Problem 5 - Advanced Engineering Mathematics

Question

As in Example 1 in the text we have \(R(r)=c_{3} r^{n}+c_{4} r^{-n}\). In order that the solution be bounded as \(r \rightarrow \infty\) we must define \(c_{3}=0 .\) Hence $$u(r, \theta)=A_{0}+\sum_{n=1}^{\infty} r^{-n}\left(A_{n} \cos n \theta+B_{n} \sin n \theta\right)$$ where $$\begin{array}{l} A_{0}=\frac{1}{2 \pi} \int_{0}^{2 \pi} f(\theta) d \theta \\ A_{n}=\frac{c^{n}}{\pi} \int_{0}^{2 \pi} f(\theta) \cos n \theta d \theta \\ B_{n}=\frac{c^{n}}{\pi} \int_{0}^{2 \pi} f(\theta) \sin n \theta d \theta \end{array}$$.

Step-by-Step Solution

Answer

Set \( c_{3}=0 \) for boundedness; express \( u(r, \theta) \) using \( A_n \) and \( B_n \) coefficients, derived as Fourier coefficients.

Step 1: Analyze Given Functions

We have the function \( R(r) = c_{3} r^{n} + c_{4} r^{-n} \) that needs to be bounded as \( r \rightarrow \infty \). This implies we must eliminate terms that grow as \( r \) becomes large, which means setting \( c_{3} = 0 \). Thus, for a bounded solution we focus on \( c_{4} r^{-n} \).

Step 2: Analyze Solution Series

Given that \( c_{3} = 0 \) for a bounded solution, the function simplifies to:\[ u(r, \theta) = A_{0} + \sum_{n=1}^{\infty} r^{-n} \left( A_{n} \cos n \theta + B_{n} \sin n \theta \right) \]Here, each term \( r^{-n} \) ensures the boundedness of the solution as \( r \to \infty \).

Step 3: Define Coefficient \(A_0\)

The coefficient \( A_{0} \) is given by the formula:\[ A_{0} = \frac{1}{2 \pi} \int_{0}^{2 \pi} f(\theta) \, d\theta \]This represents the average value of the function \( f(\theta) \) over the interval from 0 to \( 2\pi \).

Step 4: Define Coefficient \(A_n\)

The coefficient \( A_{n} \) is calculated using:\[ A_{n} = \frac{c^{n}}{\pi} \int_{0}^{2 \pi} f(\theta) \cos n \theta \, d\theta \]This integral projects \( f(\theta) \) onto \( \cos n \theta \), scaled by \( c^n \) and the factor \( \frac{1}{\pi} \).

Step 5: Define Coefficient \(B_n\)

The coefficient \( B_{n} \) is given by:\[ B_{n} = \frac{c^{n}}{\pi} \int_{0}^{2 \pi} f(\theta) \sin n \theta \, d\theta \]This integrates \( f(\theta) \) onto \( \sin n \theta \), also scaled by \( c^n \) and \( \frac{1}{\pi} \). These coefficients are part of the Fourier series representation of \( f(\theta) \).

Step 6: Formulate the Solution

The complete solution for \( u(r, \theta) \) is determined by combining the constant term \( A_0 \) and the series terms involving \( A_n \) and \( B_n \), each multiplied by \( r^{-n} \). The final expression for \( u(r, \theta) \) is:\[ u(r, \theta) = A_{0} + \sum_{n=1}^{\infty} r^{-n} \left( A_{n} \cos n \theta + B_{n} \sin n \theta \right) \]

Key Concepts

Boundary Conditions

Boundary conditions are essential constraints applied to differential equations or systems to determine a unique solution. In our exercise, we're dealing with the function \( R(r) = c_{3} r^{n} + c_{4} r^{-n} \). Boundary conditions help us ensure the solution behaves in a specific way as the variable approaches certain values, particularly at endpoints or along boundaries of a defined region. When dealing with infinite domains like \( r \rightarrow \infty \), it's crucial to control the behavior of the function.

• For the function to remain bounded as \( r \rightarrow \infty \), we need to remove any terms that grow unbounded. For instance, the term \( c_{3} r^{n} \) increases without bound as \( r \) becomes large unless \( c_{3} \) is set to zero.
• By setting \( c_{3} = 0 \), we eliminate these terms, ensuring the solution only includes diminishing terms \( c_{4} r^{-n} \), which shrink as \( r \) grows.

Understanding boundary conditions is fundamental for solving problems where the behavior of functions as they approach infinity or other limits is a concern.

Harmonic Functions

Harmonic functions are solutions to Laplace's equation \( abla^{2} u = 0 \) and arise naturally in various physical settings, including fluid flow and electrostatics. In the context of Fourier series, harmonic functions have periodic properties which are significant in simplifying complex phenomena:

• A harmonic function like \( u(r, \theta) \) can be separated into radial and angular components, which often simplifies our task by breaking down complex problems into more manageable parts.
• The exercise employs Fourier series expansion to express \( u(r, \theta) \) in terms of sine and cosine functions. These trigonometric components \( \cos n \theta \) and \( \sin n \theta \) play the crucial role of harmonics, resembling waves that exhibit the property of harmonicity.
• In simpler terms, when solving problems that involve variations in space and periodic boundary conditions, harmonic functions help represent solutions that can be easily manipulated and understood.

Recognizing the role of harmonic functions is crucial for working with phenomena that require solutions with specific oscillatory characteristics.

Integral Calculus

Integral calculus, a key area of mathematics, deals with integration and its applications. In this exercise, integration is essential for determining coefficients in the Fourier series representation of \( f(\theta) \):

• The coefficients \( A_0, A_n, \) and \( B_n \) involve integrals which help in projecting the original function onto basis functions such as cosine and sine functions.

Let's breakdown how these coefficients are calculated:

\( A_0 = \frac{1}{2 \pi} \int_{0}^{2 \pi} f(\theta) \, d\theta \) is the average value of the function over one period \( [0, 2\pi] \). This integral helps determine the constant term in the Fourier series, representing a mean level or equilibrium state of \( f(\theta) \).

\( A_n = \frac{c^{n}}{\pi} \int_{0}^{2 \pi} f(\theta) \cos n \theta \, d\theta \) and \( B_n = \frac{c^{n}}{\pi} \int_{0}^{2 \pi} f(\theta) \sin n \theta \, d\theta \) are integrals that identify how much of the function can be expressed as a combination of these harmonics.

Integral calculus in Fourier analysis provides a systematic method to dissect and reconstruct periodic functions, enabling us to comprehensively describe and analyze situations characterized by repetitive patterns.