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: $G[(t-\tau )<0]=0$. 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
  
  $$\left. k\frac{\partial T}{\partial n}\right\vert _{r_{i}}+\left. hT\right\vert _{r_{i}}=f_{i}(r_{i},t) \nonumber$$
  
  
  Here h is the heat transfer coefficient and $\overrightarrow{n}$ is the outward normal at the body surface. The homogeneous convection boundary condition is
  
  $$\left. k\frac{\partial T}{\partial n}\right\vert _{r_{i}}+\left. hT\right \vert _{r_{i}}=0.$$
  
  

• Delta function. See Dirac delta function.
• Dirac delta function. Symbol $\delta (% \overrightarrow{\mathbf{r}})$, is zero everywhere except at $\overrightarrow{% \mathbf{r}}\mathbf{=0}$ 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/meter³ or Joule/meter $^{3}/\sec$], 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
  
  $$\mbox{erf}(z)=\frac{2}{\sqrt{\pi }}\int_{z}^{\infty }e^{-t^{2}}dt.$$
  
  

• Error function (complementary). Related to the error function by
  $\mbox{erfc}(z)=1-\mbox{erf}(z).$
• 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,
  
  $$\nabla ^{2}T+\frac{1 }{k}g(r,t)=\frac{1}{\alpha} \frac{\partial T}{\partial t}$$
  
  
  Here T is temperature, k is conductivity, g is energy generation, $\alpha$ is thermal diffusivity, r is spatial coordinate, and t is time.
• Heat flux. Energy per unit time per unit area. Units Watts/meters².
• Heat transfer coefficient. Symbol h, units [W/m²/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
  
  $$\nabla ^{2}T-m^2 T = -\frac{1}{k}g(r)$$
  
  
  When m² is real this equation describes steady heat with ``side'' heat losses (fin losses). When m² is imaginary the Helmholtz equation is the heat equation in Fourier-transform space (also called the thermal-wave equation). Finally, replace -m² by real +m² 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, $\partial T/ \partial n_i = 0$;
  type 3, $k[\partial T/ \partial n_i ] + h T = 0$.
• 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 $\partial T/ \partial n_i = 0$ 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
  
  $$\left. k\frac{\partial T}{\partial n } \right\vert _{r_{i}}=f_{i}(t)$$
  
  
  where $\overrightarrow{n}$ is the outward normal on the body surface at r_i. The homogeneous Neumann condition is the insulated boundary, $\left. \partial T/\partial n\right\vert _{r_{i}}=0$.
• 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, $% (r,\phi ,\theta )$.
• Temperature. Symbol T, units Kelvin. A measure of the intensity of thermal energy present in a body.
• Thermal diffusivity. Symbol $\alpha$, units m$^{2}/\sec$. A defined property found from specific heat, density and thermal conductivity: $\alpha = k/c/\rho$.
• Thermal conductivity. Symbol k, units W/m/K. Defines the proportionality relation between heat flux and temperature gradient, defined by Fourier's law:
  
  $$q_{i}=-k\frac{\partial T}{\partial n_i}.$$
  
  

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