11.1 Fourier Series Expansion

Fourier Series Expansion is a powerful tool for breaking down complex periodic functions into simpler parts. It's like taking a complicated song and splitting it into individual notes. This method helps us understand and work with tricky functions in math and engineering.

By representing functions as sums of sines and cosines, we can solve tough problems in physics and engineering. It's super useful for analyzing things like sound waves, electrical signals, and heat transfer. Fourier Series Expansion is a key player in making sense of the world around us.

Representing periodic functions with Fourier series

Fourier series representation

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• Fourier series is a method of representing periodic functions as an infinite sum of sine and cosine functions with different frequencies and amplitudes
• A periodic function repeats its values at regular intervals, satisfying the condition $f(x) = f(x + T)$ for all $x$, where $T$ is the period
• The Fourier series representation of a periodic function $f(x)$ with period $2π$ is given by:
  • $f(x) = \frac{a₀}{2} + \sum_{n=1}^{\infty} [a_n \cos(nx) + b_n \sin(nx)]$
  • $a₀$, $a_n$, and $b_n$ are Fourier coefficients
  • $n$ represents the harmonic number
• The Fourier series allows for the decomposition of complex periodic functions into a sum of simple harmonic components, which can be analyzed and manipulated individually

Applicability and conditions

• The Fourier series representation is applicable to both continuous and discontinuous periodic functions, as long as the function satisfies certain conditions (Dirichlet conditions)
  • The function is periodic with period $2π$
  • The function is piecewise continuous on the interval $[-π, π]$
  • The function has a finite number of maxima and minima on the interval $[-π, π]$
• Examples of periodic functions that can be represented by Fourier series include:
  • Square wave
  • Sawtooth wave
  • Triangle wave
• Fourier series can be used to approximate non-periodic functions over a finite interval by extending the function periodically outside the interval

Fourier series coefficients

Calculation of Fourier coefficients

• The Fourier coefficients $a₀$, $a_n$, and $b_n$ can be calculated using the following formulas:
  • $a₀ = \frac{1}{π} \int_{-π}^{π} f(x) dx$
  • $a_n = \frac{1}{π} \int_{-π}^{π} f(x) \cos(nx) dx$
  • $b_n = \frac{1}{π} \int_{-π}^{π} f(x) \sin(nx) dx$
• To determine the Fourier coefficients, the given periodic function $f(x)$ is multiplied by the corresponding trigonometric function ($1$, $\cos(nx)$, or $\sin(nx)$) and integrated over one period
• The resulting integrals are evaluated using appropriate integration techniques, such as trigonometric substitution, integration by parts, or using known integral formulas

Piecewise defined functions

• In cases where the periodic function is defined piecewise, the Fourier coefficients are calculated by splitting the integral into subintervals corresponding to each piece of the function
• Example: For a square wave defined as $f(x) = \begin{cases} 1, & 0 < x < π \\ -1, & -π < x < 0 \end{cases}$
  • The Fourier coefficients are calculated separately for each subinterval $[0, π]$ and $[-π, 0]$
  • The results are then combined to obtain the final Fourier coefficients
• The calculated Fourier coefficients can be substituted back into the Fourier series representation to obtain the complete Fourier series expansion of the periodic function

Fourier series convergence

Types of convergence

• Convergence of a Fourier series refers to whether the series approaches the original function as the number of terms in the series increases
• Pointwise convergence occurs when the Fourier series converges to the function value at each point where the function is continuous
  • Example: The Fourier series of a continuous function converges pointwise to the function at every point in its domain
• Uniform convergence occurs when the maximum difference between the function and its Fourier series approximation approaches zero as the number of terms increases, for all points in the domain
  • Example: The Fourier series of a smooth, continuous function converges uniformly to the function over its entire domain

Gibbs phenomenon

• Gibbs phenomenon is observed when a Fourier series approximates a function with discontinuities, resulting in oscillations near the discontinuities that do not diminish as the number of terms increases
• The oscillations occur because the Fourier series tries to approximate the discontinuity with a sum of continuous functions
• Example: The Fourier series approximation of a square wave exhibits Gibbs phenomenon at the discontinuities, with overshoots and undershoots that do not disappear as more terms are added to the series
• Convergence of a Fourier series has implications for the accuracy of the approximation and the smoothness of the approximated function

Fourier series for boundary value problems

Solving PDEs with Fourier series

• Fourier series can be used to solve boundary value problems (BVPs) in partial differential equations (PDEs), such as the heat equation, wave equation, and Laplace's equation
• The general steps to solve a BVP using Fourier series are:
  1. Formulate the PDE and specify the boundary conditions and initial conditions (if applicable)
  2. Assume a solution in the form of a Fourier series with unknown coefficients
  3. Substitute the assumed solution into the PDE and boundary conditions to obtain a system of equations for the Fourier coefficients
  4. Solve the system of equations to determine the Fourier coefficients
  5. Substitute the obtained Fourier coefficients back into the assumed solution to get the final solution of the BVP
• Fourier series are particularly useful for solving BVPs in rectangular domains with homogeneous boundary conditions (Dirichlet, Neumann, or mixed)

Choosing sine or cosine functions

• The choice of sine or cosine functions in the Fourier series depends on the type of boundary conditions imposed on the problem
• For Dirichlet boundary conditions (specified function values at the boundaries), the Fourier series typically involves sine functions
  • Example: A vibrating string with fixed ends can be modeled using a Fourier sine series
• For Neumann boundary conditions (specified derivatives at the boundaries), the Fourier series typically involves cosine functions
  • Example: Heat transfer in a rod with insulated ends can be modeled using a Fourier cosine series
• Fourier series solutions provide an analytical representation of the solution, allowing for further analysis and interpretation of the physical phenomenon described by the PDE