Glossary.

- Auxiliary problem. The boundary value problem that defines a Green's function, which includes a differential equation with an impulsive
heating term and homogeneous boundary conditions.
*See*``What is a Green's Function?'' - Boundary condition. A specified value (or relation) at a surface of a body.
*See*Dirichlet, Neumann, and convection conditions. - Cartesian coordinates.
*See*rectangular coordinates. - Causality relation. The requirement that the transient GF is zero for any time before the heat pulse is released: . In control theory, a function is said to be ``causal'' if the response only appears after the effect (heat pulse) that causes it.
- Conductivity.
*See*thermal conductivity. - Convection at boundary, described by

Here*h*is the heat transfer coefficient and is the outward normal at the body surface. The homogeneous convection boundary condition is

- Delta function.
*See*Dirac delta function. - Dirac delta function. Symbol
,
is zero everywhere except at
in such a way that the integral over the volume is
unity. Also called the unit impulse function.
*See*Properties of Dirac delta function. - Dirichlet boundary condition. Specified temperature at a boundary.
The homogeneous Dirichlet boundary condition is
*T*(**r**_{i},*t*)=0. - energy generation. Symbol
*g*, units [Watts/meter^{3}or Joule/meter ], energy deposited in a body per unit volume per unit time, for example from electric heating, microwave absorption, nuclear reaction, etc. - erf
*See*error function. - erfc
*See*error function (complementary). - Error function. Symbol erf(
*z*), is defined

- Error function (complementary). Related to the error
function by

- Fin. A solid body exposed to a fluid for the purpose of exchanging heat with the fluid.
- Green's function. A fundamental solution of a linear differential equation
satisfying homogeneous boundary conditions. (other names include
influence function, impulse response, source solution).
*See*What is Green's Function? - Green's function solution equation. Formal solution to a boundary value problem in the form of one or more integrals, each of which contains a Green's function and a non-homogeneous term (``driving term''). The non-homogeneous terms may be boundary conditions, initial conditions, or volume energy generation.
- Heat equation. Also called the transient heat conduction equation. Describes the
movement of heat by diffusion (molecule-to-molecule transport) in a solid
(or motionless) medium. In vector notation,

Here*T*is temperature,*k*is conductivity,*g*is energy generation, is thermal diffusivity,*r*is spatial coordinate, and*t*is time. - Heat flux. Energy per unit time per unit area. Units Watts/meters
^{2}. - Heat transfer coefficient. Symbol
*h*, units [W/m^{2}/*K*]. Relates the surface temperature and surface heat flux with a surrounding fluid according to Newton's law of cooling:

*q*_{surface}=*h*(*T*_{surface}-*T*_{fluid}) .

- Helmholtz equation. Given by

When*m*^{2}is real this equation describes steady heat with ``side'' heat losses (fin losses). When*m*^{2}is imaginary the Helmholtz equation is the heat equation in Fourier-transform space (also called the thermal-wave equation). Finally, replace -*m*^{2}by real +*m*^{2}to give the wave equation in Fourier-transform space. - Homogeneous equation. An equation in which every non-zero term contains the independent variable. For example, in a homogeneous heat equation every term contains T (there is no energy generation term).
- Homogeneous boundary condition. A boundary condition defined by an equation
in which the temperature appears
in every non-zero term. The following boundary
conditions are homogeneous:

type 1,*T*= 0;

type 2, ;

type 3, . - Homogeneous body. A body composed of the same material all the way through.
- Initial condition. Temperature distribution at
*t*=0. - Insulated boundary. A boundary with no heat flow, defined by on the boundary. See also Neumann boundary condition.
- Laplace equation. Also called the steady heat conduction equation.
- Multiplicative property. For transient heat conduction, many rectangular and cylindrical 2-D and 3-D GF may be constructed by multiplying 1-D GF. See Beck (1992, section 4.5) for restrictions.
- Neumann boundary condition. The specified heat flux boundary condition,
defined by

where is the outward normal on the body surface at*r*_{i}. The homogeneous Neumann condition is the insulated boundary, . - Pseudo Green's function. A GF modified for use in a body with all boundaries of type 2 (specified heat flux); for these problems the usual GF does not exist. The defining auxiliary equation for the pseudo GF has an additional term.
- Rectangular coordinates. Cartesian coordinates (
*x*,*y*,*z*). - Reciprocity. A GF that is symmetric with respect to space and
asymmetric with respect to time is said to exhibit reciprocity.
*see*Properties of GF. - Specific heat. Material property with symbol
*c*and units [Joule/kg/K]. The amount of energy needed to raise a unit mass of a material one degree. - Spherical coordinates. also called spherical polar coordinates, .
- Temperature. Symbol
*T*, units Kelvin. A measure of the intensity of thermal energy present in a body. - Thermal diffusivity. Symbol , units m. A defined property found from specific heat, density and thermal conductivity: .
- Thermal conductivity. Symbol
*k*, units W/m/K. Defines the proportionality relation between heat flux and temperature gradient, defined by Fourier's law:

- Volume energy generation.
*See*energy generation. - Watt. A unit of power equal to one Joule per second.