Fourier Series

Fourier series are a powerful tool in mathematics used to break down complex periodic functions into simpler sine and cosine components. This concept is important in the study of differential equations, particularly partial differential equations (PDEs). In this lesson, we will explore in depth Fourier series, their mathematical formulation, practical examples, and applications in solving PDEs.

What is a Fourier series?

Fourier series are a way of representing a periodic function as an infinite sum of sine and cosine functions. Sinusoidal functions are important because of their periodic nature and their orthogonality, which means they can be used as building blocks for more complex functions.

The general form of the Fourier series for a function f(x) with period T is given by:

f(x) = a_0 + ∑ (a_n * cos(n * omega_0 * x) + b_n * sin(n * omega_0 * x))

Where:

• n is a positive integer (denotes the harmonic number).
• omega_0 = 2π/T is the fundamental frequency.
• a_0, a_n, b_n are the Fourier coefficients.

The Fourier coefficients are calculated as follows:

a_0 = (1/T) ∫[−T/2, T/2] f(x) dx

a_n = (2/T) ∫[−T/2, T/2] f(x) cos(n * omega_0 * x) dx

b_n = (2/T) ∫[−T/2, T/2] f(x) sin(n * omega_0 * x) dx

These integrals are evaluated over a complete period of f(x). The coefficients a_n and b_n determine the amplitudes of the cosine and sine components in the series. The coefficient a_0 represents the average or mean value of the function over a period.

Visual example: building a square wave

To understand how a Fourier series works, let's consider a square wave, a periodic function that switches between a high and a low value. Mathematically, this function can be represented approximately as a Fourier series using only a finite number of terms.

// square wave Fourier series approximation
f(x) = (4/π) * [sin(x) + (1/3) * sin(3x) + (1/5) * sin(5x) + ...]

The SVG graph above shows how a square wave can be approximated by adding sine functions of different frequencies and amplitudes. As more sine waves are added, the approximation gets closer to the actual square wave.

The role of Fourier series in partial differential equations

Partial differential equations (PDEs) describe a variety of phenomena in physics, engineering, and other fields that involve functions dependent on several variables. Solving PDEs can be very complicated, but Fourier series help simplify these problems, especially for linear PDEs with periodic boundary conditions.

For example, consider the one-dimensional heat equation:

∂u/∂t = α ∂^2u/∂x^2

Here, u(x,t) is the temperature distribution along a rod, and α is the thermal diffusivity. Solving such an equation usually involves expressing u(x,t) as a Fourier series. This turns the PDE into an ordinary differential equation (ODE) for the coefficients.

Let u(x, 0) = f(x) be a periodic function. We express f(x) in a Fourier series:

f(x) = a_0 + ∑ (a_n * cos(n * omega_0 * x) + b_n * sin(n * omega_0 * x))

Example: solving the heat equation

Solving the heat equation with initial conditions using Fourier series involves the following steps:

• Express the initial condition f(x) as a Fourier series.
• Find the solution u(x, t) in terms of a series.
• Use initial or marginal conditions to solve for the unknown coefficients.

Let us solve the heat equation with f(x) = sin(x) on the interval [-π, π] with periodic boundary conditions. We already know from previous evaluations that:

f(x) = sin(x)

The initial series expansion is as follows:

u(x, 0) = sin(x)

Due to the symmetry and nature of sin(x), the Fourier series expansion involves similar terms at later time points:

u(x, t) = ∑ b_n * e^(-n^2 * α * t) * sin(n * x)

For n = 1, and given b_1 = 1, the simplified solution becomes:

u(x, t) = e^(-α * t) * sin(x)

This solution shows that as heat dissipates over time, the initial wave becomes smaller in size and retains its shape.

Applications beyond PDEs

Fourier series are used in a variety of fields beyond mathematics and PDEs, such as signal processing, acoustics, and electrical engineering. For example, in signal processing, Fourier series are used to analyze the frequency components of complex signals. They help to isolate and study specific frequencies, which are important for sound and image compression technologies.

In electrical engineering, Fourier analysis of circuits can determine the behavior of circuits subjected to periodic inputs, informing design decisions for filtering out unwanted frequencies.

Practical example: audio signal analysis

Consider analysing an audio signal representing a musical output. This signal is typically a complex waveform that can be decomposed into multiple sine and cosine waves using Fourier series.

// audio signal Fourier series
audio_signal(t) = a_0 + a_1 * cos(ω_1 t) + b_1 * sin(ω_1 t) + a_2 * cos(ω_2 t) + b_2 * sin(ω_2 t) + ...

The coefficients of each term represent the amplitude of the translated signals, allowing audio engineers to extract essential musical features.

Example: electric current analysis

An alternating current (AC) signal can be viewed as a sum of sinusoidal signals. An engineer can represent this signal as a Fourier series to study its frequency components. By calculating the important frequency components, unexpected harmonics can be isolated.

Conclusion: the power of Fourier series

Fourier series are invaluable in mathematics and other scientific fields, providing important insights into periodic functions and differential equations, which extend to wide applications such as signal analysis. Through their expansion, Fourier series enable these complex derivations to be reduced to a lightweight set of routine calculations, paving the way for new solutions to scientific and engineering problems.

Mastering Fourier series improves your perspective, prepares you to tackle complex PDEs, and demonstrates the beauty of mathematical abstractions in real-world applications.

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