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[edit]
Numbers have long been used to identify types of boundary conditions.[1][2][3] The Green's function number system was proposed by Beck and Litkouhi in 1988[4] and has seen increasing use since then.[5][6][7][8] The number system has been used to catalog a large collection of Green's functions and related solutions.[9][10][11][12]
Although the examples given below are for the heat equation, this number system applies to any phenomena described by differential equations such as diffusion, acoustics, electromagnetics, fluid dynamics, etc.
Notation[edit]
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.
Name | Boundary condition | Number |
---|---|---|
No physical boundary | G is bounded | 0 |
Dirichlet | 1 | |
Neumann | 2 | |
Robin | 3 |
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[edit]
X11[edit]
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 is the thermal diffusivity (m2/s) and is the Dirac delta function. This GF is developed elsewhere. [13] [14]
X20[edit]
As another Cartesian example, number X20 denotes the Green's function in the semi-infinite body () with a Neumann (type 2) 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 (boundedness) at . The boundary value problem for the X20 Green's function is given by
This GF is published elsewhere. [15] [16]
X10Y20[edit]
As a two-dimensional example, number X10Y20 denotes the Green's function in the quarter-infinite body (, ) with a Dirichlet (type 1) boundary at x = 0 and a Neumann (type 2) boundary at y = 0. The boundary value problem for the X10Y20 Green's function is given by
Applications of related half-space and quarter-space GF are available. [17]
Examples in cylindrical coordinates[edit]
R03[edit]
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 is thermal conductivity (W/(m K)) and is the heat transfer coefficient (W/(m2 K)). See [18] [19] for this GF.
R10[edit]
As another example, number R10 denotes the Green's function in a large body containing a cylindrical void (a < r < ) with a type 1 (Dirichlet) 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 (boundedness) at large values of r. The boundary value problem for the R10 Green's function is given by
This GF is available elsewhere. [20] [21]
R01đ00[edit]
As a two dimensional example, number R0100 denotes the Green's function in a solid cylinder with angular dependence, with a type 1 (Dirichlet) 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
Both a transient [22] and steady form [23] of this GF are available.
Example in spherical coordinates[edit]
RS02[edit]
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
This GF is available elsewhere. [24]
See also[edit]
- Fundamental solution
- Dirichlet boundary condition
- Neumann boundary condition
- Robin boundary condition
- Heat equation
References[edit]
- ^ Luikov, A. V. (1968). Analytical Heat Diffusion Theory. doi:10.1016/B978-0-12-459756-3.X5001-9. ISBN 0124597564.
- ^ ĂzıĆık, M. Necati (1980). Heat conduction (1st ed.). New York: Wiley. ISBN 047105481X.
- ^ Nowak, A.; BiaĆecki, R.; Kurpisz, K. (February 1987). "Evaluating eigenvalues for boundary value problems of heat conduction in rectangular and cylindrical co-ordinate systems". International Journal for Numerical Methods in Engineering. 24 (2): 419â445. doi:10.1002/nme.1620240210.
- ^ Beck, James V.; Litkouhi, Bahman (March 1988). "Heat conduction numbering system for basic geometries". International Journal of Heat and Mass Transfer. 31 (3): 505â515. doi:10.1016/0017-9310(88)90032-4.
- ^ Al-Nimr, M. A.; Alkam, M. K. (19 September 1997). "A generalized thermal boundary condition". Heat and Mass Transfer. 33 (1â2): 157â161. doi:10.1007/s002310050173. S2CID 119549322.
- ^ de Monte, Filippo (September 2006). "Multi-layer transient heat conduction using transition time scales". International Journal of Thermal Sciences. 45 (9): 882â892. doi:10.1016/j.ijthermalsci.2005.11.006.
- ^ Lefebvre, G. (December 2010). "A general modal-based numerical simulation of transient heat conduction in a one-dimensional homogeneous slab". Energy and Buildings. 42 (12): 2309â2322. doi:10.1016/j.enbuild.2010.07.024.
- ^ Toptan, A.; Porter, N. W.; Hales, J. D. (2020). "Construction of a code verification matrix for heat conduction with finite element code applications". Journal of Verification, Validation and Uncertainty Quantification. 5 (4): 041002. doi:10.1115/1.4049037.
- ^ Cole, Kevin; Beck, James; Haji-Sheikh, A.; Litkouhi, Bahman (16 July 2010). Heat Conduction Using Greens Functions. doi:10.1201/9781439895214. ISBN 9781439813546.
- ^ Green's Function Library, https://www.engr.unl.edu/~glibrary/home/index.html
- ^ "Green's Function Library". Retrieved November 19, 2020.
- ^ "Exact Analytical Conduction Toolbox". Retrieved March 4, 2021.
- ^ Luikov, A. V. (1968). Analytical Heat Diffusion Theory. Academic Press. p. 388. ISBN 0124597564.
- ^ Cole, K. D.; Beck, J. V.; Haji-Sheikh, A.; Litkouhi, B. (2011). Heat Conduction using Green's Functions (2nd ed.). Boca Rotan, FL: Taylor and Francis. p. 119. doi:10.1201/9781439895214. ISBN 9780429109188.
- ^ Luikov, A. V. (1968). Analytical Heat Diffusion Theory. Academic Press. p. 387. ISBN 0124597564.
- ^ Carslaw, H. S.; Jaeger, J. C. (1959). Conduction of Heat in Solids. Oxford University Press. p. 276. ISBN 9780198533689.
- ^ Beck, J. V.; Wright, N..; Haji-Sheikh, A.; Cole, K. D; Amos. D. (2008). "Conduction in rectangular plates with boundary temperatures specified". International Journal of Heat and Mass Transfer. 52 (19â20): 4676â4690. doi:10.1016/j.ijheatmasstransfer.2008.02.020. S2CID 12677235.
- ^ Carslaw, H. S.; Jaeger, J. C. (1959). Conduction of Heat in Solids. Oxford University Press. p. 369. ISBN 9780198533689.
- ^ Cole, K. D.; Beck, J. V.; Haji_Sheikh, A.; Litkouhi, B. (2011). Heat Conduction Using Green's Functions (2nd ed.). Boca Rotan, FL: Taylor and Francis. p. 543. doi:10.1201/9781439895214. ISBN 9780429109188.
- ^ Carslaw, H. S.; Jaeger, J. C. (1959). Conduction of Heat in Solids. Oxford University Press. p. 378. ISBN 9780198533689.
- ^ Thambynayagam, R. K. M. (2011). The Diffusion Handbook. McGraw Hill. p. 432. ISBN 9780071751841.
- ^ Cole, K. D.; Beck, J. V.; Haji_Sheikh, A.; Litkouhi, B. (2011). Heat Conduction Using Green's Functions (2nd ed.). Boca Rotan, FL: Taylor and Francis. p. 554. doi:10.1201/9781439895214. ISBN 9780429109188.
- ^ Melnikov, Y. A. (1999). Influence Functions and Matrices. New York: Marcel Dekker. p. 223. ISBN 9780824719418.
- ^ Cole, K. D.; Beck, J. V.; Haji_Sheikh, A.; Litkouhi, B. (2011). Heat Conduction Using Green's Functions (2nd ed.). Boca Rotan, FL: Taylor and Francis. p. 309. doi:10.1201/9781439895214. ISBN 9780429109188.