Problem 16 - Applied Partial Differential Equations

Question

Fourier series can be defined on other intervals besides \(-L \leqslant x \leqslant L\). Suppose that \(g(y)\) is defined for \(a \leqslant y \leqslant b\). Represent \(g(y)\) using periodic trigonometric functions with period \(b-a\). Determine formulas for the coefficients. [Hint: Use the linear transformation $$ \left.y=\frac{a+b}{2}+\frac{b-a}{2 L} x .\right] $$

Step-by-Step Solution

Answer

The function \(g(y)\) can be represented as a Fourier series \(g(x) =\frac{a_0}{2} + \sum \limits _{n=1} ^{\infty} [a_n cos(\frac{nπx}{L}) + b_n sin(\frac{nπx}{L})]\) on the interval \[-L \leq x \leq L\]. The Fourier coefficients \(a_0\), \(a_n\), and \(b_n\) can be calculated using the formulas \(a_0 = \frac{1}{L} \int _{-L} ^{L} g(x) dx, a_n = \frac{1}{L} \int _{-L} ^{L} g(x) cos(\frac{nπx}{L}) dx\), and \(b_n = \frac{1}{L} \int _{-L} ^{L} g(x) sin(\frac{nπx}{L}) dx\), for \(n \geq 1\).

Step 1: Linear Transformation

Using the linear transformation \(y=\frac{a+b}{2}+\frac{b-a}{2L}x\), substitute \(y\) in \(g(y)\) to get a function \(g(x)\) defined on the interval \[-L \leq x \leq L\]. This enables one to use the standard definition of the Fourier series.

Step 2: Shifting Interval

The previous step shifts the interval from \[a \leq y \leq b\] to \[-L \leq x \leq L\]. This is paved by changing the variables inside the function using the linear transformation.

Step 3: Define the Fourier Series

The function \(g(x)\) can now be represented by a Fourier series on the interval \[-L \leq x \leq L\]. The Fourier series is given by \(g(x) =\frac{a_0}{2} + \sum \limits _{n=1} ^{\infty} [a_n cos(\frac{nπx}{L}) + b_n sin(\frac{nπx}{L})]\) where \(a_0\), \(a_n\), and \(b_n\) are the Fourier coefficients to be determined.

Step 4: Determine the Fourier Coefficients

The coefficients of the Fourier series are determined using the following standard formulas, derived from the orthogonality relationships of the sine and cosine functions: \(a_0 = \frac{1}{L} \int _{-L} ^{L} g(x) dx, a_n = \frac{1}{L} \int _{-L} ^{L} g(x) cos(\frac{nπx}{L}) dx\), and \(b_n = \frac{1}{L} \int _{-L} ^{L} g(x) sin(\frac{nπx}{L}) dx\), for \(n \geq 1\).

Key Concepts

Linear Transformation

In the world of mathematics, a linear transformation is essentially a bridge that allows us to translate one set of points to another within a space. This concept is fundamental in the Fourier series context, as it helps us adapt functions that are defined on various intervals to a standard form that's easier to work with.

For example, if we have a function defined on an interval \[a, b\], but our Fourier analysis tools are designed for an interval \[ -L, L \], a linear transformation can be applied to adjust the initial function's domain. The transformation \( y = \frac{a+b}{2} + \frac{b-a}{2L}x \) mentioned in our exercise is a classic example. This formula shifts and scales the original interval without altering its inherent properties, preparing it for further analysis through Fourier series—a powerful technique for studying functions and their behaviors.

Periodic Trigonometric Functions

Consider the heartbeat of mathematics, periodic trigonometric functions, which repeat their values in regular intervals, over and over, much like the ebb and flow of tides. Sine and cosine functions are the stars of this cyclical universe, instrumental in forming the backbone of the Fourier series.

These functions are defined with periods, the length over which the function's values repeat. The Fourier series takes advantage of this periodicity, allowing any piecewise function, through a meticulous orchestration of sines and cosines, to be represented as an infinite sum of these periodic components. The period of trigonometric functions in a Fourier series is directly related to the length of the interval on which they are defined, demonstrating the central role of periodicity in Fourier analysis.

Fourier Coefficients

Diving into the recipe of any Fourier series, the ingredients that give it its unique flavor are the Fourier coefficients. These coefficients, denoted as \(a_0\), \(a_n\), and \(b_n\), play the pivotal role of determining the amplitude - that is, the 'volume' or 'intensity' - of the periodic trigonometric functions within the series.

To find these coefficients, one harnesses the power of integration, essentially 'averaging' the original function with the sine and cosine functions over the interval. This process, akin to finding the perfect balance in a complex equation, extracts the essence of the function, preserving its identity within the Fourier series. It's a delicate process, as each coefficient is carefully calculated to ensure that the series faithfully represents the original function.

Orthogonality Relationships

Imagine two strangers walking down a path, oblivious to each other—they intersect but never really 'interact.' In math, we describe such relationships as orthogonal. Within the realm of Fourier series, this concept of orthogonality is a cornerstone, particularly when discussing sine and cosine functions.

These functions are orthogonal over any given interval, meaning that when you multiply them together and integrate over that interval, their product vanishes, unless they are the same function. This peculiar relationship is the bedrock for computing Fourier coefficients, as it allows for the isolation of each coefficient during the integration process. By exploiting this orthogonality, one can elegantly unravel the coefficients that express a function as a Fourier series—like finding the solo in a grand symphony of mathematical elements.