Problem 13 - Advanced Engineering Mathematics

Question

$$\begin{array}{l} a_{0}=\frac{1}{5} \int_{-5}^{5} f(x) d x=\frac{1}{5}\left(\int_{-5}^{0} 1 d x+\int_{0}^{5}(1+x) d x\right)=\frac{9}{2} \\ a_{n}=\frac{1}{5} \int_{-5}^{5} f(x) \cos \frac{n \pi}{5} x d x=\frac{1}{5}\left(\int_{-5}^{0} \cos \frac{n \pi}{5} x d x+\int_{0}^{5}(1+x) \cos \frac{n \pi}{5} x d x\right)=\frac{5}{n^{2} \pi^{2}}\left[(-1)^{n}-1\right] \\ b_{n}=\frac{1}{5} \int_{-5}^{5} f(x) \sin \frac{n \pi}{5} x d x=\frac{1}{5}\left(\int_{-5}^{0} \sin \frac{n \pi}{5} x d x+\int_{0}^{5}(1+x) \cos \frac{n \pi}{5} x d x\right)=\frac{5}{n \pi}(-1)^{n+1} \\ f(x)=\frac{9}{4}+\sum_{n=1}^{\infty}\left[\frac{5}{n^{2} \pi^{2}}\left[(-1)^{n}-1\right] \cos \frac{n \pi}{5} x+\frac{5}{n \pi}(-1)^{n+1} \sin \frac{n \pi}{5} x\right] \end{array}$$

Step-by-Step Solution

Answer

Calculate coefficients using integrals, write Fourier series with these coefficients.

Step 1: Understanding the a0 coefficient

The coefficient \(a_0\) is the average value of the function \(f(x)\) over the interval \([-5, 5]\). It is calculated by integrating first \(f(x) = 1\) over the interval \([-5, 0]\) and \(f(x) = 1 + x\) over the interval \([0, 5]\). This results in \(a_0 = \frac{1}{5}\left(\int_{-5}^{0} 1 dx + \int_{0}^{5} (1+x) dx \right) = \frac{9}{2}\).

Step 2: Calculating the an coefficients

The coefficients \(a_n\) are calculated by finding the cosine Fourier coefficients over the interval. This involves evaluating the integral \(a_n = \frac{1}{5} \left(\int_{-5}^{0} \cos\left(\frac{n \pi}{5} x\right) dx + \int_{0}^{5} (1+x) \cos\left(\frac{n \pi}{5} x\right) dx \right)\). Solving gives \(a_n = \frac{5}{n^2 \pi^2} \left[(-1)^n - 1 \right]\). This formula shows the coefficients only depend on whether \(n\) is odd or even as the expression \((-1)^n\) switches signs.

Step 3: Calculating the bn coefficients

The coefficients \(b_n\) are calculated by determining the sine Fourier coefficients over the interval. This involves the integral \(b_n = \frac{1}{5} \left(\int_{-5}^{0} \sin\left(\frac{n \pi}{5} x\right) dx + \int_{0}^{5} (1+x) \sin\left(\frac{n \pi}{5} x\right) dx \right)\) yielding the formula \(b_n = \frac{5}{n \pi}(-1)^{n+1}\). The sine coefficients depend on the parity of \(n\) similarly influencing the sign.

Step 4: Writing the Fourier Series

The Fourier series for the function \(f(x)\) is built using the coefficients \(a_0\), \(a_n\), and \(b_n\) as follows: \(f(x) = \frac{9}{4} + \sum_{n=1}^{\infty} \left[ \frac{5}{n^2 \pi^2} \left[(-1)^n - 1 \right] \cos\left(\frac{n \pi}{5} x\right) + \frac{5}{n \pi}(-1)^{n+1} \sin\left(\frac{n \pi}{5} x\right) \right]\). This expresses \(f(x)\) as a sum of sine and cosine functions with coefficients that depend on \(n\).

Key Concepts

Fourier Coefficients

Fourier coefficients are the building blocks of a Fourier series, which is used to express a function as a sum of sine and cosine terms. These coefficients determine the weights of the sine and cosine functions in the series.

The formula for finding these coefficients involves integrating the product of the function and the sine or cosine function over a specified interval.

• The constant term or the average value of the function over an interval is given by the coefficient \(a_0\). If you perform the calculation, you'll find that \(a_0\) provides a measure of the function's overall magnitude.
• The coefficients \(a_n\), associated with cosine terms, are determined by integrating the product of the function and the cosine function over the interval.
• Similarly, the coefficients \(b_n\), tied to sine terms, are found by integrating the product of the function and the sine function over the interval. They capture the periodicity and oscillatory behavior of the function being analyzed.

Understanding these coefficients allows you to decompose a function into simpler harmonic components, greatly aiding in its analysis and study.

Cosine Series

A cosine series is a type of Fourier series in which a function is expressed purely in terms of cosine functions. This is particularly useful when the function is even, meaning it has symmetry about the y-axis.

In the context of Fourier analysis, a cosine series captures the essence of the function's even symmetry.

• The coefficients \(a_n\) define the amplitude of each cosine term. The series starts with a constant term \(a_0/2\), representing the function's average or DC component.
• Each subsequent term \(a_n \cos(n\pi x/T)\) represents the function further using harmonic frequencies.

By efficiently capturing the even oscillations, cosine series are key in applications like signal processing, where encoding only the even parts of a waveform is needed without the sine terms.

Sine Series

A sine series is another valuable Fourier series form, comprising only sine functions. It's mainly used to represent functions that are odd, meaning they have symmetry about the origin. This makes sine series especially useful when dealing with boundary value problems.

The sine series relies heavily on the sine coefficients \(b_n\). These coefficients are computed in a similar fashion as the cosine coefficients, through integration over the required interval.

• Each \(b_n\) term dictates the amplitude of the corresponding sine term in the series.
• As sine functions naturally capture the odd symmetry of a function, they contribute by encoding the behaviors that arise at the function's zero crossing points.

Employing a sine series allows you to effectively reconstruct functions that exhibit odd symmetry, thus playing a crucial role in numerous mathematical and engineering applications.