Question

Find the Green's function for the three-dimensional Neumann problem $$ \nabla^{2} \phi=0 \quad \text { for } z>0 \quad \text { and } \quad \frac{\partial \phi}{\partial z}=f(x, y) \quad \text { on } z=0 $$ Determine \(\phi(x, y, z)\) if $$ f(x, y)= \begin{cases}\delta(y) & \text { for }|x|

Short answer

The solution is \ \phi(x, y, z) = \begin{cases} \delta(y) z &\text{ for } |x| < a, \ 0 &\text{ for } |x| \geq a. \end{cases}.

Step-by-step solution

General Solution of Laplace's Equation

The general form for the solution to Laplace's equation \( abla^{2} \phi = 0 \) in three dimensions can be written as \[ \phi(x, y, z) = A(x, y) z + B(x, y) \]. Since \( \abla^{2} \phi = 0 \), \( A \) and \( B \) must independently satisfy \( \abla_{\text{2D}} \) when applied to them.

Applying the Neumann Boundary Condition

We have the boundary condition \( \frac{\text{\partial} \phi}{\text{\partial} z} = f(x, y) \) on \( z = 0 \). Differentiating the general solution \phi(x, y, z) \ with respect to \( z \) gives us \( A(x, y) \). Therefore, we have \( A(x, y) = f(x, y) \. \)

Substituting the Specific Boundary Condition

According to the specific boundary condition, \( f(x, y) \) is \[ f(x,y) = \begin{cases} \delta(y) &\text{ for } |x| < a. \ 0 &\text{ for } |x| \geq a. \end{cases} \]. Therefore, \( A(x, y) \) is exactly this piecewise function.

Constructing the Solution

Substitute \( A(x, y) \) back into the general form. Thus, the specific solution is \( \phi(x, y, z) = f(x, y) z + B(x, y). \) To satisfy the boundary conditions fully, \( B(x, y) \) must vanish. This gives \[ \phi(x, y, z) = f(x, y)z. \]

Piecewise Solution for \( \phi(x, y, z)\)

Thus, using the specific form of \( f(x, y) \), the solution \phi(x, y, z)\ is \ \phi(x, y, z) = \begin{cases} \delta(y) z &\text{ for } |x| < a, \ 0 &\text{ for } |x| \geq a. \end{cases} \.

Key concepts

Green's function

Green's function is a fundamental solution used to solve differential equations, particularly in physics and engineering. Understanding Green's function is crucial for solving problems involving boundary conditions. It essentially allows you to find a particular solution to inhomogeneous differential equations based on the given boundary conditions. For the Neumann problem in three dimensions, Green's function helps in finding the potential \( \phi \) that satisfies the given Laplace's equation and boundary conditions. This method essentially converts a complex boundary problem into an easier-to-handle problem.

Laplace's equation

Laplace's equation, denoted as \( abla^{2} \phi = 0 \), is a second-order partial differential equation. It's widely used in physics and engineering to describe the behavior of electric, gravitational, and fluid potentials. In this problem, you solve Laplace's equation to find the potential function \( \phi(x, y, z) \). In a three-dimensional space, this equation implies that the sum of the second partial derivatives in all three coordinate directions is zero. The general solution involves finding functions \( A(x, y) \) and \( B(x, y) \) that satisfy this condition.

Boundary conditions

Boundary conditions are essential in solving differential equations because they specify the solution's behavior on the boundary of the domain. In the Neumann problem, the boundary condition is given as \( \frac{\partial \phi}{\partial z} = f(x, y) \) on \( z = 0 \). This condition informs us about the rate of change of the potential \( \phi \) with respect to \( z \) at the boundary. To solve the problem, you need to apply this condition to the general solution of Laplace's equation. By doing so, you ensure that the derived solution conforms to the physical or geometrical constraints of the problem.

Three-dimensional solutions

Three-dimensional solutions to differential equations provide a more comprehensive understanding of physical phenomena. In this context, you are dealing with a three-dimensional Neumann problem where the potential function \( \phi(x, y, z) \) depends on all three spatial coordinates. The general solution is a linear combination of functions \( A(x, y) \) and \( B(x, y) \), which need to satisfy Laplace's equation. By including the specific boundary conditions, you can determine the exact form of these functions, leading to a complete solution that describes the system's behavior in three dimensions.