Question

Use Theorem \(11.3 .5(\mathbf{d})\) or where applicable, Exercise \(50(\mathbf{b}),\) to find the mixed Fourier sine series of the \(f\) on \([0, L]\). $$ f(x)=x\left(x^{3}-2 L x^{2}+2 L^{3}\right) $$

Short answer

Question: Find the mixed Fourier sine series representation of the function \(f(x) = x(x^3 - 2Lx^2 + 2L^3)\) on the interval \([0, L]\). Answer: The mixed Fourier sine series representation of the function \(f(x)\) is given by: $$ f(x) = \sum_{n=1}^{\infty} \frac{2}{n\pi L} \left[(-1)^n - \frac{2L^2}{n\pi} + \frac{2L^4}{(n\pi)^3}\right]\sin\left(\frac{n\pi x}{L}\right) $$

Step-by-step solution

Identify the given function

We are given a function: $$ f(x) = x(x^3 - 2Lx^2 + 2L^3) $$ Our task is to find the mixed Fourier sine series representation of this function on the interval \([0, L]\).

Derive the Fourier sine coefficients formula

According to Theorem 11.3.5(d) or Exercise 50(b), the Fourier sine coefficients \(B_n\) for a function \(f(x)\) are defined as: $$ B_n = \frac{2}{L} \int_0^L f(x) \sin\left(\frac{n\pi x}{L}\right) dx $$ Where \(n\in\mathbb{N}\)

Plug in the given function and compute \(B_n\)

Substitute \(f(x)\) into the formula for \(B_n\) and integrate to find the Fourier sine coefficients: $$ B_n = \frac{2}{L} \int_0^L x(x^3 - 2Lx^2 + 2L^3) \sin\left(\frac{n\pi x}{L}\right) dx $$ This integral can be solved by splitting it into three parts, and applying integration by parts twice for each part: $$ B_n = \frac{2}{L} \left[\int_0^L x^4 \sin\left(\frac{n\pi x}{L}\right) dx - 2L\int_0^L x^3 \sin\left(\frac{n\pi x}{L}\right) dx + 2L^3\int_0^L x \sin\left(\frac{n\pi x}{L}\right) dx\right] $$ Let us calculate each integral and express \(B_n\) after simplifications: $$ B_n = \frac{2}{n\pi L} \left[(-1)^n - \frac{2L^2}{n\pi} + \frac{2L^4}{(n\pi)^3}\right] $$

Write down the mixed Fourier sine series representation

Having found the Fourier sine coefficients \(B_n\), we can now write the mixed Fourier sine series representation for the function \(f(x)\): $$ f(x) = \sum_{n=1}^{\infty} B_n \sin\left(\frac{n\pi x}{L}\right) = \sum_{n=1}^{\infty} \frac{2}{n\pi L} \left[(-1)^n - \frac{2L^2}{n\pi} + \frac{2L^4}{(n\pi)^3}\right]\sin\left(\frac{n\pi x}{L}\right) $$ This is the mixed Fourier sine series representation of the given function \(f(x) = x(x^3 - 2Lx^2 + 2L^3)\) on the interval \([0, L]\).

Key concepts

Mixed Fourier Series

In mathematics, a Mixed Fourier Series is an expansion of a function that combines both sine and cosine functions and possibly a constant term, allowing for a representation of functions that may not be periodic. However, when the function is defined only on a half interval, like \(0, L\), and satisfies specific boundary conditions, we can use a Fourier sine series to represent it.

The Fourier sine series is particularly useful for solving boundary value problems in partial differential equations, where the solution is required to satisfy certain conditions on the boundary of the domain, often zero (hence, 'sine' due to the zero value at the endpoints in the sine function). It represents the odd extension of a function and is defined by an infinite sum of sine terms, each with a specific Fourier coefficient, that captures the function’s behavior on the interval. This series is immensely practical in analyzing vibrating strings, heat transfer, and other physical phenomena.

Boundary Value Problems

A Boundary Value Problem (BVP) is a mathematical situation where we're looking to find a solution to a differential equation that must satisfy set conditions at the boundaries of its domain. These problems are essential in various fields of engineering and physics where the solution needs to satisfy physical constraints.

For example, when modeling the vibrations of a string with fixed ends, the displacement of the string at the ends must be zero—this is a boundary condition. These conditions are what make the Fourier sine series so crucial, as its elements inherently satisfy the zero boundary condition at both ends, which is why functions that describe physical phenomena exhibiting these boundary behaviors are often expanded into their Fourier sine series.

Integration by Parts

The method of Integration by Parts is a powerful integration technique that comes from the product rule of differentiation. It can be used quite effectively to integrate products of functions where the integral of the complete function is complex or unknown.

Mathematically, it is given by:\[\int u dv = uv - \int v du\]
This technique simplifies the process of integration when dealing with polynomial functions multiplied by trigonometric, exponential, or logarithmic functions—commonly encountered in calculating Fourier coefficients for a Fourier series. Integration by parts often needs to be applied iteratively to reduce intricate integrals to more manageable forms.

Fourier Coefficients

The Fourier coefficients are the weights assigned to each sine or cosine function in the Fourier series, and they play a crucial role in defining the shape and amplitude of the series representation of a function. They are essentially determined by integration, capturing how much of each sine or cosine wave is needed to reproduce the original function.

For a Fourier sine series, these coefficients are calculated through the integral:\[B_n = \frac{2}{L} \int_0^L f(x) \sin\left(\frac{n\pi x}{L}\right) dx\]
This formula ensures that each coefficient is extracted by 'projecting' the function onto the corresponding sine function over the given interval. The accuracy of the resulting Fourier series in representing the original function largely depends on the proper calculation of these coefficients—highlighting their significance in Fourier analysis and its applications to real-world problems.