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.
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 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, for cylindrical coordinates; and, RS,, for spherical coordinates. Designations for several boundary conditions are given in Table 1. The zeroth boundary condition is important for identifying the presence of a coordinate boundary where no physical boundary exists, for example, far away in a semi-infinite body or at the center of a cylindrical or spherical body.
Examples in Cartesian coordinates
X11
As an example, number X11 denotes the Green's function that satisfies the heat equation in the domain for boundary conditions of type 1 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 is the thermal diffusivity and is the Dirac delta function.
X20
As another Cartesian example, number X20 denotes the Green's function in the semi-infinite body with a Neumann boundary at x = 0. Here X denotes the Cartesian coordinate, 2 denotes the type 2 boundary condition at x = 0 and 0 denotes the zeroth type boundary condition at. The boundary value problem for the X20 Green's function is given by
X10Y20
As a two-dimensional example, number X10Y20 denotes the Green's function in the quarter-infinite body with a Dirichlet boundary at x = 0 and a Neumann boundary at y = 0. The boundary value problem for the X10Y20 Green's function is given by
Examples in cylindrical coordinates
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 with a boundary condition of type 3 at r = a. Here letter R denotes the cylindrical coordinate system, number 0 denotes the zeroth boundary condition at the center of the cylinder, and number 3 denotes the type 3 boundary condition at r = a. The boundary value problem for R03 Green's function is given by Here is thermal conductivity and is the heat transfer coefficient.
R10
As another example, number R10 denotes the Green's function in a large body containing a cylindrical void with a type 1 boundary condition at r = a. Again letter R denotes the cylindrical coordinate system, number 1 denotes the type 1 boundary at r = a, and number 0 denotes the type zero boundary at large values of r. The boundary value problem for the R10 Green's function is given by
R0100
As a two dimensional example, number R0100 denotes the Green's function in a solid cylinder with angular dependence, with a type 1 boundary condition at r = a. Here letter denotes the angular coordinate, and numbers 00 denote the type zero boundaries for angle; here no physical boundary takes the form of the periodic boundary condition. The boundary value problem for the R0100 Green's function is given by
Example in spherical coordinates
RS02
As an example in the spherical coordinate system, number RS02 denotes the Green's function for a solid sphere with a type 2 boundary condition at r = b. Here letters RS denote the radial-spherical coordinate system, number 0 denotes the zeroth boundary condition 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
