Problem 2 - Applied Partial Differential Equations

Question

For the following functions, sketch the Fourier sine series of \(f(x)\) and determine its Fourier coefficients. (a) \(f(x)=\cos \pi x / L\) (Verify the formula on p. 89 ) (b) \(f(x)= \begin{cases}1 & xL / 2\end{cases}\) (c) \(f(x)= \begin{cases}0 & xL / 2\end{cases}\) (d) \(f(x)= \begin{cases}1 & xL / 2\end{cases}\)

Step-by-Step Solution

Answer

To acquire the Fourier Sine series of a function, understand the periodicity and odd function. Apply the formula for the Fourier Sine series which involves calculating Fourier coefficients. The Sine series should contain only sine terms.

Step 1: Identify the right Fourier Series.

The task states to use the Fourier Sine Series, and given that it’s used for odd functions, ensure that each function fits this parameter is important.

Step 2: Definition of Fourier Sine Series

The general Fourier Sine series is given by \[f(x)=\frac{a_{0}}{2}+\sum _{n=1} ^{\infty} (a_{n}\cos n\omega x+b_{n}\sin n\omega x)\] where \(\omega = \frac{2\pi}{L}\) and the Fourier coefficients \(a_{n}\) and \(b_{n}\) can be expressed as \[a_{n}=\frac{2}{L}\int_{0}^{L}f(x)\cos n\omega x dx, n = 0, 1, 2, ....\] \[b_{n}=\frac{2}{L}\int_{0}^{L}f(x)\sin n\omega x dx, n = 1, 2, 3, ....\] The task requires checking the formula on page 89, which will be the basis for calculating Fourier coefficients.

Step 3: Calculate the Fourier Coefficients for each function.

Start off by calculating the \(a_{0}\) coefficient for each function, according to the formula stated above. Ensure to understand the periodicity of each function before applying the formula. This would be the more complex part which requires to be broken down further.

Step 4: Analyze and Sketch the Fourier Sine series of each function.

After calculating the coefficients, replace \(a_{n}\) and \(b_{n}\) in the Fourier series equation with the computed results. Analyze the series and sketch the Fourier Sine series on a graph. With practice, you should be able to observe that cosine terms disappear in the Fourier Sine series for odd functions.

Key Concepts

Fourier coefficients

Fourier coefficients are crucial when dealing with Fourier Sine Series. They determine the weight of each sine and cosine component in the series. When working with the Fourier Sine Series, you'll often see these coefficients noted as \(a_n\) and \(b_n\). However, primarily in a sine series, you focus on \(b_n\) coefficients, since these multiply the sine terms that specifically describe the function.

• Calculate the coefficients using integrals, where \(b_n\) is given by:

\[b_{n} = \frac{2}{L} \int_{0}^{L} f(x) \sin \left( \frac{n \pi x}{L} \right) dx\] This integral finds how much of the sine component \(\sin \left( \frac{n \pi x}{L} \right)\) is present in the function \(f(x)\).
To compute these, understand what happens when the function integrates with the sine term over one full period. This integration considers how your function aligns or differs from the periodic sine wave, which helps break down the function into a series that sums those elements.
By carefully calculating these coefficients, we can reconstruct the function accurately using the sine components.

odd functions

Odd functions have a special property—when you take the value of the function at a point and its negative, they are opposites: \(f(-x) = -f(x)\). This symmetry about the origin means these functions are well-suited for representation by a Fourier Sine Series.

• The Fourier Sine Series inherently covers these functions perfectly because sine functions themselves are odd. This means they naturally exhibit the symmetry that matches the function type.
• By representing the function as a sum of odd sine functions, we can ensure our series representation perfectly epitomizes the function's nature over its interval.

This is why Fourier Sine Series is the go-to method for finding solutions to problems involving odd functions.
You must check whether your original function is odd to determine the suitability of representing it using the Fourier Sine Series. If it is, you can safely omit any cosine terms, focusing purely on the sine components.

periodicity

Periodicity refers to how functions repeat over regular intervals. In the context of Fourier Sine Series, it's crucial to understand how the function repeats over its period, denoted as \(L\).

• The series is designed to represent the function over a specific interval \([-L, L]\), usually extended by the properties of sine functions.
• Sine functions have a natural periodicity of \(2\pi\), but by adjusting the input term as \(\sin \left( \frac{n \pi x}{L} \right)\), we can fit sine waves to the desired period of the function.

Periodic extension of a function means that although defined over a small interval, using the Fourier Sine Series allows it to be represented over all integers—a crucial aspect in physics and engineering applications.
Understanding the periodic nature of the function and accurately calculating it using Fourier coefficients allows for precise modeling of phenomena in a cyclical environment.

sketching series

Sketching the Fourier Sine Series is a valuable skill for visualizing how the sine components add to give you the original function. Once you have calculated the coefficients \(b_n\), you can visualize the series.

• Begin by plotting the first few terms of the series individually.
• Look at how the individual sine waves approximate different parts of the function \(f(x)\).
• As you incorporate more terms, your sketch should start to resemble the original function more closely.

To facilitate this, use graph paper or graphing software to superimpose terms. This method will help you see the convergence of the infinite series to the function.
Keep in mind that initially, the series might look rough, but as you increase \(n\), the details of the function get clearer. This step validates the accuracy of your coefficients and demonstrates the power of Fourier Series in function approximation.