PoissonGleichungssystem

The Poisson equation is a second-order partial differential equation that arises in various fields of physics and engineering, including electrostatics, fluid dynamics, and heat transfer. It is named after the French mathematician Siméon Denis Poisson. The Poisson equation in its most general form is given by:

∇²φ = f

where ∇² is the Laplacian operator, φ is the unknown function, and f is a given function. The Laplacian operator in Cartesian coordinates is defined as:

∇²φ = ∂²φ/∂x² + ∂²φ/∂y² + ∂²φ/∂z²

In the context of electrostatics, the Poisson equation describes the relationship between the electric potential φ and the charge density ρ, with f = -ρ/ε₀, where ε₀ is the permittivity of free space. In this case, the equation becomes:

∇²φ = -ρ/ε₀

The Poisson equation can be solved using various methods, including analytical solutions, numerical methods, and Green's functions. The solution to the Poisson equation provides valuable insights into the behavior of physical systems and is essential for understanding and predicting the behavior of various phenomena in nature.