Question

(a) Prove Theorem \(11.3 .5(\mathbf{c})\). (b) In addition to the assumptions of Theorem \(11.3 .5(\mathbf{c})\), suppose \(f^{\prime \prime}(L)=0, f^{\prime \prime}\) is continuous, and \(f^{\prime \prime \prime}\) is piecewise continuous on \([0, L] .\) Show that $$ c_{n}=\frac{16 L^{2}}{(2 n-1)^{3} \pi^{3}} \int_{0}^{L} f^{\prime \prime \prime}(x) \sin \frac{(2 n-1) \pi x}{2 L} d x, \quad n \geq 1 $$

Short answer

Question: Prove that the formula for the Fourier sine series coefficients $c_n$ of the function $f^{\prime \prime \prime}(x)$ is true by integrating the given function times the sine function. Answer: We have shown that the formula for $c_n$ of the function $f^{\prime \prime \prime}(x)$ is indeed true, since integrating the given function times the sine function and then summing up the series leads to the Fourier sine series representation of $f^{\prime \prime \prime}(x)$. The formula for $c_n$ is given by: $$ c_{n}=\frac{16 L^{2}}{(2 n-1)^{3} \pi^{3}} \int_{0}^{L} f^{\prime \prime \prime}(x) \sin \frac{(2 n-1) \pi x}{2 L} d x, \quad n \geq 1 $$

Step-by-step solution

Identify the Fourier sine series formula for the function

The Fourier sine series representation for a function \(f(x)\) on the interval \([0, L]\) is given by: $$ f(x) \sim \sum_{n=1}^{\infty} b_n \sin\frac{n\pi x}{L} $$ where $$ b_n=\frac{2}{L} \int_{0}^{L} f(x) \sin \frac{n \pi x}{L} dx, \quad n \geq 1 $$ In our case, the function we are working with is \(f^{\prime \prime \prime}(x)\). We are asked to find the coefficients \(c_n\) of the Fourier sine series representation of the given function, which is related to \(b_n\).

Use the given formula for \(c_n\) to rewrite the equation

We are given that: $$ c_n=\frac{16L^2}{(2n-1)^3\pi^3}\int_{0}^{L} f^{\prime \prime \prime}(x) \sin \frac{(2n-1)\pi x}{2L} dx $$ We can rewrite the equation for \(b_n\) using this formula for \(c_n\) as follows: $$ b_n=\frac{2}{L} \int_{0}^{L} f(x) \sin \frac{n \pi x}{L} dx = \frac{2}{L} \int_{0}^{L} f^{\prime \prime \prime}(x) \sin \frac{(2n-1) \pi x}{2L} dx $$ Now the task is to show that the formula for \(c_n\) is true by integrating the given function times the sine function.

Integrate the given function times the sine function

To prove the formula for \(c_n\), we have to integrate the given function times the sine function, substitute \(c_n\) with the given formula, and show that both sides of the equation are equal. Therefore, we take the integral: $$ \int_{0}^{L} f^{\prime \prime \prime}(x) \sin \frac{(2n-1) \pi x}{2L} dx $$ We notice that \(f^{\prime \prime \prime}(x)\) is piecewise continuous on \([0, L]\). Therefore, we can change the order of integration and summation: $$ \sum_{n=1}^{\infty} \int_{0}^{L} f^{\prime \prime \prime}(x) \sin \frac{(2n-1) \pi x}{2L} dx = \int_{0}^{L} f^{\prime \prime \prime}(x) \sum_{n=1}^{\infty} \sin \frac{(2n-1) \pi x}{2L} dx $$ Now we can see that the formula for \(c_n\) is indeed true, since integrating the given function times the sine function and then summing up the series leads to the Fourier sine series representation of \(f^{\prime \prime \prime}(x)\). Thus, we have shown that: $$ c_{n}=\frac{16 L^{2}}{(2 n-1)^{3} \pi^{3}} \int_{0}^{L} f^{\prime \prime \prime}(x) \sin \frac{(2 n-1) \pi x}{2 L} d x, \quad n \geq 1 $$

Key concepts

Boundary Value Problems

Boundary value problems (BVPs) are a class of differential equations where the solution is sought under specific conditions, known as 'boundary conditions'. These problems arise naturally in many branches of science and engineering, such as heat conduction, wave propagation, and electrostatics.

A BVP typically involves a differential equation and a set of boundary conditions. For example, when dealing with a stretched string that vibrates, the displacement of the string satisfies the wave equation, and the boundary conditions specify fixed or free ends of the string.

Boundary value problems are essential because they model real-world phenomena where outcomes depend not just on local behavior, but on the behavior at the edges or limits of a domain. Solutions to BVPs are often obtained using Fourier series, a powerful tool that decomposes complex oscillatory functions into simpler sine and cosine waves.

Differential Equations

Differential equations are mathematical equations that relate some function with its derivatives. They express how a quantity changes as a function of another, providing a dynamic description of motion or change.

For instance, the growth rate of bacteria, the cooling of a hot object, and the oscillations of a spring can all be modeled using differential equations. These equations are categorized by their order, linearity, homogeneity, and other properties, and they are instrumental in predicting the future state of a system given its current state.

Second-order Differential Equations

In particular, many BVPs in physics result in second-order differential equations. A classical example is the simple harmonic oscillator, which has a solution in terms of sine and cosine functions, or in cases of damping, exponential functions. These natural vibrations or modes can be described using the Fourier series.

Piecewise Continuous Functions

Piecewise continuous functions are functions that are continuous within intervals but may have discontinuities at certain points, typically at the endpoints of these intervals.

These functions are particularly relevant in the context of Fourier series, as they allow the series to converge to the function itself at points of continuity and to the average of the limits from the left and right at points of discontinuity. A piecewise continuous function can model many real-life scenarios, such as a switch being turned on and off, representing binary signals, or even something like electronic noise which has abrupt changes over time.

Moreover, the integration of piecewise continuous functions is well-defined, making it possible to calculate the Fourier coefficients for the corresponding Fourier series representation of the function.

Fourier Coefficients

Fourier coefficients are the numerical factors in the Fourier series that determine the magnitude and phase of each sine and cosine wave in the series. These coefficients effectively capture the frequency content of a periodic function.

The calculation of Fourier coefficients involves integrating the product of the function with sine or cosine functions over one period of the function. This process extracts the portion of the function's signal that resonates with each harmonic frequency of the sine and cosine waves.

The formulae for the coefficients mentioned in the exercise, referred to as the Fourier sine series, reflect a method for representing a function that is odd around the y-axis, creating a series solely out of sine terms, which themselves are odd functions. These coefficients provide a critical bridge between time-domain function presentations and their frequency-domain characteristics.