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Introduction to Fourier Series
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Today we will talk about Fourier series and their importance in solving PDEs. The Fourier series allows us to express functions as a sum of sine functions. Can anyone tell me what a sine function looks like?
Isnβt it a wave-like function that oscillates?
Exactly! These sine waves can be combined to approximate more complex shapes. When we solve problems in physics, we often need to express initial conditions. For instance, how can we represent a function at time zero?
We can use Fourier series, right?
Correct! We can expand the initial condition into a Fourier series to make it easier to work with.
Applying Superposition
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Now, letβs discuss superposition. This principle states that if we can create a solution from simpler parts, we can add those parts together. How does this apply to the Fourier series?
So, we can take each sine function component and treat it separately?
Precisely! Each eigenfunction behaves independently. When we add all components, we get a complete solution to our differential equation.
Wait, so the overall solution is just the summation of these individual parts?
Thatβs right! By using superposition, we construct complex solutions from simpler sine terms.
Fourier Series Example
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Let's look at an example. If we have a function represented as $u(x, 0) = f(x)$, can anyone write down how we would approach this using Fourier series?
We would express it using a summation of sine functions, something like: $f(x) = \sum_{n=1}^{\infty} A_n \sin\left(\frac{n\pi x}{L}\right)$.
Exactly! And each coefficient $A_n$ is calculated from the initial condition. What does this allow us to do?
It helps us find the solution for $u(x, t)$ over time!
Great! Remember, we can take each $A_n$ and create our overall function with superposition.
Key Points Recap
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So, to recap, weβve learned about Fourier series and how to express initial conditions. We also talked about superposition and how it contributes to solving PDEs.
So, we sum individual sine terms to get the complete solution?
Correct! And thatβs a powerful concept in both mathematics and physics.
Thanks! I feel more confident about using Fourier series now.
Introduction & Overview
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Quick Overview
Standard
In this section, we explore how initial conditions can be expressed through Fourier series to represent solutions of differential equations. The method utilizes superposition, allowing solutions to evolve over time based on eigenfunctions determined from boundary conditions.
Detailed
Fourier Series and Superposition
In the study of partial differential equations (PDEs), specifically when applying the method of separation of variables, it becomes essential to incorporate initial conditions into our general solutions. This is where Fourier series play a significant role.
When we have an initial condition expressed as
$$ u(x, 0) = f(x) $$
we can represent the function $f(x)$ as a Fourier sine series:
$$ f(x) = \sum_{n=1}^{\infty} A_n \sin\left(\frac{n\pi x}{L}\right) $$
where $A_n$ are the Fourier coefficients determined by the boundary conditions of the problem.
These Fourier series allow us to form a solution to the PDE by substituting back into the separated equations:
$$ u(x, t) = \sum_{n=1}^{\infty} A_n \sin\left(\frac{n\pi x}{L}\right)e^{-k\left(\frac{n\pi}{L}\right)^2t} $$
This expression illustrates the superposition of solutions, where each eigenfunction component behaves independently and contributes to the overall solution behavior over time.
Therefore, the section highlights the vital concept of superposition, allowing us to construct complex solutions from simpler, manageable parts.
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Introduction to Fourier Series
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Often, the solution is a sum of eigenfunctions. The initial condition is expanded using a Fourier series.
Detailed Explanation
In this section, we learn that when solving partial differential equations (PDEs) using the method of separation of variables, the solutions to the equations often involve sums of functions called eigenfunctions. These sums can be represented through a mathematical tool known as a Fourier series. A Fourier series allows us to express complex periodic functions as sums of simpler sine and cosine functions, making it easier to analyze and calculate the values of the function at various points.
Examples & Analogies
Think of a musical chord made up of different notes. Just like multiple notes combine to create a harmonious sound, different eigenfunctions combine in a Fourier series to create a complex solution for a PDE. Imagine trying to capture the sound of a guitar. Instead of trying to play every note individually, you can blend together certain frequencies (notes) to recreate that sound in a way that's easier to manage.
Mathematical Representation of the Fourier Series
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If π’(π₯,0) = π(π₯), then:
β β
πππ₯ πππ₯ ππ 2
βπ( ) π‘
π(π₯) = βπ΄ sin( ) β π’(π₯,π‘) = βπ΄ sin( )π πΏ
π πΏ π πΏ
π=1 π=1
Detailed Explanation
The mathematical expression states that if the initial condition of our function is π(π₯) at time 0 (i.e., when t = 0), then we can describe how the function evolves over time by summing up multiple eigenfunctions. Each term in the series corresponds to a sine wave multiplied by a coefficient (π¨), and an exponential decay factor (e^{-ππ‘}). This means that weβre able to take the initial shape of the function and see how it changes over time using the Fourier series.
Examples & Analogies
Imagine you are baking a cake. The recipe calls for a mix of various ingredients (like flour, sugar, eggs), much like how the Fourier series calls for a mix of sine functions. Just as the right combination of ingredients leads to a delicious cake, the correct coefficients (π΄) lead to an accurate representation of the function over time. Each sine function can be thought of as an ingredient that contributes to the overall flavor of the dish β our final solution to the PDE.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Fourier Series: A mathematical representation of a function as an infinite sum of sine and cosine terms.
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Superposition: Using the principle that individual solution components can be added together for the overall solution.
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Initial Condition Expansion: The process of representing initial conditions of a PDE using Fourier series.
Examples & Real-Life Applications
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Examples
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To solve a heat equation with initial condition u(x,0)=f(x), we express f(x) using Fourier series and derive u(x,t) via superposition.
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In a vibrating string problem, the position of the string at t=0 can be described by a Fourier series, which helps in determining its movement over time.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π΅ Rhymes Time
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Fourier's series, wave on a leaf, summing together brings us relief.
π Fascinating Stories
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Imagine a musician creating a symphony by layering different notes; that's similar to how superposition works in Fourier series!
π§ Other Memory Gems
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F for Fourier, S for Series, S for Superposition β 'FSS' helps remember!
π― Super Acronyms
FOSS β Fourier, Overlap, Sine, Superposition.
Flash Cards
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Glossary of Terms
Review the Definitions for terms.
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Term: Fourier Series
Definition:
An infinite series that expresses a function in terms of sine and cosine functions.
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Term: Superposition
Definition:
The principle that the sum of multiple solutions to linear systems forms another valid solution.
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Term: Eigenfunction
Definition:
A non-zero function that changes at most by a scalar factor when a linear transformation is applied.
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Term: Boundary Conditions
Definition:
Conditions that specify the behavior of a function at the boundaries of its domain.