Question

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

Short answer

Question: Find the solution u(x, y) for the given boundary conditions and Laplace's equation in the rectangular region, where a=3 and b=1, and compare it with Problem 1's solution. Solution: For a=3 and b=1, we can substitute these values and the calculated \(A_n\) in the Fourier series to find the solution \(u(x, y)\) as follows: $$ u(x, y) = \sum_{n=1}^{\infty}A_n\sin{\frac{n\pi x}{3}}\cosh{\frac{n\pi y}{3}} $$ To compare the solution with Problem 1, plot both solutions and observe the differences in their behavior. The comparison will show how the solutions differ in terms of their shape, amplitude, and other characteristics, thus highlighting the impact of the different boundary conditions and the rectangular domain's size.

Step-by-step solution

Convert to a Dirichlet Problem

First, we'll double the rectangle by reflecting the region with respect to the x-axis (i.e., \(0<y<-b\)), and extend the function \(g(x)\) to be an odd function in terms of y in the new region. This transformed problem has boundary conditions as follows: $$ u(x, 0)= u(x, b) = u(x, -b) = 0\\ u(0, y) = u(a, y) = 0 $$ Now, we have a Dirichlet problem in the extended region. The solution in the original region will still satisfy the original boundary conditions.

Write the Solution as a Fourier Series

The solution for the Laplace's equation with these boundary conditions can be written as a sum of Fourier Series: $$ u(x, y) = \sum_{n=1}^{\infty}A_n\sin{\frac{n\pi x}{a}}\cosh{\frac{n\pi y}{a}} $$ where $$ A_n = \frac{2}{a}\int_{0}^{a}g(x) \sin{\frac{n\pi x}{a}} dx $$

Find the Series Coefficient for Given g(x)

For the given \(g(x)\), we can find the Fourier coefficients \(A_n\): $$ g(x)=\left\\{\begin{array}{ll}{x,} & {0 \leq x \leq a / 2} \\\ {a-x,} & {a / 2 \leq x \leq a}\end{array}\right. $$ Therefore, \(A_n\) will be given by: $$ A_n = \frac{2}{a}\left[\int_{0}^{a/2}x\sin{\frac{n\pi x}{a}}dx + \int_{a/2}^{a}(a-x)\sin{\frac{n\pi x}{a}}dx\right] $$ Evaluating these integrals, we can find \(A_n\) for the given \(g(x)\).

Write Down the Full Solution u(x, y)

Now that we have the Fourier coefficients \(A_n\), we can write down the solution \(u(x, y)\) for the given conditions: $$ u(x, y) = \sum_{n=1}^{\infty}A_n\sin{\frac{n\pi x}{a}}\cosh{\frac{n\pi y}{a}} $$

Find the Solution for a=3 and b=1 and Compare with Problem 1

Let \(a=3\) and \(b=1\), the solution \(u(x, y)\) can be found by substituting these values and the calculated \(A_n\) in the above solution. To compare the solution with Problem 1, we can plot both solutions and observe the differences in their behavior.

Key concepts

Dirichlet Problem

The Dirichlet problem is a classic boundary value problem where the solution to a partial differential equation is sought under the condition that the solution is equal to a given function on the boundary of the domain. Specifically in the context of Laplace's equation, the Dirichlet problem involves finding a function \(u(x, y)\) such that \(u\) satisfies Laplace's equation inside a region, and on the boundary, the function \(u\) equals a specified function.

For the problem at hand, the challenge presented is not a straightforward Dirichlet problem because the function \(u\) is specified on only part of the boundary. However, the step-by-step solution converts it into a Dirichlet problem by extending the domain and the conditions to include the reflections, which allows for the entire boundary to have specified values of \(u\), thus fulfilling the Dirichlet condition.

Neumann Problem

In contrast to the Dirichlet problem, the Neumann problem is another type of boundary value problem where the solution is determined not by the values of the function at the boundary, but by the normal derivative of the function at the boundary. This condition effectively specifies the gradient or the rate of change of the function as it approaches the boundary rather than its actual value there.

The initial problem presents a mix of Dirichlet and Neumann conditions, demanding the normal derivative of \(u\) to be specified on part of the boundary (\(u_y(x, 0) = 0\)). It is often more challenging to deal with such mixed conditions directly, which is why the exercise provided involves clever reformulation into a full Dirichlet problem to simplify the solution approach.

Fourier Series

Fourier series are an immensely powerful tool in solving partial differential equations, particularly those with periodic boundary conditions. A Fourier series represents a periodic function as an infinite sum of sines and cosines, each multiplied by a coefficient that represents the strength of that particular harmonic component. In solving Laplace's equation for a rectangular region, Fourier series decomposition allows us to consider complex vibrational modes for each dimension as superpositions of simpler harmonic functions.

In the provided exercise solution, after converting the problem to a Dirichlet problem, the solution within the rectangular domain is represented as a Fourier series. This series directly relates to the physical notion of standing waves in the rectangular membrane, where each term in the series can be seen as one of the infinite possible vibrational modes.

Boundary Value Problem

Boundary value problems (BVPs) pertain to solving differential equations along with a set of constraints called boundary conditions. The problems are termed 'value' because of the conditions set on the solution at the boundaries of the domain. BVPs are pivotal in the context of physical and engineering problems because they model the behavior of systems constrained within finite boundaries.

The exercise demonstrates a boundary value problem with mixed boundary conditions — prescribing function value on some parts and specifying the normal derivative on other parts of the boundary. To target such a complex BVP, we often use a strategic approach, such as extending the problem to a configuration that simplifies the boundary conditions, as shown in the exercise solution, to facilitate finding the solution through established methods like Fourier series expansion.