# Poisson equation with a dirac delta source.

• Coffee_
In summary, the differential equation disappears by taking the Fourier transform and transforming back to find something like a function. However, the homogeneous solutions are not square integrable and so are lost when taking the Fourier transform.
Coffee_
Consider:

##\nabla^{2} V(\vec{r})= \delta(\vec{r})##

By taking the Fourier transform, the differential equation dissapears. Then by transforming that expression back I find something like ##V(r) \sim \frac{1}{r}##.

I seem to have lost the homogeneous solutions in this process. Where does this happen?

Taking the Fourier transform you are essentially saying that your domain is infinite and that your function (##V##) is square integrable. The homogeneous solution is not square integrable.

Thanks. Maybe you could get this one as well since you already bothered to answer so I don't have to make another post:

What's the difference in the evolution of ##f## for ##t>0## between saying

1) ##\partial_{x}^{2} f - \frac{1}{c^{2}} \partial_{t}^{2}f = -\delta(x) \delta(t)## where nothing is happening before ##t=0##

2) ##\partial_{x}^{2} f - \frac{1}{c^{2}} \partial_{t}^{2}f = 0 ## and to impose a delta peak as initial condition?

Essentially nothing, but depending a bit on which initial condition you put a delta in (the wave equation is second order in t and so needs two initial conditions).

I was talking about the condition on ##f##, the derivative condition is zero everywhere.

Is it correct to say that the first expression is more general. It can encompass the state of the system at ##t<0##. For example, if an oscillatory motion already exists before this delta peak, this expression can still be in agreement with it.

The second one is only a description for ##t>0## of a special case of the above description (namely, when everything is zero before the delta peak).

Well, if you have initial conditions, you are encoding anything that happened before that time in those conditions. So for solving it for future ##t## it does not really matter. But the domain of the other problem is larger, yes.

Well, it's all correct with your calculation of the Green function of the Laplace operator. The solution is not square integrable, because it's a distribution and not a function since the source is already a distribution. The Green's function in fact is
$$G(\vec{x})=-\frac{1}{4\pi |\vec{x}|} \; \Leftrightarrow \; \Delta G(\vec{x})=\delta^{(3)}(\vec{x}).$$

## 1. What is the Poisson equation with a Dirac delta source?

The Poisson equation with a Dirac delta source is a partial differential equation that describes the distribution of a scalar field in a region where there is a localized source of a given strength.

## 2. How is the Poisson equation with a Dirac delta source used in physics?

The Poisson equation with a Dirac delta source is used to model a point charge in electrostatics, a point mass in gravitation, or a point vortex in fluid mechanics. It can also be used to model the distribution of a point-like impurity in a material.

## 3. What is the mathematical form of the Poisson equation with a Dirac delta source?

The mathematical form of the Poisson equation with a Dirac delta source is ∇²U = -4πδ(x-x₀), where U is the scalar field, x is the position vector, x₀ is the position of the source, and δ(x-x₀) is the Dirac delta function.

## 4. How is the Dirac delta function related to the Poisson equation with a Dirac delta source?

The Dirac delta function is a mathematical tool used to represent a point source or a point-like distribution. In the Poisson equation with a Dirac delta source, the Dirac delta function represents the strength or magnitude of the point source at a given position.

## 5. What are the boundary conditions for the Poisson equation with a Dirac delta source?

The boundary conditions for the Poisson equation with a Dirac delta source depend on the specific problem being solved. In general, the boundary conditions should ensure that the scalar field U is continuous and has a well-defined value at the boundaries of the region of interest.

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