Question

(a) Find the solution \(u(x, y)\) of Laplace's equation in the rectangle \(0

Short answer

Short answer: The solution of Laplace's equation in a rectangular domain with the given boundary conditions can be found using the separation of variables technique. By solving two separate problems with specific boundary conditions and adding their solutions, we obtain the final solution for \(u(x, y)\). For the given functions \(h(x) = (x/a)^2\) and \(f(y) = 1 - (y/b)\), after evaluating the integrals, we get \(A_n = \frac{2b(-1)^{n+1}}{n\pi}\) and \(B_m = \frac{8a}{m^3 \pi^3}\). To plot the solution for \(a = 2\) and \(b = 2\), you will need to use a numerical software package like MATLAB, Python, etc., and create 2D, 3D, and contour plots of \(u(x, y)\) as a function of \(x\) and \(y\).

Step-by-step solution

Solve the first problem with homogeneous boundary conditions except for \(u(a, y) = f(y)\)

We will first solve the problem with the given boundary conditions: \(u(0, y) = 0,\ u(a, y) = f(y),\ u(x, 0) = 0,\ u(x, b) = 0.\) We assume the solution \(u(x, y) = X(x)Y(y)\), and substitute it back into Laplace's equation \((\nabla^2 u = 0)\). We will then separate the variables and find the solutions to the resulting ordinary differential equations. After setting up the equations and solving them, we obtain the solution: $$ u_1(x, y) = \sum_{n=1}^{\infty} A_n \sin\left(\frac{n\pi x}{a}\right)\sinh\left(\frac{n\pi y}{a}\right), $$ where: $$ A_n = \frac{2}{a \sinh\left(\frac{n\pi b}{a}\right)} \int_0^a f(y) \sin\left(\frac{n\pi y}{a}\right) dy $$

Solve the second problem with homogeneous boundary conditions except for \(u(x, 0) = h(x)\)

Next, we will solve the problem with the given boundary conditions: \(u(0, y) = 0,\ u(a, y) = 0,\ u(x, 0) = h(x),\ u(x, b) = 0.\) We assume the solution \(u(x, y) = X(x)Y(y)\), and substitute it back into Laplace's equation \((\nabla^2 u = 0)\). We will then separate the variables and find the solutions to the resulting ordinary differential equations. After setting up the equations and solving them, we obtain the solution: $$ u_2(x, y) = \sum_{m=1}^{\infty} B_m \sin\left(\frac{m\pi x}{a}\right)\left[\cosh\left(\frac{m\pi y}{a}\right) - \frac{\cosh\left(\frac{m\pi b}{a}\right)}{\sinh\left(\frac{m\pi b}{a}\right)}\sinh\left(\frac{m\pi y}{a}\right)\right], $$ where: $$ B_m = \frac{2}{a} \int_0^a h(x) \sin\left(\frac{m\pi x}{a}\right) dx. $$

Adding the two solutions

Now, we will add the solutions \(u_1(x, y)\) and \(u_2(x, y)\) to get the final solution for \(u(x, y)\): $$ u(x, y) = u_1(x, y) + u_2(x, y). $$ #Part (b): Find the solution for specific functions#

Substitute the given functions and calculate the coefficients \(A_n\) and \(B_m\)

Now, we will substitute the given functions: \(h(x) = (x/a)^2\) and \(f(y) = 1 - (y/b)\), and calculate the coefficients \(A_n\) and \(B_m\) by evaluating the integrals. After evaluating the integrals, we get: $$ A_n = \frac{2b(-1)^{n+1}}{n\pi} \quad\text{and}\quad B_m = \frac{8a}{m^3 \pi^3}. $$ Now, we will substitute these coefficients into the expressions for \(u_1(x, y)\) and \(u_2(x, y)\). #Part (c): Plotting the solution for \(a = 2, b = 2\)# At this point, we have the solution for \(u(x, y)\) with the necessary coefficients. For this part, you will need to use a numerical software package (like MATLAB, Python, etc.) to plot \(u(x, y)\) for the given values of \(a = 2\) and \(b = 2\). Plot the solution as a function of \(x\), \(y\), and both \(x\) and \(y\), as well as a contour plot. As assistants can't produce graphs, create plots with 2D and 3D view as well as contour plot using a suitable programming language or software, based on the given values of \(a\) and \(b\).

Key concepts

Laplace's Equation

Laplace's Equation is a second-order partial differential equation fundamental in physics and engineering. It is expressed as \( abla^2 u = 0 \), where \( u \) is a twice-differentiable function, and \( abla^2 \) denotes the Laplacian operator. The equation describes phenomena such as heat distribution, electric potential, and fluid flow.

In the context of this exercise, solving Laplace’s equation involves finding a potential \( u(x, y) \) in a specified rectangular domain with given boundary conditions. The boundary conditions provided help determine the unique solution by specifying the behavior of \( u \) on the edges of the rectangle.

Boundary Value Problems

Boundary Value Problems (BVPs) are problems where the solution to a differential equation is required to satisfy certain conditions at the boundaries of the domain. For our exercise, we deal with a rectangle where some boundaries have specified conditions.

The exercise splits into two smaller BVPs:

• One with a non-homogeneous boundary condition at \( u(a, y) = f(y) \)
• Another with a non-homogeneous boundary condition at \( u(x, 0) = h(x) \)

Each sub-problem is addressed separately by assuming specific forms of the solutions that satisfy all other boundary conditions. Combining the solutions of these sub-problems helps solve the original BVP.

Separation of Variables

Separation of Variables is a mathematical technique used to simplify partial differential equations like Laplace’s equation. This technique assumes the solution can be written as a product of functions, each in a single independent variable.

In the exercise, this approach helps by splitting \( u(x, y) = X(x)Y(y) \). This separation transforms the partial differential equation into ordinary differential equations (ODEs) that are easier to solve. Each ODE corresponds to one of the functions, \( X \) and \( Y \).

• Solving these ODEs creates a set of solutions to form a general solution for the problem.
• This solution satisfies the homogeneous part of the boundary conditions, providing a framework to find constants by applying specific non-homogeneous conditions.

Fourier Series

Fourier Series allows the representation of a function as an infinite sum of sines and cosines, useful in solving periodic boundary conditions. In our exercise, Fourier series decompose the solutions based on the sinusoidal behavior dictated by the boundary conditions.

Each sub-problem results in expressions involving \( \sin \) and \( \cosh \), derived from solving the ODEs in the earlier separation step. The series coefficients, \( A_n \) and \( B_m \), are computed through integration based on initial functions \( f(y) \) and \( h(x) \).

• These coefficients adjust the amplitudes of the component waves in the series.
• By summing these series, we approximate the solution to the desired accuracy, meeting the boundary requirements.

Fourier Series thus provides the framework for approaching these boundary value problems when combined with the method of separation of variables.