Question

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

Short answer

Answer: The solution u(x, y) is given by: $$ u(x, y) = c_0 + \sum_{n=1}^{\infty} c_n \cosh (n \pi x / b) \cos (n \pi y / b) $$ The arbitrary constant c_0 allows the solution to be determined only up to an additive constant, which is a property of all Neumann problems. The specific value of c_0 does not affect the general behavior of the solution, but it influences the exact displacement of the solution.

Step-by-step solution

Simplify Laplace's Equation and Boundary Conditions

For a rectangle with dimensions \(a\) and \(b\), Laplace's equation is given by: $$ \dfrac{\partial^2 u}{\partial x^2} + \dfrac{\partial^2 u}{\partial y^2} = 0 $$ with the boundary conditions: \begin{gather*} \dfrac{\partial u}{\partial x}(0,y) = 0 \quad \dfrac{\partial u}{\partial x}(a,y)= f(y), \quad 0<y<b \\ \dfrac{\partial u}{\partial y}(x,0) = 0 \quad \dfrac{\partial u}{\partial y}(x,b)= 0, \quad 0 \leq x \leq a \end{gather*}

Find the Fundamental Set of Solutions u_n(x, y)

To find the solutions, we can try a separation of variables technique. Let \(u(x, y) = X(x) Y(y)\). We then have: $$ \dfrac{X''(x)}{X(x)} + \dfrac{Y''(y)}{Y(y)} = 0 $$ We can rewrite this as two separate equations: $$ X''(x) = -\lambda^2 X(x) \quad \text{and} \quad Y''(y) = \lambda^2 Y(y) $$ Now we proceed to find the fundamental set of solutions u_n(x, y): $$ u_0(x, y) = c_0 \\ u_n(x, y) = c_n \cosh (n \pi x / b) \cos (n \pi y / b), \quad n=1,2,3, \ldots $$

Superpose the Fundamental Solutions

By superposing the solutions found in step 2, we formally determine a function \(u\) satisfying the nonhomogeneous boundary condition: $$ u(x, y) = c_0 + \sum_{n=1}^{\infty} c_n \cosh (n \pi x / b) \cos (n \pi y / b) $$ Now, let's calculate the derivative of \(u(x, y)\) with respect to \(x\) at \((a, y)\): $$ u_x(a,y) = \sum_{n=1}^{\infty} \dfrac{n \pi}{b} \sinh (n \pi a / b) \cos (n \pi y / b) $$ We are given that \(u_x(a,y) = f(y)\). To satisfy this condition, we must expand \(f(y)\) as a Fourier cosine series with period \(2b\) which doesn't have a constant term. Also, the integral of \(f(y)\) must be zero for the problem to be solvable: $$ \int_{0}^{b} f(y) dy = 0 $$ The constant term \(c_0\) remains arbitrary, making the solution determined only up to this additive constant. This is a property of all Neumann problems. Overall, our solution for the problem is given by: $$ u(x, y) = c_0 + \sum_{n=1}^{\infty} c_n \cosh (n \pi x / b) \cos (n \pi y / b) $$

Key concepts

Neumann Boundary Conditions

In mathematics, Neumann boundary conditions specify the derivative of a function on a boundary. Essentially, they define how the slope of a solution behaves at the boundaries of the domain. Neumann conditions can be expressed as:

• At the vertical boundaries: \( \frac{\partial u}{\partial x}(0, y) = 0 \) and \( \frac{\partial u}{\partial x}(a, y) = f(y) \).
• At the horizontal boundaries: \( \frac{\partial u}{\partial y}(x, 0) = 0 \) and \( \frac{\partial u}{\partial y}(x, b) = 0 \).

In a Neumann problem, the function's behavior on the boundary is controlled by these derivatives rather than its direct values. A common feature is that solutions are often unique only up to an additive constant. This is because the derivative doesn't fix the starting value of the function inside the area.

Fourier Series

A Fourier series is a way to represent a function as an infinite sum of sines and cosines. This is particularly useful when dealing with periodic functions. For our exercise, after finding the solution form, we relate \( f(y) \) to a Fourier cosine series:

• The series representation requires that \( \int_{0}^{b} f(y) \, dy = 0 \) so no constant term is present.
• \( f(y) \) should be expressible as a series like \( f(y) = \sum_{n=1}^{\infty} a_n \cos(n \pi y / b) \).

This expansion is integral because it allows us to match the boundary conditions with the form of the solution. The absence of the constant term implies symmetry and periodicity, essential in certain physical models.

Separation of Variables

Separation of Variables is a powerful method often used to solve partial differential equations like Laplace's equation. It involves assuming the solution can be written as a product of functions, each depending on a single coordinate. For this problem:

• Start by assuming \( u(x, y) = X(x)Y(y) \), separating dependent variables \( x \) and \( y \).
• This assumption reformulates Laplace’s equation into two ordinary differential equations: \( X''(x) = -\lambda^2 X(x) \) and \( Y''(y) = \lambda^2 Y(y) \).

The beauty of this method is that it simplifies a complex multidimensional problem into more manageable one-dimensional problems. After solving for \( X(x) \) and \( Y(y) \), they can be recombined into a complete solution, such as the form \( u_n(x, y) = c_n \cosh(n \pi x / b) \cos(n \pi y / b) \), allowing for further analysis and solution synthesis.