Poisson's equation

Poisson's equation is an elliptic partial differential equation in which a scalar field's Laplacian equals a specified source function. It generalizes Laplace's equation to nonzero sources.

Laplacian: A differential operator that sums a function's second spatial derivatives. It measures the local curvature on the left side of Poisson's equation.
Partial differential equation: An equation involving an unknown function and its partial derivatives with respect to multiple variables. Poisson's equation belongs to this broad family of equations.
Electrostatics: The study of electric charges and fields that are stationary or vary slowly. Electric potential satisfies Poisson's equation when charge density is present.
Convolution: An operation combining two functions by integrating one against shifted values of the other. Translation-invariant Green's functions turn a source into a solution by convolution.
Helmholtz equation: A partial differential equation in which the Laplacian is proportional to the field plus a source term. Its field-dependent term distinguishes wave and resonance problems from Poisson's static source equation.
Green's function: A kernel that gives a linear differential operator's response to a point source. Convolving it with the source constructs solutions under specified boundary conditions.
Elliptic partial differential equation: A partial differential equation whose principal part has elliptic character, typically governing spatial equilibrium. This classification explains Poisson's smoothing behavior and boundary-value structure.
Newtonian gravity: The classical theory describing gravitational attraction between masses. Gravitational potential obeys Poisson's equation with mass density as its source.
Fourier transform: A transform expressing a function in terms of its spatial or temporal frequencies. It converts constant-coefficient Poisson equations into algebraic relations in frequency space.
Wave equation: A hyperbolic partial differential equation describing propagation of disturbances at finite speed. Unlike Poisson's elliptic equation, it evolves fields through time.

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