In mathematical heat conduction, the Green's function number is used to uniquely categorize certain fundamental solutions of the heat equation to make existing solutions easier to identify, store, and retrieve.
Contents
Background
Numbers have long been used to identify types of boundary conditions. The Green's function number system was proposed by Beck and Litkouhi in 1988 and has seen increasing use since then. The number system has been used to catalog a large collection of Green’s functions and related solutions.
Although described here for solutions of the heat equation, this number system could also be used for any phenomena described by differential equations such as diffusion, acoustics, electromagnetics, fluid dynamics, etc.
Notation
The Green's function number specifies the coordinate system and the type of boundary conditions that a Green's function satisfies. The Green's function number has two parts, a letter designation followed by a number designation. The letter(s) designate the coordinate system while the numbers designate the type of boundary conditions that are satisfied.
Some of the designations for the Greens function number system are given next. Coordinate system designations include: X, Y, and Z for Cartesian coordinates; R, Z,
X11
As an example, number X11 denotes the Green's function that satisfies the heat equation in the domain (0 < x < L) for boundary conditions of type 1 (Dirichlet) at both boundaries x = 0 and x = L. Here X denotes the Cartesian coordinate and 11 denotes the type 1 boundary condition at both sides of the body. The boundary value problem for the X11 Green’s function is given by
Here
X20
As another Cartesian example, number X20 denotes the Green's function in the semi-infinite body (
X10Y20
As a two-dimensional example, number X10Y20 denotes the Green's function in the quarter-infinite body (
R03
As an example in the cylindrical coordinate system, number R03 denotes the Green's function that satisfies the heat equation in the solid cylinder (0 < r < a) with a boundary condition of type 3 (Robin) at r = a. Here letter R denotes the cylindrical coordinate system, number 0 denotes the zeroth boundary condition (boundedness) at the center of the cylinder (r = 0), and number 3 denotes the type 3 (Robin) boundary condition at r = a. The boundary value problem for R03 Green's function is given by
Here
R10
As another example, number R10 denotes the Green’s function in a large body containing a cylindrical void (a < r <
R01 ϕ {displaystyle phi } 00
As a two dimensional example, number R01
RS02
As an example in the spherical coordinate system, number RS02 denotes the Green’s function for a solid sphere (0 < r < b ) with a type 2 (Neumann) boundary condition at r = b. Here letters RS denote the radial-spherical coordinate system, number 0 denotes the zeroth boundary condition (boundedness) at r=0, and number 2 denotes the type 2 boundary at r = b. The boundary value problem for the RS02 Green’s function is given by