Problem 4 - Applied Partial Differential Equations

Question

(a) Using Parseval's equality, express the error in terms of the tail of a series. (b) Redo part (a) for a Fourier sine series on the interval \(0 \leqslant x \leqslant L\). (c) If \(f(x)\) is piecewise smooth, estimate the tail in part (b). (Hint: Use integration by parts.)

Step-by-Step Solution

Answer

The error in terms of tail can be expressed using Parseval's equality as \(E_{N} = |f(x)|^{2} - \left(\frac{1}{2}a_{0}^{2} + \sum_{n=1}^{N} (a_{n}^{2} + b_{n}^{2})\right)\) for a given series and as \(E_{N} = |f(x)|^{2} - \sum_{n=1}^{N} B_{n}^{2} \) for the Fourier Sine series. If \(f(x)\) is piecewise smooth, the tail can be estimated using integration by parts as \(E_{N} \approx \sum_{n=N+1}^{\infty} \frac{|f^{(k)}(x)|^{2}}{n^{2k}} \).

Step 1: Parseval's equality

Parseval's equality for a function \( f \) that is square-integrable over a period \( P \) can be denoted as: \[ \int_{0}^{P} |f(x)|^{2} \, dx = \frac{1}{2} a_{0}^{2} + \sum_{n=1}^{\infty} (a_{n}^{2} + b_{n}^{2}) \] where \( a_{n} \) and \( b_{n} \} are Fourier coefficients for the function \( f \). Given this, to express error in terms of tail of series, it can be written as \[ E_{N} = |f(x)|^{2} - \left(\frac{1}{2}a_{0}^{2} + \sum_{n=1}^{N} (a_{n}^{2} + b_{n}^{2})\right) \]

Step 2: Fourier Sine Series

The Fourier Sine series for \( f(x) \)on interval \(0 \leqslant x\leqslant L\) can be represented as \[ f(x) = \sum_{n=1}^{\infty} B_{n} \sin\left(\frac{n\pi x}{L}\right) \] where \( B_{n} = \frac{2}{L} \int_{0}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) dx \) is the Fourier coefficient. For this function, Parseval's theorem will translate to: \[ \int_{0}^{L} |f(x)|^{2}dx = \sum_{n=1}^{\infty} B_{n}^{2} \] and error can be expressed as \[ E_{N} = |f(x)|^{2} - \sum_{n=1}^{N} B_{n}^{2} \]

Step 3: Estimating the tail in part (b)

If \(f(x)\) is piecewise smooth, the \(N^{th}\) term of the sine series can be estimated as \(B_{N} \approx \frac{|f^{(k)}(x)|}{N^{k}}\) for some \(k\). Therefore, the tail can be estimated as \(E_{N} \approx \sum_{n=N+1}^{\infty} \frac{|f^{(k)}(x)|^{2}}{n^{2k}} \). The sum on the right is a p-series with \(p=2k>1\), which converges to a finite value, therefore, the error decreases as \(N\) increases.

Key Concepts

Fourier Series

A Fourier series is a way to represent a function as a sum of sine and cosine functions. It is particularly useful for periodic functions, meaning functions that repeat their values at regular intervals. In mathematical terms, any reasonably well-behaved periodic function can be expressed as an infinite sum of sines and cosines. This is incredibly helpful in understanding complex signals, such as sound waves or electrical signals, by breaking them down into simpler components.

When you have a function defined over a period, say from 0 to some length \(L\), the Fourier series helps you to approximate it using a set of coefficients. These coefficients are determined by integrating the function multiplied by sine and cosine terms over this interval. For example, for a Fourier sine series on \(0 \leq x \leq L\), the formula is given by:

• \(f(x) = \sum_{n=1}^{\infty} B_{n} \sin\left(\frac{n\pi x}{L}\right)\)

Where \(B_{n}\) is a Fourier coefficient dependent on the shape of \(f(x)\). Each term in the series represents a harmonic or frequency component of \(f(x)\). The more terms you include, the closer the series approximates the original function.

Error Estimation

Error estimation in the context of Fourier series refers to how accurately the series can represent a function. Imagine you're watching a video online. If the quality is low, you see a lot of pixelation—this is the error. Higher quality means less error. In a similar way, when representing a function using its Fourier series, not all infinite terms are usually used in practice. Instead, we often stop at a finite number of terms, denoted as \(N\).

The error then becomes the difference between the original function and the approximation given by the partial sum of the series up to \(N\) terms. Using Parseval's equality, this error \(E_{N}\) can be expressed as the sum of the remaining terms, which is also called the tail of the series:

• \(E_{N} = \left| f(x) \right|^{2} - \sum_{n=1}^{N} B_{n}^{2}\)

Understanding and estimating this error is important in both theoretical analysis and practical applications because it tells us how "close" our truncated Fourier series is to the actual function. It can also guide us on how many terms are needed to achieve a desired level of accuracy.

Piecewise Smooth Functions

Piecewise smooth functions are functions that may not be completely smooth or continuous but can be divided into sections where each section is smooth. For example, imagine a piece of paper that is folded in some areas but lies perfectly flat in others—this is akin to a piecewise smooth function. These functions often occur in real-world scenarios such as signal processing and image analysis.

When dealing with Fourier series and such functions, the concept of piecewise smoothness plays a crucial role in estimating the coefficients and the resulting error. The Fourier coefficients, in this case, can be tricky because of the discontinuities. The estimate of the error or the tail of the series can often be approximated by considering the rate at which these coefficients decrease. Typically, if \(f(x)\) is piecewise smooth, we use integration by parts to track the decay rate of the Fourier coefficients.

One useful approximation involves the derivative of the function, symbolized as \(f^{(k)}(x)\). If \(f^{(k)}(x)\) is continuous and bounded, the:

• \(B_{N} \approx \frac{|f^{(k)}(x)|}{N^{k}}\)

This helps us calculate the error for large \(N\) as the series converges, noting that the convergence is assured by the convergence of the related p-series. The smaller the error, the better your Fourier series approximation of a piecewise smooth function.