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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

(1a)
(1b)

where Γi 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

(2)

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

(3a)

and the condition

(3b)

or, as generalized function, the equation

(4)

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

(5)

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

(6)

where {φk} is the functional system of solutions of a homogeneous equation L(φ) = 0, Ck 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

(7a)
(7b)

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

(8)

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

(9a)
(9b)

and the initial condition

where Γi 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

(10)

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

(11)

(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.

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