Problem 15 - Advanced Engineering Mathematics

Question

Expand the given function in an appropriate cosine or sine series. $$ f(x)=x^{2},-1

Step-by-Step Solution

Answer

The cosine series expansion is \( f(x) = \frac{1}{3} + \sum_{n=1}^{\infty} \frac{4(-1)^n}{n^2 \pi^2} \cos(n \pi x) \).

Step 1: Identify the Type of Series

We're given a function, \( f(x) = x^2 \) within the interval \(-1 < x < 1\). We are asked to expand it in a cosine or sine series. Since \( x^2 \) is an even function, it will be expanded in a cosine series, as cosine functions are also even.

Step 2: Write the Cosine Series Formula

The general formula for a cosine series expansion for a function \( f(x) \) over \(-L < x < L\) is \( f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos\left(\frac{n\pi x}{L}\right) \), where \( a_n = \frac{2}{L} \int_{0}^{L} f(x) \cos\left(\frac{n\pi x}{L}\right) \, dx \). In our case, \( L = 1 \).

Step 3: Compute the Coefficient \(a_0\)

For the \(a_0\) term, we compute the integral: \[ a_0 = \frac{2}{1} \int_{0}^{1} x^2 \, dx = 2 \left[ \frac{x^3}{3} \right]_{0}^{1} = 2 \times \frac{1}{3} = \frac{2}{3}. \] This implies \( \frac{a_0}{2} = \frac{1}{3} \).

Step 4: Compute the Coefficients \(a_n\)

For \(n \geq 1\), \( a_n = 2 \int_{0}^{1} x^2 \cos(n \pi x) \, dx \). Using integration by parts may be needed multiple times to solve this integral, eventually leading to \[ a_n = \frac{4(-1)^n}{n^2 \pi^2} \].

Step 5: Write the Final Series

The series expansion is \( f(x) = \frac{1}{3} + \sum_{n=1}^{\infty} \frac{4(-1)^n}{n^2 \pi^2} \cos(n \pi x) \). This is the cosine series representation of \( f(x) = x^2 \) for \(-1 < x < 1\).

Key Concepts

Cosine series

A cosine series is a particular type of Fourier series that is used to represent even functions. In a cosine series, we only use cosine terms because cosine functions are inherently even. This means that they are symmetric about the y-axis. In mathematical terms, an even function satisfies the condition \(f(x) = f(-x)\). Using only cosine terms allows us to efficiently represent any even function over a specified interval.For example, the cosine series expansion for a function \(f(x)\) over the interval \([-L, L]\) is given by:

• \(f(x) = \frac{a_0}{2} + \sum_{n=1}^{finity} a_n \cos\left(\frac{n\pi x}{L}\right)\)

Here, \(a_0\) and \(a_n\) are the coefficients determined by integrating the function over the interval. In our example, \(L = 1\), simplifying the integration process. Cosine series are a fundamental concept in harmonic analysis and are crucial for solving partial differential equations in engineering and physics.

Even function

An even function is a type of mathematical function that exhibits symmetry about the y-axis. This means that for any given point \(x\), the function satisfies the relation \(f(x) = f(-x)\). This property is fundamental in many areas, such as signal processing and control systems, because it simplifies the analysis and computation of function properties.In the context of Fourier series, knowing whether a function is even can help you decide which type of series to use, either sine or cosine. Even functions pair well with cosine series, due to cosine's own even nature. This significant property allows us to only consider cosine terms when expanding the function, significantly reducing computational effort.

Integration by parts

Integration by parts is a mathematical technique that is widely used to solve integrals, especially those that involve products of functions. The rule is based on the product rule for differentiation, and it is expressed as:

• \[\int u \, dv = uv - \int v \, du\]

Here, one selects parts of the integrand to differentiate and integrate strategically to simplify the integral. It is particularly useful in calculating the coefficients in a Fourier series expansion when direct integration is complex, as is the case with \(a_n = \int_{0}^{1} x^2 \cos(n \pi x) \, dx\).In our example, integration by parts was used multiple times to derive the formula for \(a_n\). Problems of this sort often require careful choice of \(u\) and \(dv\) to ultimately solve the integral.

Trigonometric series

Trigonometric series is a series of terms that are trigonometric functions, like sines and cosines. These series are extremely important in mathematical analysis, particularly in the Fourier series which uses them to represent periodic functions.Fourier series decompose complex periodic functions into sums of simpler sine and cosine functions. They are invaluable tools in many fields, such as acoustics, signal processing, and quantum physics.In the given problem, we represent \(x^2\) — a non-periodic and even function — into its periodic cosine series form, showcasing the utility of trigonometric series in general, and the versatility of Fourier series in particular. Each term in the series contributes a specific frequency, amplitude, and phase shift, providing a detailed description of the function's behavior over its interval.