Problem 10 - Advanced Engineering Mathematics

Question

In Problems \(1-10\), solve Laplace's equation (1) for a rectangular plate subject to the given boundary conditions. \(u(0, y)=10 y,\left.\frac{\partial u}{\partial x}\right|_{x=1}=-1\) \(u(x, 0)=0, u(x, 1)=0\)

Step-by-Step Solution

Answer

The solution is a Fourier series matching boundary conditions: \[ u(x,y)= \sum_{n=1}^{\infty} \left(B_n\sinh(n\pi x) + C_n\cosh(n\pi x)\right)\sin(n\pi y) \]

Step 1: Recognize the Laplace Equation

Laplace's equation is given by \( abla^2 u = 0 \), or in two dimensions, \( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \). Our task is to find a function \( u(x, y) \) that satisfies this equation within the given domain, and adheres to the specified boundary conditions.

Step 2: Identify the Boundary Conditions

The given boundary conditions are: - \( u(0, y) = 10y \) indicating a linear variation along the \( y \)-boundary at \( x = 0 \).- \( \left.\frac{\partial u}{\partial x}\right|_{x=1} = -1 \), a Neumann condition specifying the derivative along \( x = 1 \).- \( u(x, 0) = 0 \) and \( u(x, 1) = 0 \), which are Dirichlet conditions along the \( y = 0 \) and \( y = 1 \) boundaries.

Step 3: Formulate the General Solution

Assume a separable solution \( u(x, y) = X(x)Y(y) \), substitute into Laplace's equation, resulting in: \[ \frac{X''(x)}{X(x)} = - \frac{Y''(y)}{Y(y)} = \lambda \] This results in two ordinary differential equations (ODEs) to solve separately:(i) \( X''(x) = \lambda X(x) \) and (ii) \( Y''(y) = -\lambda Y(y) \).

Step 4: Solve the Associated ODEs

Depending on the sign of \( \lambda \), different forms of solutions arise. Since \( u(x, 0) = 0 \) and \( u(x, 1) = 0 \), assume \( \lambda = (n\pi)^2 \), solving yields: - \( Y(y) = A\sin(n\pi y) \). - Solving \( X''(x) = (n\pi)^2 X(x) \) gives \( X(x) = B\sinh(n\pi x) + C\cosh(n\pi x) \). This form satisfies the periodic conditions in the \( y \)-direction and the requirements of \( x \) on Neumann boundaries.

Step 5: Apply Boundary Conditions

Impose the conditions on \( u(x, y) \): 1. From \( u(0, y) = 10y \), identify it as a Fourier sine series\(\Rightarrow X(x)\) doesn't contribute at \( x = 0 \). 2. Using \( \frac{\partial u}{\partial x}(1, y) = -1 \), solve for \( X'(1)Y(y) = -1 \). Repeat this for terms in \( X(x) \) and \( Y(y) \), check compatibility.3. Boundary \( Y(y) \) conditions give the series form, combine, solve coefficients for matching.

Step 6: Construct the Solution

The final expression aligning with conditions and series is a sum over modes determined by: \[ u(x,y)= \sum_{n=1}^{\infty} \left(B_n\sinh(n\pi x) + C_n\cosh(n\pi x)\right)\sin(n\pi y) \] Solve coefficients by forcing matches with boundary, deploy Fourier methods iteratively if direct match absent; partial results due to real boundary replacement.

Key Concepts

Boundary Conditions

Boundary conditions in the context of solving Laplace's equation are crucial as they dictate the behavior of the solution at the edges of the domain. For the given problem, there are four boundary conditions: two are Dirichlet conditions and one is a Neumann condition.
- **Dirichlet Conditions:** These specify the value of the function at the boundary. Here, we have \( u(x, 0) = 0 \) and \( u(x, 1) = 0 \), which tell us that along the lines \( y = 0 \) and \( y = 1 \), the function value is zero. This often represents a physical constraint like temperature being zero at the edges.
- **Neumann Condition:** This specifies the derivative (rate of change) of the function at a boundary. In this exercise, \( \left.\frac{\partial u}{\partial x}\right|_{x=1} = -1 \) specifies how the function changes along the line \( x = 1 \), indicating non-zero flux or gradient at this boundary.
Understanding these types of boundary conditions helps decide how to approach solving the Laplace equation, ensuring that any derived solutions from the differential equations will inherently satisfy these conditions.

Separation of Variables

Separation of Variables is a mathematical method used to solve partial differential equations like Laplace's equation by separating into simpler, single-variable problems. In this approach, we assume the solution can be expressed as a product of functions, each dependent on different variables. For instance, we assume \( u(x, y) = X(x)Y(y) \).
Plug this expression into Laplace’s equation: \( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \), splits into two ordinary differential equations where we can handle each independently:

• One for \( X(x) \), derived as \( X''(x) = \lambda X(x) \)
• Another for \( Y(y) \), resulting in \( Y''(y) = -\lambda Y(y) \)

The constant \( \lambda \) separates the variables and allows us to tackle these equations separately. The separation relies on identifying specific solutions that respect boundary conditions, typically leading to trigonometric (sines and cosines) solutions for \( Y(y) \) due to periodic conditions given in the problem.

Fourier Series

A Fourier series is a tool used to express a function as a sum of sines and cosines. This concept becomes particularly useful when solving boundary value problems like those involving Laplace's equation.
The appearance of periodic boundary conditions naturally lends itself to the application of Fourier series. In our problem:

• **Fourier Sine Series** was utilized to match the Dirichlet boundary conditions \( u(x, 0) = 0 \) and \( u(x, 1) = 0 \), which implied sinusoidal variation along the \( y \)-direction

When asserting the boundary condition \( u(0, y) = 10y \), the function \( u(x, y) \) gets expanded as a series with terms like \( \sin(n\pi y) \) forming the basis of periodic functions. The coefficients in this expansion are calculated to ensure the function respects the conditions across the entire boundary, achieving a solution that combines the orthogonality of sines and natural boundary constraints.
Using Fourier series effectively abstracts real boundary conditions into manageable components, especially helpful when directly solving differential equations involves complex integrations.