Screened Poisson equation
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Statement of the equation
The equation is
where is the Laplace operator, λ is a constant, f is an arbitrary function of position (known as the "source function") and u is the function to be determined.
In the homogeneous case (f=0), the screened Poisson equation is the same as the time-independent Klein–Gordon equation. In the inhomogeneous case, the screened Poisson equation is very similar to the inhomogeneous Helmholtz equation, the only difference being the sign within the brackets.
Without loss of generality, we will take λ to be non-negative. When λ is zero, the equation reduces to Poisson's equation. Therefore, when λ is very small, the solution approaches that of the unscreened Poisson equation, which, in dimension , is a superposition of 1/r functions weighted by the source function f:
On the other hand, when λ is extremely large, u approaches the value f/λ², which goes to zero as λ goes to infinity. As we shall see, the solution for intermediate values of λ behaves as a superposition of screened (or damped) 1/r functions, with λ behaving as the strength of the screening.
The screened Poisson equation can be solved for general f using the method of Green's functions. The Green's function G is defined by
Assuming u and its derivatives vanish at large r, we may perform a continuous Fourier transform in spatial coordinates:
where the integral is taken over all space. It is then straightforward to show that
The Green's function in r is therefore given by the inverse Fourier transform,
This may be evaluated using contour integration. The result is:
The solution to the full problem is then given by
As stated above, this is a superposition of screened 1/r functions, weighted by the source function f and with λ acting as the strength of the screening. The screened 1/r function is often encountered in physics as a screened Coulomb potential, also called a "Yukawa potential".
In two dimensions: In the case of a magnetized plasma, the screened Poisson equation is quasi-2D:
The 2D screened Poisson equation yields: