# Exact solution to Poisson equation in 2D

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• hikari1987
In summary, the Poisson equation in 2D is a partial differential equation that relates the second derivatives of a function to its source term. It is important to find an exact solution to the Poisson equation in 2D because it has many applications in physics and engineering and serves as a benchmark for numerical methods. The exact solution can be found using various mathematical techniques, but the boundary conditions depend on the specific problem being solved. However, there are limitations to using the exact solution, such as unknown source terms and complexity of geometries and boundary conditions, making numerical methods necessary.
hikari1987
Hi all ,

Could you please help me solve Poisson equation in 2D for heat transfer with Dirichlet and Neumann conditions analytically?
Thank you

1. In order to do that you must specify the domain on which you want to do it. You cannot just say "Poisson equation" and have a unique solution.

2. This would be a thread for the homework forums, where you should post using the homework template and include your own attempts at finding a solution.

## 1. What is the Poisson equation in 2D?

The Poisson equation in 2D is a partial differential equation that relates the second derivatives of a function to its source term. In mathematical notation, it is written as ∇²u = f, where u is the unknown function and f is the source term.

## 2. Why is finding an exact solution to the Poisson equation in 2D important?

Exact solutions to the Poisson equation in 2D have many applications in physics and engineering. They can be used to model and understand various physical phenomena, such as electrostatics, heat transfer, and fluid dynamics. Additionally, they serve as a benchmark for numerical methods used to solve the equation.

## 3. How is the exact solution to the Poisson equation in 2D found?

The exact solution to the Poisson equation in 2D can be found using various mathematical techniques, such as separation of variables, Fourier transforms, and Green's functions. These methods involve solving a system of equations to determine the coefficients in the solution.

## 4. What are the boundary conditions for the exact solution to the Poisson equation in 2D?

The boundary conditions for the exact solution to the Poisson equation in 2D depend on the specific problem being solved. In general, they specify the behavior of the solution at the boundaries of the domain and are necessary to obtain a unique solution. Common boundary conditions include Dirichlet, Neumann, and mixed boundary conditions.

## 5. Are there any limitations to using the exact solution to the Poisson equation in 2D?

Yes, there are limitations to using the exact solution to the Poisson equation in 2D. In many practical applications, the source term f is not known exactly and can only be estimated. Additionally, the exact solution may not be feasible to compute for complex geometries and boundary conditions, making it necessary to use numerical methods instead.

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