Problem 14 - Advanced Engineering Mathematics

Question

Letting \(u(r, \theta)=v(r, \theta)+\psi(\theta)\) we obtain \(\psi^{\prime \prime}(\theta)=0\) and so \(\psi(\theta)=c_{1} \theta+c_{2} .\) From \(\psi(0)=0\) and \(\psi(\pi)=u_{0}\) we find, in turn, \(c_{2}=0\) and \(c_{1}=u_{0} / \pi .\) Therefore \(\psi(\theta)=\frac{u_{0}}{\pi} \theta . \quad\) Now \(u(1, \theta)=v(1, \theta)+\psi(\theta)\) so that \(v(1, \theta)=u_{0}-\frac{u_{0}}{\pi} \theta .\) From $$v(r, \theta)=\sum_{n=1}^{\infty} A_{n} r^{n} \sin n \theta \quad \text { and } \quad v(1, \theta)=\sum_{n=1}^{\infty} A_{n} \sin n \theta$$ we obtain $$A_{n}=\frac{2}{\pi} \int_{0}^{\pi}\left(u_{0}-\frac{u_{0}}{\pi} \theta\right) \sin n \theta d \theta=\frac{2 u_{0}}{\pi n}$$ Thus $$u(r, \theta)=\frac{u_{0}}{\pi} \theta+\frac{2 u_{0}}{\pi} \sum_{n=1}^{\infty} \frac{r^{n}}{n} \sin n \theta$$.

Step-by-Step Solution

Answer

The function \( u(r, \theta) = \frac{u_0}{\pi} \theta + \frac{2u_0}{\pi} \sum_{n=1}^{\infty} \frac{r^n}{n} \sin n\theta \).

Step 1: Start with the given expression

We begin with the given expression for function \( u(r, \theta) \) which is written as \( u(r, \theta)=v(r, \theta)+\psi(\theta) \).

Step 2: Solve the differential equation for \( \psi(\theta) \)

The equation \( \psi''(\theta) = 0 \) implies that \( \psi(\theta) \) is a linear function. Solving gives \( \psi(\theta) = c_1 \theta + c_2 \).

Step 3: Apply boundary conditions

Using the boundary conditions \( \psi(0) = 0 \) and \( \psi(\pi) = u_0 \), we solve for \( c_1 \) and \( c_2 \). This results in \( c_2 = 0 \) and \( c_1 = \frac{u_0}{\pi} \). Therefore, \( \psi(\theta) = \frac{u_0}{\pi} \theta \).

Step 4: Express \( v(1, \theta) \)

Since \( u(1, \theta) = v(1, \theta) + \psi(\theta) \), substituting the expression for \( \psi(\theta) \) yields \( v(1, \theta) = u_0 - \frac{u_0}{\pi} \theta \).

Step 5: Fourier series representation of \( v(r, \theta) \)

Using \( v(r, \theta) = \sum_{n=1}^{\infty} A_n r^n \sin n\theta \) and \( v(1, \theta) = \sum_{n=1}^{\infty} A_n \sin n\theta \), equate it to the expression found in Step 4 to determine \( A_n \).

Step 6: Find coefficients \( A_n \)

Calculate \( A_n \) using the formula \( A_n = \frac{2}{\pi} \int_{0}^{\pi} \left( u_0 - \frac{u_0}{\pi} \theta \right) \sin n \theta \, d\theta = \frac{2u_0}{\pi n} \).

Step 7: Construct final expression for \( u(r, \theta) \)

Substitute \( A_n \) and \( \psi(\theta) \) into the expression for \( u(r, \theta) \). Thus, \( u(r, \theta) = \frac{u_0}{\pi} \theta + \frac{2u_0}{\pi} \sum_{n=1}^{\infty} \frac{r^n}{n} \sin n\theta \).

Key Concepts

Boundary Conditions

Boundary conditions are crucial in solving differential equations, especially in problems involving Fourier series. They provide constraints that help us find the unknown constants in our solution. For example, in our problem, we have a function

• \( \psi(\theta) \) satisfying the differential equation \( \psi^{\prime\prime}(\theta) = 0 \),
• The general solution is a linear function: \( \psi(\theta) = c_1 \theta + c_2 \).
• Boundary conditions give us specific values: \( \psi(0) = 0 \) and \( \psi(\pi) = u_0 \).

By applying these, we found \( c_2 = 0 \) and \( c_1 = \frac{u_0}{\pi} \). These steps ensure our solution fits the physical or geometric constraints of the problem. Understanding how these conditions modify our solutions is key to mastering boundary value problems.

Differential Equations

Differential equations are equations that involve the derivatives of a function. They provide a powerful way to model and solve real-world phenomena. In the given exercise, the differential equation for \( \psi(\theta) \) is very simple:

• \( \psi^{\prime\prime}(\theta) = 0 \).

This indicates that the second derivative of \( \psi(\theta) \) is zero, meaning that:

• \( \psi(\theta) \) is a linear function of \( \theta \).

Differential equations often appear alongside boundary conditions, as they together help determine the precise solution. Solving differential equations can range from straightforward to highly complex, depending on the equation's form and the boundary or initial conditions applied.

Coefficients Calculation

The calculation of coefficients is central to the solution of problems involving Fourier series. Fourier series represent functions as infinite sums of sines and cosines, allowing complex periodic functions to be expressed in simpler trigonometric terms. In this exercise:

• We find the coefficients \( A_n \) by matching the Fourier series to our function \( v(1, \theta) \).
• The expression \( v(1, \theta) = \sum_{n=1}^{\infty} A_n \sin n\theta \) aligns with \( v(1, \theta) = u_0 - \frac{u_0}{\pi} \theta \).
• To compute \( A_n \), we use the formula: \[A_n = \frac{2}{\pi} \int_{0}^{\pi} \left(u_0 - \frac{u_0}{\pi} \theta\right) \sin n \theta \, d\theta = \frac{2u_0}{\pi n}\]

Coefficients like \( A_n \) show how much of each sine function contributes to the overall series. Calculating them correctly is vital, as errors can lead to inaccurate representations of the function.