Fundamental solution

A fundamental solution of a differential operator is a distribution whose image under that operator is the Dirac delta, representing its response to a point source.

Dirac delta: The distribution that evaluates test functions at a specified point and models a concentrated unit source. It is the point source on the right-hand side of the defining equation.
Poisson equation: The equation relating the Laplacian of a potential to a prescribed source. Its fundamental solution gives the potential produced by a point source.
Green's representation: An integral identity expressing a solution of a differential equation through boundary data and source terms. Fundamental solutions supply the kernels used to represent solutions by integrals.
Green's function: A kernel that expresses solutions of a linear differential equation subject to specified boundary conditions. Unlike a fundamental solution, it is tailored to a particular domain and boundary condition.
Distribution theory: A framework extending functions to continuous linear functionals on spaces of test functions. Fundamental solutions may be distributions even when no ordinary function represents them.
Newtonian potential: A potential obtained by convolving a source with the fundamental solution of the Laplacian. It applies the Laplacian's point-source response to distributed sources.
Layer potential: A solution of an elliptic equation represented by an integral over a boundary or surface. Layer potentials build boundary-value solutions from fundamental-solution kernels.
Homogeneous solution: A solution of a differential equation with zero forcing term. Adding one to a particular solution preserves the same point-source equation.
Convolution: An operation combining two functions or distributions by integrating products over translated arguments. Convolving a fundamental solution with a source often constructs a solution to the corresponding equation.
Heat kernel: A function that propagates initial data under the heat equation. It is the fundamental solution of the heat operator and describes diffusion from a point.

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