Poisson equation with three boundary conditions

In summary, the conversation discusses the 2D Poisson equation with three boundary conditions defined on a triangular region. The problem is to find the easiest way to solve this equation, preferably analytically. One suggestion is to use separation of variables.
  • #1
azzaz
4
0
I have the following 2D Poisson equation (which can also be transformed

to Laplace) defined on a triangular region (refer to plot):

\begin{equation}

\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}=C\end{equation}

with the following three boundary conditions:

\begin{equation}

\frac{\partial u}{\partial y}=0\,\,\,\,\,\,\,\mathrm{at}\, y=0\end{equation}

\begin{equation}

u=0\,\,\,\,\,\,\,\mathrm{at}\, y=ax+b\end{equation}

\begin{equation}

c\frac{\partial u}{\partial x}+d\frac{\partial u}{\partial y}=0\,\,\,\,\,\,\,\mathrm{at}\, y=ex+f\end{equation}

where C,a,b,c,d,e,f are constants.

What is the easiest way to solve this problem (preferably analytically)?

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  • #2
Can you use separation of variables?
 
  • #3
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  • #4
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upload_2014-11-20_18-26-23.png
 

1. What is the Poisson equation with three boundary conditions?

The Poisson equation with three boundary conditions is a partial differential equation used in physics and engineering to model the distribution of a scalar field, such as temperature or electric potential, in a region with three known boundary conditions.

2. What are the three boundary conditions in the Poisson equation?

The three boundary conditions in the Poisson equation are typically specified as the value of the scalar field at three distinct points on the boundary of the region of interest. These values are used to solve for the unknown distribution of the scalar field within the region.

3. How is the Poisson equation with three boundary conditions solved?

The Poisson equation with three boundary conditions can be solved using various numerical methods, such as finite difference or finite element methods. These methods involve discretizing the region of interest and solving the resulting system of equations to approximate the distribution of the scalar field.

4. What are some applications of the Poisson equation with three boundary conditions?

The Poisson equation with three boundary conditions has many applications in physics and engineering, including modeling heat transfer, electrostatics, and fluid flow. It is also commonly used in numerical simulations and computer graphics to generate realistic images.

5. What are the limitations of the Poisson equation with three boundary conditions?

One limitation of the Poisson equation with three boundary conditions is that it assumes the scalar field is continuous and differentiable within the region of interest. It also does not take into account any time-dependent or non-linear effects. Additionally, the accuracy of the solution depends on the chosen discretization and numerical method.

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