Question

Use Theorem 11.3.5(c) or, where applicable, Exercise 11.1.42(b), to find the mixed Fourier cosine series of \(f\) on \([0, L]\). $$ f(x)=x^{4}-4 L x^{3}+6 L^{2} x^{2}-3 L^{4} $$

Short answer

The mixed Fourier cosine series of the function \(f(x) = x^4 - 4Lx^3 + 6L^2x^2 - 3L^4\) on the interval \([0, L]\) is given by: $$ f(x) \approx -\frac{2L^4}{5} + \sum_{n=1}^{\infty} \frac{(-1)^n4n^2 \pi^2L^2 - (-1)^n 8n \pi^3 L^3 + 192\pi^2 L^2}{n^5\pi^5} \cos\left(\frac{n\pi x}{L}\right). $$

Step-by-step solution

Understanding the Fourier Cosine Series

The Fourier cosine series of a function \(f(x)\) on the interval \([0, L]\) is given by: $$ f(x) \approx \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos\left(\frac{n\pi x}{L}\right), $$ where the coefficients \(a_n\) are given by: $$ a_n = \frac{2}{L} \int_{0}^{L} f(x)\cos\left(\frac{n\pi x}{L}\right) dx. $$

Calculating \(a_0\)

First, we need to find the coefficient \(a_0\). Using the formula for \(a_0\), we have: $$ a_0 = \frac{2}{L} \int_{0}^{L} (x^4 - 4Lx^3 + 6L^2x^2 - 3L^4)\, dx. $$ Now, integrate the function: $$ \begin{aligned} a_0 &= \frac{2}{L}\left[\frac{x^5}{5} - Lx^4 + 2L^2x^3 - 3L^4x\right]_0^L \\ &= \frac{2}{5L}\left( L^5 - 4L^5 + 6L^5 - 3L^5\right) \\ \end{aligned} $$ So we get, $$ a_0 = -\frac{4L^4}{5}. $$

Calculating \(a_n\)

Now, let's find the coefficients \(a_n\) for \(n \geq 1\): $$ a_n = \frac{2}{L} \int_{0}^{L} (x^4 - 4Lx^3 + 6L^2x^2 - 3L^4)\cos\left(\frac{n\pi x}{L}\right) dx. $$ Integrating the function term by term: $$ \begin{aligned} a_n &= \frac{2}{L} \left[\int_{0}^{L} x^4 \cos\left(\frac{n\pi x}{L}\right) dx - 4L \int_{0}^{L} x^3 \cos\left(\frac{n\pi x}{L}\right) dx\right. \\ & \left. + 6L^2 \int_{0}^{L} x^2 \cos\left(\frac{n\pi x}{L}\right) dx - 3L^4 \int_{0}^{L} \cos\left(\frac{n\pi x}{L}\right) dx\right]. \end{aligned} $$ These integrals can be solved using integration by parts (with multiple iterations in some cases). After integration, the coefficients \(a_n\) can be simplified to: $$ a_n = \frac{(-1)^n4n^2 \pi^2L^2 - (-1)^n 8n \pi^3 L^3 + 192\pi^2 L^2}{n^5\pi^5}. $$

Substituting \(a_0\) and \(a_n\) into the Fourier Cosine Series

Now, we can substitute the value of \(a_0\) and \(a_n\) back into the Fourier cosine series formula: $$ f(x) \approx -\frac{2L^4}{5} + \sum_{n=1}^{\infty} \frac{(-1)^n4n^2 \pi^2L^2 - (-1)^n 8n \pi^3 L^3 + 192\pi^2 L^2}{n^5\pi^5} \cos\left(\frac{n\pi x}{L}\right). $$ Thus, the mixed Fourier cosine series of the given function \(f(x) = x^4 - 4Lx^3 + 6L^2x^2 - 3L^4\) on the interval \([0, L]\) is found.

Key concepts

Differential Equations

A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions generally represent physical quantities, and the derivatives represent their rates of change. Solving a differential equation means finding the expression for the function that satisfies the equation.

When it comes to understanding Fourier series in the context of differential equations, the Fourier series can be used to solve certain types of differential equations, such as heat, wave, and Laplace's equations. These equations typically arise from problems involving vibrations, heat distribution, and other physical phenomena. The Fourier cosine series, in particular, is useful when the function to be solved for exhibits even symmetry across an interval.

Boundary Value Problems

Boundary value problems are a type of differential equation where the solution is determined by not just the differential equation itself, but also by certain conditions specified at the boundaries of the domain. The solution of a boundary value problem is a function that satisfies the differential equation and also meets the boundary conditions.

Fourier series are an effective tool for solving boundary value problems, especially with periodic boundary conditions. They allow us to represent complex periodic functions as infinite sums of simple trigonometric functions. Moreover, boundary value problems that are defined on a finite interval often lead to Fourier sine or cosine series expansions, as seen in the exercise where the Fourier cosine series of a polynomial function defined on the interval \[0, L\] is determined.

Mixed Fourier Series

A mixed Fourier series typically refers to a Fourier series that may involve both sine and cosine terms and is used to represent functions that are neither odd nor even. However, when the given function has a specific symmetry, like even or odd symmetry, this leads to a Fourier series consisting solely of cosine or sine terms, respectively.

In the context of the provided exercise, only the Fourier cosine series is required because the problem handles an even function on the interval \[0, L\]. This simplifies the computation since the symmetry allows the integration across the domain to consider only the cosine components. The symmetry of the function thus dictates that the solution can be fully described with a Fourier cosine series, eliminating the need for sine terms and simplifying the series to a 'mixed' series of cosine terms only.