Poisson's equation is

Δ φ = f

Physically this equation describes how a function diffuses in space. They can be seen everywhere in science and thus also in real life. Some examples include

Poisson's equation for gravity,

Δϕ = 4 π G ρ .

Poisson's equation for electrostatics, which is

Δφ = − ρ ε .

Poisson's equation for pressure field in incompressible fluid flow,

Δp = − f(v,V) .

Sarmad, Poisson Equation is applied in Fluid Dynamics for computing pressure field when velocity field is known (e.g. in a numerical iterative algorithm one computes velocity field from Navier-Stokes equations, then it can be computed pressure by Poisson eqn).

Perhaps the more known application of Poisson eqn is the computation of scalar electric potential, known the charges distribution.

Gianluca

Poisson's equation is

Δ φ = f

Physically this equation describes how a function diffuses in space. They can be seen everywhere in science and thus also in real life. Some examples include

Poisson's equation for gravity,

Δϕ = 4 π G ρ .

Poisson's equation for electrostatics, which is

Δφ = − ρ ε .

Poisson's equation for pressure field in incompressible fluid flow,

Δp = − f(v,V) .

A really good application is the reconstruction of a closed surface from a point and normal sample.

http://hhoppe.com/poissonrecon.pdf

Over the years, much development have been done in improving this technique:

http://lgg.epfl.ch/publications/2014/reconstar/paper.pdf

Wiki gives good, simple examples, that appear to be carefully written:

https://en.wikipedia.org/wiki/Poisson%27s_equation

His simplest example gravitation!

My only comment is: his approach with Green's function is too formal! Integration by parts is how I do this in class, to reduce it to the variational formulation.

It determines the displacement u of a membrane loaded by a distributed force f. Freeze the wave equation of vibrational motion.

Many application available in Electrostatics, Newtonian gravity, hydrodynamics, diffusion etc

Poisson equation has applications in plasma to find the electric potential.

Any physical model that is second order, frame invariant, Galilean invariant and homogeneous must be a combination of div, grad and curl. The only scalar combination is div grad . Thus NOT getting a Laplacian is the exceptional case.

When we use the Poisson equation, we are within the class of PDE called elliptic equations that governs the equilibrium problems in physics. The answer above provided several example, I can add the case of the stream-function when a vorticity field is known (in 3D you get 3 Poissone equations) but also in the elastic field you can get a system of two Poisson equations (a deforming panel under the action of a normal stress)