Problem 1 - Applied Partial Differential Equations

Question

For the following functions, sketch \(f(x)\), the Fourier series of \(f(x)\), the Fourier sine series of \(f(x)\), and the Fourier cosine series of \(f(x)\). (a) \(f(x)=1\) (b) \(f(x)=1+x\) (c) \(f(x)= \begin{cases}x & x<0 \\ 1+x & x>0\end{cases}\) (d) \(f(x)=e^{x}\) (e) \(f(x)= \begin{cases}2 & x<0 \\ e^{-x} & x>0\end{cases}\)

Step-by-Step Solution

Answer

The sketches for each function and their Fourier series, sine and cosine series, would be as described in the steps. The patterns of oscillation change depending on the specifics of the function. All the Fourier series try to mimic the original function by introducing oscillations around the shape of the function.

Step 1 Title

First, we analyze the function \(f(x) = 1\). The Fourier series for a constant function is trivially just the function itself. So \(f(x)\), its Fourier series, Fourier sine series and Fourier cosine series, will all just be straight horizontal line at \(y=1\) in a sketch.

Step 2 Title

Second, we analyze \(f(x) = 1 + x\). We calculate the Fourier series as the sum of sine and cosine functions that oscillate around the average value of the function, which is a straight line with a slope. For this function, the Fourier, sine and cosine series will all dodge around the line \(y=1+x\) with increasingly higher frequency terms cancelling out more of the deviations.

Step 3 Title

Third, we examine \(f(x) = x\) for \(x<0\) and \(f(x) = 1+x\) for \(x>0\). This is a piecewise function, which breaks at \(x=0\). Thus the Fourier series, and the sine and cosine series will switch behaviour at this point. On the left, they will oscillate around the line \(y=x\) and on the right they will oscillate around the line \(y=1+x\).

Step 4 Title

Fourth, we consider \(f(x) = e^x\). This function is exponential with a base \(e\), and it grows rapidly. Its Fourier series, and the sine and cosine series, are hard to visualize but they will involve oscillations around this growing curve, with the oscillations attempting to mimic the shape of the exponential growth.

Step 5 Title

Lastly, we take \(f(x) = 2\) for \(x<0\) and \(f(x) = e^{-x}\) for \(x>0\). This is another piecewise function, which also breaks at \(x=0\). The Fourier series, and the sine and cosine series, will switch behaviour at this point. On the left, they will oscillate around the line \(y=2\) and on the right they will oscillate around the curve \(y=e^{-x}\), a decreasing curve.

Key Concepts

Fourier Sine Series

The Fourier sine series is an expansion of a function into a series of sine waves. It enables us to represent odd periodic functions and non-periodic functions over a certain interval. In mathematics, an odd function is one that satisfies the property that \(f(-x) = -f(x)\). The Fourier sine series is given by:
\[ f(x) = \sum_{n=1}^{\infty}b_n \sin(\frac{n\pi x}{L}) \]
where \(b_n\) is the coefficient found by integrating the original function multiplied by the corresponding sine function over the function's period and \(L\) is half of the period of the function. For instance, for the function \(f(x) = 1+x\), although it is not odd, we can still compute a sine series over a symmetric interval around the origin by extending the function to be odd. This will result in a Fourier sine series that osculates around the original function, approximating it within the defined interval.

Fourier Cosine Series

Conversely, the Fourier cosine series pertains to the representation of even periodic functions and non-periodic functions over a specific interval using a series of cosine waves. An even function meets the condition \(f(x) = f(-x)\). The formula for the Fourier cosine series is:
\[ f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty}a_n \cos(\frac{n\pi x}{L}) \]
where \(a_0\) is the average value of the function over its period, \(a_n\) are the cosine coefficients computed by integrating the product of the original function and the corresponding cosine function, and \(L\) represents half the period of the function. For the constant function \(f(x) = 1\), the Fourier cosine series reduces to a horizontal line, as the function is already even and does not need further oscillations to be represented.

Piecewise Functions

Piecewise functions are defined by different expressions based on different intervals of the domain. They are often used to describe scenarios that change behavior based on certain conditions or thresholds. For example, the function:
\[ f(x) = \begin{cases} x & x<0 \ 1+x & x>0 \end{cases} \]
is a piecewise function that has one formula for negative values and another for positive values. When we depict Fourier series for piecewise functions, we notice that the sine and cosine series will have distinctive patterns on each interval, adapting to match the function's different segments. This nature of piecewise functions can sometimes lead to more complex Fourier series, as they must accommodate changes in behavior at the boundaries where the function's formula changes.

Exponential Functions

Exponential functions, such as \(f(x) = e^x\) and \(f(x) = e^{-x}\), grow or decay at a rate proportional to their value. They are characterized by the presence of an exponent in the power of a constant base, commonly the number \(e\), Euler's number. In Fourier analysis, exponential functions are noteworthy because they are not periodic and their growth or decay rate can dramatically affect the appearance of their Fourier series. When we calculate the Fourier series of an exponential function, we end up with an infinite series of sine and cosine terms that try to replicate the exponential growth or decay pattern. This series is often complex and may not converge rapidly, highlighting the fact that Fourier series are better suited for functions that resemble periodic behavior over the interval of interest.