Problem 4 - Applied Partial Differential Equations

Question

Suppose that \(f(x)\) and \(d f / d x\) are piecewise smooth. (a) Prove that the Fouricr sine series of a continuous function \(f(x)\) can only be differentiated term by term if \(f(0)=0\) and \(f(L)=0\). (b) Prove that the Fourier cosine series of a continuous function \(f(x)\) can be differentiated term by term.

Step-by-Step Solution

Answer

For a Fourier sine series of a continuous function, term-by-term differentiation is possible if \(f(0)=0\) and \(f(L)=0\). For a Fourier cosine series of a continuous function, term-by-term differentiation can always be done, irrespective of the function's values at any points.

Step 1: Understanding Fourier Series

A Fourier sine series for function \(f(x)\) is a representation of the function as an infinite sum of sine terms. A Fourier cosine series is the same, but with cosine terms. The coefficients of these series are calculated using integration over a segment [0, L] and depend on the initial function \(f(x)\). Now it's necessary to investigate whether the series can be differentiated term-by-term.

Step 2: Case of Fourier Sine Series

For a Fourier sine series, which has the form \(f(x) = \sum a_n \sin(\frac{n\pi x}{L})\), it can be seen that when \(f(0) = 0\) and \(f(L) = 0\), the coefficients \(a_n\) are given by \(a_n = \frac{2}{L}\int_0^L f(x)\sin(\frac{n\pi x}{L}) dx\). It is proved by differentiating these expressions term by term, then substituting the limits at 0 and L. After differentiation and substitution, since at \(x=0\) and \(x=L\), \(f(x)\) equals zero, the resulting series become identical with the differentiated initial one. It is hence proved that series can be differentiated term-by-term, granted that both sides equal zero.

Step 3: Case of Fourier Cosine Series

For the Fourier cosine series, the term-by-term differentiation is easier to prove. The Fourier cosine series takes the form \(f(x) = a_0 + \sum a_n \cos(\frac{n\pi x}{L})\). The coefficients \(a_n\) are given by \(a_n = \frac{2}{L}\int_0^L f(x)\cos(\frac{n\pi x}{L}) dx\), with \(a_0 = \frac{1}{L}\int_0^L f(x) dx\). Upon term-by-term differentiation and using the properties of cosine, the derived series again matches with the original function, hence proving that it can be differentiated term-by-term under any circumstances.

Key Concepts

Fourier Sine Series

The Fourier sine series is a powerful mathematical tool used to represent a piecewise smooth function as an infinite sum of sine waves. Each sine term in the series corresponds to a harmonic of a fundamental frequency. For a function defined on the interval \(0,L\), the Fourier sine series takes the form:

\[f(x) = \sum_{n=1}^{\infty} a_n \sin(\frac{n\pi x}{L})\]
where the coefficients \(a_n\) are determined by:

\[a_n = \frac{2}{L}\int_0^L f(x)\sin(\frac{n\pi x}{L})dx\]
These coefficients encapsulate the amplitude of each sine component in the series, where \(n\) signifies the nth harmonic. This series is particularly useful for functions that are odd about their midpoint or have certain boundary conditions, such as \(f(0) = 0\) and \(f(L) = 0\). This allows for the function to be represented solely in terms of sine waves which are themselves odd functions.

Fourier Cosine Series

In contrast to the Fourier sine series, the Fourier cosine series represents a function with an infinite series of cosine waves. It's especially apt for even, piecewise smooth functions defined on \(0,L\). The general expression for the cosine series is:

\[f(x) = a_0 + \sum_{n=1}^{\infty} a_n \cos(\frac{n\pi x}{L})\]
The calculation of coefficients differs slightly with \(a_0\), representing the average value of the function over \(0, L\), being:

\[a_0 = \frac{1}{L}\int_0^L f(x) dx\]
And for \(n > 0\):

\[a_n = \frac{2}{L}\int_0^L f(x)\cos(\frac{n\pi x}{L})dx\]
These coefficients reflect the contribution of each cosine component to the overall series, and since cosine waves are even functions, this series effectively captures the symmetry in even functions or functions with specific symmetry about the y-axis.

Term-by-Term Differentiation

Term-by-term differentiation is a technique used when dealing with Fourier series, allowing us to differentiate each term of the series individually. This process is valid under specific conditions tied to the smoothness and continuity of the function and its derivatives. For the Fourier sine series, this is permissible only when the function satisfies the boundary conditions \(f(0) = 0\) and \(f(L) = 0\). These conditions ensure that, upon differentiating each term, the resulting sum converges to the derivative of the original function, \(\frac{df}{dx}\).

In the case of the Fourier cosine series, term-by-term differentiation is generally always valid because the resulting series after differentiation will inherently satisfy the boundary conditions for cosine terms. This process is important for solving various types of differential equations and for analyzing the behavior of waves and vibrations in physics and engineering.

Piecewise Smooth Functions

Piecewise smooth functions are continuous functions that can be divided into a finite number of intervals, within which the function is smooth—meaning it has a continuous derivative. These functions may have a finite number of discontinuities in their first derivative, such as sharp corners or points where the slope changes abruptly, which are typical in sawtooth waves, square waves, and other similar functions. Fourier series are particularly useful for representing such functions because they naturally accommodate the jumps and discontinuities within a finite domain. Being able to apply Fourier series to such functions expands our ability to analyze and reconstruct signals and functions which are common in real-world applications, such as in signal processing and harmonic analysis.