Stories
Imagine constructing a sound wave. Each note you play is a sine or cosine that together creates a beautiful symphony, just like Fourier series composing a complex function.
Memory Tools
Sine and cosine can be easy as A-B-C: A (always), B (balance), C (coefficients)!
Acronyms
F.C.C. - Fourier Series, Coefficients, Continuous Functions.
Flash Cards
Glossary
- Fourier Series
A way to represent a periodic function as an infinite sum of sine and cosine functions.
- Piecewise Continuous Function
A function that is continuous on each of its subintervals and has a finite number of discontinuities.
- Coefficient
A numerical factor in a term of a series, representing the weight of a particular function in the decomposition.
- Boundary Conditions
Constraints necessary to solve differential equations describing physical systems.
Reference links
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