Problem 10 - Advanced Engineering Mathematics

Question

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 series of hyperbolic sine and sine functions satisfying all boundary conditions.

Step 1: State Laplace's Equation

Laplace's equation in two dimensions is given by: \[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \] where \( u(x, y) \) is the function we need to solve.

Step 2: Assume a Solution Form

Assume a solution of the form \( u(x, y) = X(x)Y(y) \). Substitute \( u(x, y) \) into Laplace's equation to separate the variables.

Step 3: Derive the Ordinary Differential Equations (ODEs)

Substitute \( u(x,y) = X(x)Y(y) \) into the Laplace equation and separate the terms to get: \[ \frac{X''(x)}{X(x)} = -\frac{Y''(y)}{Y(y)} = \lambda \] This gives two ODEs: \( X''(x) = \lambda X(x) \) and \( Y''(y) = -\lambda Y(y) \).

Step 4: Solve the Y component ODE

Solve \( Y''(y) = -\lambda Y(y) \) subject to the boundary conditions \( Y(0) = 0 \) and \( Y(1) = 0 \). This leads to \( Y(y) = C \sin(n\pi y) \) with \( \lambda = (n\pi)^2 \).

Step 5: Solve the X component ODE

Solve \( X''(x) = (n\pi)^2 X(x) \) subject to the boundary condition \( X'(1) = -1 \). The general solution is \( X(x) = A e^{n \pi x} + B e^{-n \pi x} \).

Step 6: Apply Remaining Boundary Condition

Apply the boundary condition \( u(0,y) = 10y \) to find the constants. This will involve substituting \( x=0 \) and matching the resulting function to \( 10y \).

Step 7: Construct Full Solution

Combine the solutions into one expression for \( u(x, y) \) by summing the solutions for different values of \( n \) that satisfy the boundary conditions. Each term will look like \( (A_n e^{n \pi x} + B_n e^{-n \pi x}) \sin(n\pi y) \).

Step 8: Determine Constants

Use the method of Fourier coefficients to determine the constants \( A_n \) and \( B_n \). These are found by integrating the boundary conditions over the interval \([0,1]\).

Step 9: Final Solution

From the previous steps, derive the final form of the solution for the plate with the given boundary conditions. The constants \( A_n \) and \( B_n \) will be calculated based on these boundary conditions.

Key Concepts

Boundary Conditions

Boundary conditions are crucial for solving partial differential equations like Laplace's equation. They specify the behavior of the solution on the boundary of the domain. In this exercise, we are given several boundary conditions for a rectangular plate:

• At the left edge, at x = 0, the condition is that the solution is a linear function of y, specifically, \( u(0, y) = 10y \).
• At the right edge, the rate of change with respect to x is constant: \( \left. \frac{\partial u}{\partial x} \right|_{x=1} = -1 \).
• Along the bottom and top edges, \( u(x, 0) = 0 \) and \( u(x, 1) = 0 \), implying the solution touches zero at these boundaries.

These conditions ensure that the solution not only solves the differential equation but also fits the physical or geometrical constraints given by the problem. They help uniquely determine the solution by preventing infinitely many solutions that Laplace's equation allows without such constraints.

Separation of Variables

The method of separation of variables is a powerful technique for solving partial differential equations, such as Laplace's equation. The first step involves assuming a solution that can be written as a product of functions, each depending on a single coordinate. In our case, we assume:\[ u(x, y) = X(x)Y(y) \]This assumption allows us to substitute into Laplace's equation and separate the equation into two ordinary differential equations.This technique simplifies the problem significantly, transforming the original partial differential equation into simpler, solvable ordinary differential equations. Each ODE, corresponding to one variable, can be solved independently. Once solved, these solutions are combined to form the complete solution to the original problem.

Ordinary Differential Equations (ODEs)

In the separation of variables approach, once substituted, Laplace's equation yields two ordinary differential equations (ODEs):

• \( X''(x) = \lambda X(x) \)
• \( Y''(y) = -\lambda Y(y) \)

These ODEs arise from the separation of variables and are crucial in solving for each component independently. For the y-component, the boundary conditions \( Y(0) = 0 \) and \( Y(1) = 0 \) lead us to a sinusoidal solution \( Y(y) = C \sin(n\pi y) \), where \( \lambda = (n\pi)^2 \).For the x-component, solving \( X''(x) = (n\pi)^2 X(x) \) under the condition \( X'(1) = -1 \) results in an exponential form. Understanding the derivation of these solutions is key to constructing the solution of the original problem.

Fourier Series

After solving the ODEs, the solutions are usually expressed as an infinite series, known as a Fourier series. These series sum solutions for different values of n that fit the boundary conditions. In particular, each term in our series solution will look like:\[ (A_n e^{n \pi x} + B_n e^{-n \pi x}) \sin(n\pi y) \]Fourier series are critical for representing complex periodic functions as sums of simpler trigonometric functions. Here, the Fourier series allows us to incorporate the solution to meet all specified boundary conditions, specifically \( u(0, y) = 10y \).To figure out the constants \( A_n \) and \( B_n \), we use the orthogonality property of sine functions over an interval, applying the boundary condition through integration. Understanding how Fourier series work gives insights into how complex solutions can be pieced together from more straightforward parts.