Question

Find the Fourier series of \(f\) on \([-L, L]\) and determine its sum for \(-L \leq x \leq L .\) Where indicated by \(C\), graph \(f\) and $$ F_{m}(x)=a_{0}+\sum_{n=1}^{m}\left(a_{n} \cos \frac{n \pi x}{L}+b_{n} \sin \frac{n \pi x}{L}\right) $$ on the same axes for various values of \(m\). $$ L=\pi ; \quad f(x)=2 x-3 x^{2} $$

Short answer

Question: Determine the Fourier series of the function \(f(x) = 2x - 3x^2\) defined on the interval \([-L, L]\) with \(L=\pi\), and graph the function together with its Fourier series for different values of \(m\). Answer: The Fourier series of the function \(f(x) = 2x - 3x^2\) on the interval \([-L, L]\) with \(L=\pi\) is given by: $$ F_m(x) = \sum_{n=1}^{m} \left(\frac{2}{n^2\pi}\left((-1)^n - 1 \right) \cos \frac{n \pi x}{\pi} + \frac{6}{n^3\pi}\left((-1)^n - 1 \right) \sin \frac{n \pi x}{\pi}\right). $$ To graph the function and its Fourier series, plot the equations side by side on the same axis for different values of \(m\). As \(m\) increases, the Fourier series better approximates the function \(f(x)\).

Step-by-step solution

Find the coefficients \(a_0\), \(a_n\), and \(b_n\)

To find the coefficients of the Fourier series, we use the following formulas: $$ a_0 = \frac{1}{2L} \int_{-L}^{L} f(x)\, dx, \\ a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos\left(\frac{n \pi x}{L}\right) dx, \\ b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin\left(\frac{n \pi x}{L}\right) dx. $$ For our function \(f(x)=2x - 3x^2\) and \(L=\pi\), calculate \(a_0\): $$ a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} (2x - 3x^2) dx $$ Calculate \(a_n\) and \(b_n\): $$ a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} (2x - 3x^2) \cos\left(\frac{n \pi x}{\pi}\right) dx,\\ b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} (2x - 3x^2) \sin\left(\frac{n \pi x}{\pi}\right) dx. $$

Calculate the coefficients \(a_0\), \(a_n\), and \(b_n\)

Computing the integrals, we obtain the coefficients: $$ a_0 = 0, \\ a_n = \frac{2}{n^2\pi}\left((-1)^n - 1 \right), \\ b_n = \frac{6}{n^3\pi}\left((-1)^n - 1 \right). $$

Write down the Fourier series \(F_m(x)\)

Using the obtained coefficients, we can write the Fourier series, where the sum is taken up to the \(m\)-th term: $$ F_m(x) = a_0 + \sum_{n=1}^{m} \left(a_n \cos \frac{n \pi x}{L} + b_n \sin \frac{n \pi x}{L}\right) = \sum_{n=1}^{m} \left(\frac{2}{n^2\pi}\left((-1)^n - 1 \right) \cos \frac{n \pi x}{\pi} + \frac{6}{n^3\pi}\left((-1)^n - 1 \right) \sin \frac{n \pi x}{\pi}\right). $$

Graph \(f(x)\) and \(F_m(x)\) for different values of \(m\)

To graph the function \(f(x)\) and the Fourier series \(F_m(x)\) using a mathematical software or calculator, plot the equations side by side on the same axis for different values of \(m\). This will give a visual representation of the convergence of the Fourier series to the original function. Note that as \(m\) increases, the Fourier series better approximates the function \(f(x)\).

Key concepts

Differential Equations

Differential equations are mathematical equations that describe the relationship between a function and its derivatives. They are pivotal in mapping the behavior of dynamic systems across many fields such as physics, engineering, biology, and economics. A differential equation often represents physical phenomena, like the movement of waves or the oscillation of a pendulum, by correlating rates of change to the state of the system itself.

For instance, consider a moving particle with its position determined by the function of time. A differential equation can relate its velocity (the first derivative of position) and acceleration (the second derivative of position) to describe its motion completely. Solving these equations often leads to finding a function that satisfies the set conditions or initial values, providing insightful predictions about the system's behavior.

Boundary Value Problems

Boundary value problems (BVP) are specific types of differential equations where the solution is determined by the values of the function at the boundaries of the domain, instead of initial values. They are crucial in the study of physical systems that are constrained by conditions at the edges, like the vibrational modes of a string or temperature distribution in a rod.

A classic example of a BVP is the Sturm-Liouville problem, which arises in the separation of variables in partial differential equations. These problems are associated with eigenfunctions and eigenvalues, pivotal in quantum mechanics and other areas of theoretical physics. Solving a BVP provides the specific solution of a differential equation that not only adheres to the laws of physics but also to the physical constraints imposed by the boundaries.

Fourier Coefficients

Fourier coefficients are the weights assigned to the sine and cosine terms in a Fourier series, which together provide the series representation of a periodic function. They are determined by integral formulas that constitute the heart of Fourier analysis. The coefficients are a direct translation of the function's behavior into the language of trigonometric functions, revealing the oscillatory components within the original function.

More specifically, the coefficients are calculated with formulas that involve integrating the product of the function with the sine and cosine functions over a full period. The computed coefficients, like the ones obtained in the exercise, tell us exactly how much of each sine and cosine wave is needed to reconstruct the original function faithfully across its domain.

Convergence of Fourier Series

The convergence of Fourier series is crucial in understanding how well the series approximates the original function over time. If a function is periodic and satisfies certain conditions, the Fourier series converges to the function at most points as the number of terms in the series increases. This convergence is a central topic in mathematical analysis since it establishes the conditions under which the Fourier series faithfully reconstructs the original function.

While observing the plot for different values of the truncation point ‘m’, we notice that with a higher 'm', the Fourier series graph begins to resemble the shape of the original function more closely, showcasing its property of convergence. This visualization strengthens the conceptual understanding that as 'm' goes to infinity, the series converges to the actual function, barring points of discontinuity where convergence might be approached in alternative, more nuanced ways.