Laplace Differential Equation of a Half-Annulus

In summary, the given DE and boundary conditions suggest a half annulus shape and the use of the 2D Laplace equation for polar coordinates to find the solution u(r,θ). Separating variables into R(r) and Θ(θ), with the constant of separation C potentially being any value, the boundary conditions require Θ(0) = Θ(π) = 0 and the boundary condition at r = 1 suggests Θ = sin θ as a good choice. The type of Laplace equation being used is not specified.
  • #1
mrkevelev
5
0
Here is the DE:
Δu(r,θ)=0, 1 ≤ r ≤ 2, 0 ≤ θ ≤ pi
and here are the Boundary Conditions:
u(1,θ)=sin(θ), u(2,θ)=0, u(r,0)=0, u(r,pi)=0

Based on the Boundary Conditions I believe this is half of an annulus.
Using the 2D Laplace equation for polar coordinates, find the solution u(r,θ).

I've begun to separate variables, R(r) and Theta(θ), but I'm confused with whether I should use as +,-, or 0. The steps after that I am stuck on as well. Also, what kind of Laplace equation is this (Dirichlet, Neumann, etc.)?
 
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  • #2
mrkevelev said:
Here is the DE:
Δu(r,θ)=0, 1 ≤ r ≤ 2, 0 ≤ θ ≤ pi
and here are the Boundary Conditions:
u(1,θ)=sin(θ), u(2,θ)=0, u(r,0)=0, u(r,pi)=0

Based on the Boundary Conditions I believe this is half of an annulus.
Using the 2D Laplace equation for polar coordinates, find the solution u(r,θ).

I've begun to separate variables, R(r) and Theta(θ), but I'm confused with whether I should use as +,-, or 0. The steps after that I am stuck on as well. Also, what kind of Laplace equation is this (Dirichlet, Neumann, etc.)?

You have [itex]\Theta'' = C\Theta[/itex] where in principle the constant of separation [itex]C[/itex] can be anything, because the domain restricts [itex]\theta[/itex] to only half a circle. Thus [itex]u[/itex] need not be periodic and you are not for that reason compelled to take [itex]C = -n^2[/itex] for [itex]n \in \mathbb{Z}^{+}[/itex].

However you do need to satisfy the boundary conditions [itex]\Theta(0) = \Theta(\pi) = 0[/itex], and the boundary condition at [itex]r = 1[/itex] certainly suggests that [itex]\Theta = \sin \theta[/itex] would be a good choice.
 

1. What is the Laplace Differential Equation of a Half-Annulus?

The Laplace Differential Equation of a Half-Annulus is a partial differential equation that describes the behavior of a function in a half-annular region. It is a second-order differential equation that is widely used in mathematical physics and engineering.

2. What is the mathematical form of the Laplace Differential Equation of a Half-Annulus?

The mathematical form of the Laplace Differential Equation of a Half-Annulus is:

2u/∂x2 + ∂2u/∂y2 = 0

where u is the unknown function and x and y are the independent variables.

3. What are the boundary conditions for the Laplace Differential Equation of a Half-Annulus?

The boundary conditions for the Laplace Differential Equation of a Half-Annulus depend on the specific problem being solved. However, typically the boundary conditions include specifying the value of the unknown function u at the boundaries of the half-annulus, as well as any necessary continuity or symmetry conditions.

4. What are some applications of the Laplace Differential Equation of a Half-Annulus?

The Laplace Differential Equation of a Half-Annulus has many practical applications. It is commonly used to model heat transfer in a half-annular region, as well as fluid flow and electrostatics problems. It is also used in the study of potential theory and harmonic functions.

5. What are some techniques for solving the Laplace Differential Equation of a Half-Annulus?

There are several techniques for solving the Laplace Differential Equation of a Half-Annulus, including separation of variables, the method of images, and the use of Green's functions. Numerical methods, such as finite difference or finite element methods, can also be used to approximate solutions to this equation.

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