Green's function is a function of many variables associated with integral representation of solution of a boundary problem for a differential equation.

In the general case of a linear boundary problem with homogeneous boundary conditions

where Γ_{i} φ(x) are linear homogeneous functions of φ(x) and its derivatives on the boundary S of domain D. An inverse transformation (if it exists) of the form

uses Green's function G(x, ξ) as a kernel for the given problem, Eq. (1).

Equation (2) describes the solution as a superposition of elementary solutions which can be interpreted as point sources or power pulses f(ξ) δ(x, ξ) at the point x = ξ (where δ(x, ξ) is the Dirac delta function).

The function G(x, ξ) of the argument x must satisfy the homogeneous boundary condition (1b), and also the equation

and the condition

or, as generalized function, the equation

If the operator L is self-conjugate, Green's function G(x, ξ) is symmetric, i.e., G(x, ξ) = G(ξ, x). For a boundary problem for a linear ordinary differential equation

the general solution on the section [a, b] can be presented in the form

where {φ_{k}} is the functional system of solutions of a homogeneous equation L(φ) = 0, C_{k} are arbitrary constants obtaind from boundary conditions.

It often appears possible to determine Green's function so that a particular solution

satisfies the given boundary conditions. Such Green's function must have a jump of (n – 1)th derivative for x = ξ

Further Green's function for linear differential equations with partial derivatives concerns

Elliptic equations. The solution of *Dirichlet's problem* for the *Poisson equation*

can be written with the help of Green's function G(x, ξ) as

where n is the outer normal to the surface S. Green function for the given problem is represented in the form

where N is the problem dimensionality, r is the distance between the points x and ξ, g(x, ξ) is a harmonic function of (x, ξ) D, chosen so that Green's function satisfies boundary condition (7b).

Parabolic equations. The solution of a boundary problem for the equation of thermal conductivity with homogeneous boundary conditions

and the initial condition

where Γ_{i} are the linear boundary operators with coefficients which depend on t and x, can be written with the help of Green's function G(t, x, τ, ξ) as

Green's function for the given problem as a function t, x satisfies the equation (3a) for (t, x) ≠ (τ, ξ) and for t > τ ≥ 0, x D condition (9b).

For instance, the solution of Eq. (9a) on the entire infinite space can be expressed in the form (10) with the help of Green's function

(t > τ; n = 1, 2, 3), where r is the distance between the points x and ξ.

Hyperbolic equations. In a number of cases the solution of a two-dimensional Cauchy problem with boundary conditions specified on a boundary curve can be obtained employing an integral relation based on the Green-Riemann function which has a more complex character than in the case of elliptic and parabolic equations.