12.4 - Fourier Series and Initial Condition
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Understanding Fourier Series
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Today, we'll discuss how we can use Fourier series to solve the one-dimensional heat equation. Can anyone tell me what a Fourier series is?
Isn't it a way to represent functions as sums of sine and cosine?
Exactly! Fourier series allow us to express complex periodic functions in terms of simple sine and cosine functions. This is especially useful when dealing with our initial conditions in differential equations.
So we can represent the initial temperature distribution using a Fourier series?
That's correct! We will also find coefficients, B_n, to express the initial condition accurately. Let's delve into how to determine these coefficients.
Deriving Fourier Coefficients
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To calculate the coefficients B_n for our initial condition f(x), we use this formula: B_n = (2/L) * ∫(f(x)*sin(nπx/L))dx. Who can explain why we multiply by sin?
Because we're only dealing with the sine series due to the boundary conditions, right?
Spot on! The sine functions fulfill the Dirichlet boundary conditions. Now, could someone illustrate how we would apply this to a specific f(x)?
If f(x) = x(L−x), we'd integrate that over the interval to find B_n.
Perfect! Customizing our initial condition is crucial for proper setups. Now let’s summarize our method.
To summarize, we find B_n through integration, which allows us to build our solution for the heat equation.
Application of Fourier Series in the Heat Equation
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Having derived our coefficients, how do we use them in the solution to the heat equation?
We place the B_n values back into the series form to describe temperature over time, right?
Exactly, the final temperature solution integrates all these series together. What’s crucial about the decay of high-frequency terms?
They decay faster due to the exponential term.
Correct! This explains how we can anticipate the behavior of the heat distribution over time. Fantastic teamwork today!
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
The section elaborates on how to express an initial condition as a Fourier series to find the coefficients needed to solve the one-dimensional heat equation. The relationship between the initial temperature distribution and the Fourier coefficients is mathematically detailed.
Detailed
The Fourier series is a powerful tool to express periodic functions as a sum of sine and cosine terms. In the context of solving the one-dimensional heat equation, the initial condition defined by the temperature distribution, denoted as u(x,0) = f(x), needs to be represented in this form to determine the Fourier coefficients, B_n. Specifically, the coefficients are calculated by integrating the product of the initial condition and the sine function over the length of the rod, leading to the formula: B_n = (2/L) * ∫(f(x)*sin(nπx/L))dx from 0 to L. This process is crucial for solving PDEs as it transforms the problem into a solvable series form, allowing for the evaluation of how temperature evolves over time.
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Fourier Coefficient Calculation
Chapter 1 of 1
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Chapter Content
To find 𝐵_n, expand 𝑓(𝑥) in sine series:
𝐵_n = rac{2}{L} rac{n ext{π}x}{L} ext{ with the integral }
B_n = rac{2}{L} imes rac{n ext{π}}{L} imes ext{Integrate from 0 to L}
∫ f(x)sin( rac{n ext{π}x}{L})dx
Detailed Explanation
In order to solve the heat equation, we need to determine the coefficients (B_n) of the Fourier series expansion of the initial temperature distribution (f(x)). The formula for B_n involves integrating the product of the function f(x) and the sine function over the length of the rod, multiplied by a coefficient that relates to the length of the rod. This calculation effectively gives us the contribution of each sine function in representing the initial temperature distribution.
Examples & Analogies
Consider a musical instrument, like a guitar, which produces sound through vibrating strings. The vibration creates a combination of sound waves (harmonics). Similarly, the Fourier series breaks down the initially complex temperature distribution (the sound of the guitar) into simpler sine waves (each representing a note), allowing us to understand how each component contributes to the overall heat distribution over time.
Key Concepts
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Fourier Series: A mathematical tool to decompose functions into sine and cosine waves.
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Fourier Coefficients (B_n): Calculated from the initial conditions to determine the behavior of the solutions over time.
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Initial Condition: The temperature distribution function required to solve the heat equation.
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Dirichlet Boundary Conditions: Applied at the boundaries of the rod to guide the solution behavior.
Examples & Applications
If f(x) = x(L−x), then B_n = (2/L) * ∫(x(L−x) * sin(nπx/L))dx from 0 to L. This integral gives the necessary Fourier coefficients to solve our heat equation.
For a steady-state heat distribution, if the initial temperature follows a triangular profile, the Fourier coefficients will reflect that shape, affecting how the temperature decays over time.
Memory Aids
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Rhymes
For every n we take a look, we find B_n with just one hook. Integrate and let it sit, over the rod you'll get a fit.
Stories
Imagine a conductor heating up from two ends. The temperature story lives in how we calculate B_n, the bridge connecting the past heating to future glow.
Memory Tools
Remember 'B Is Found': the initial condition leads to Fourier coefficients through integration.
Acronyms
B.I.F (B_n Initial Function)
Segregate initial conditions to find the Fourier piece.
Flash Cards
Glossary
- Fourier Series
A way to represent periodic functions as a sum of sine and cosine functions.
- Fourier Coefficient (B_n)
Coefficients calculated from the initial condition used in a Fourier series representation.
- Initial Condition (IC)
The temperature distribution at time t=0, denoted as u(x,0) = f(x).
- Dirichlet Boundary Conditions
Boundary conditions that specify the value of the function at the boundaries.
- Sine Series
A series representation that only employs sine functions, typically used for odd-function expansions in a given interval.
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