Poissonegyenlet

Poissonegyenlet, also known as Poisson's equation, is a fundamental partial differential equation in mathematical physics and engineering. It is named after the French mathematician Siméon Denis Poisson, who introduced it in 1813. The equation is widely used to describe a variety of physical phenomena, including electrostatics, gravitation, fluid flow, and heat transfer.

In its most general form, Poisson's equation is written as:

∇²φ = f

where ∇² represents the Laplacian operator, φ is a scalar field, and f is a given function of space and time. The Laplacian operator is defined as the divergence of the gradient of φ, and it measures the second spatial derivatives of φ.

In electrostatics, Poisson's equation describes the relationship between the electric potential φ and the charge density ρ. The equation is given by:

∇²φ = -ρ/ε₀

where ε₀ is the permittivity of free space.

In gravitation, Poisson's equation relates the gravitational potential φ to the mass density ρ. The equation is given by:

∇²φ = 4πGρ

where G is the gravitational constant.

Poisson's equation is a generalization of Laplace's equation, which is obtained by setting f = 0. Laplace's equation is used to describe phenomena where the source term is absent, such as the steady-state temperature distribution in a homogeneous medium.

Poisson's equation is a linear partial differential equation, and its solutions can be found using various analytical and numerical methods. The equation is widely used in scientific and engineering applications, and it plays a crucial role in the development of mathematical models for physical systems.