- Green's Function Solution to Poisson/Helmholtz equations

In summary, the conversation discusses the use of Green's function to solve Poisson and Helmholtz equations, specifically with Dirichlet or Neumann conditions. The solution is expressed in integral form and there is a condition on the Green's function for the surface in a Poisson equation with Neumann boundary conditions. The conversation also mentions the need for a specific integral solution for the Helmholtz equation, assuming it acts in 3D and is spherically symmetric. However, a comprehensive tutorial on this topic is not possible and specific questions may be answered.
  • #1
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URGENT - Green's Function Solution to Poisson/Helmholtz equations

hey, i have an exam pretty soon and couldn't find any answers/hints on how to do this:

1.How do you express the solution f(x') of the Helmholtz equation in terms of the green function g(x,x') in integral form, with dirichlet or neumann conditions?

2.What is the condition on the Green's function on the surface S for a poisson equation with neumann bc on S? and how do you get the integral solution up to an undetermined constant?

would be great if someone can help, ml
 
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  • #2
There are whole books written on these topics! We might be able to answer specific questions but I don't think anyone can give you a whole tutorial on this.
 
  • #3
hey, i just need a specific integral solution for the helmholtz equation using greens function, assuming that it acts in 3d and is spherically symmetric (only dependent on the distance of the centre of the dirac function)
 

1. What is a Green's function and how is it used to solve Poisson/Helmholtz equations?

A Green's function is a mathematical tool used to solve differential equations, specifically the Poisson/Helmholtz equations in this case. It is a function that represents the response of a system to an impulse or point source. By using the Green's function, the solution to the Poisson/Helmholtz equations can be written as a convolution of the Green's function with the source term. This allows for solving these equations in a more efficient and systematic manner.

2. What are the advantages of using Green's function in solving these equations?

One of the main advantages of using Green's function is that it reduces the problem of solving a complicated differential equation to a simpler convolution operation. This makes it easier to find solutions for a wide range of boundary conditions. Additionally, the use of Green's function allows for the superposition of solutions, making it useful for solving problems with multiple sources or complex geometries.

3. What is the relationship between the Green's function and the boundary conditions of the Poisson/Helmholtz equations?

The Green's function is directly related to the boundary conditions of the Poisson/Helmholtz equations. The boundary conditions determine the behavior of the Green's function, and in turn, the behavior of the solution. Therefore, choosing appropriate boundary conditions is crucial in obtaining an accurate solution.

4. Can Green's function be used to solve other types of differential equations?

Yes, Green's function can be used to solve other types of differential equations, such as the heat equation and the wave equation. It is a powerful mathematical tool that can be applied to a wide range of problems in physics, engineering, and other fields.

5. Are there any limitations to using Green's function to solve Poisson/Helmholtz equations?

While Green's function is a useful tool in solving these equations, it does have some limitations. It may not be applicable in cases where the source term is not localized or when the boundary conditions are too complex. Additionally, the calculation of the Green's function may be computationally intensive, especially for higher dimensions or more complex geometries.

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