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
![](/content/5249/eqn181.gif)
![](/content/5249/eqn182.gif)
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
![](/content/5249/eqn183.gif)
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
![](/content/5249/eqn184.gif)
and the condition
![](/content/5249/eqn185.gif)
or, as generalized function, the equation
![](/content/5249/eqn186.gif)
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
![](/content/5249/eqn187.gif)
the general solution on the section [a, b] can be presented in the form
![](/content/5249/eqn188.gif)
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
![](/content/5249/eqn189.gif)
satisfies the given boundary conditions. Such Green's function must have a jump of (n – 1)th derivative for x = ξ
![](/content/5249/eqn190.gif)
Further Green's function for linear differential equations with partial derivatives concerns
Elliptic equations. The solution of Dirichlet's problem for the Poisson equation
![](/content/5249/eqn191.gif)
![](/content/5249/eqn192.gif)
can be written with the help of Green's function G(x, ξ) as
![](/content/5249/eqn193.gif)
where n is the outer normal to the surface S. Green function for the given problem is represented in the form
![](/content/5249/eqn194.gif)
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
![](/content/5249/eqn195.gif)
![](/content/5249/eqn196.gif)
and the initial condition
![](/content/5249/eqn197.gif)
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
![](/content/5249/eqn198.gif)
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
![](/content/5249/eqn199.gif)
(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.