Introduction - Heat Conduction: Mathematical Models and Analytical Solutions - page 17

Equation (1.21) is called the three-dimensional heat-conduction equation with
heat generation . It is second-order, linear and nonhomogeneous.
For a medium with uniform thermal conductivity and no heat generation,
Eq. (1.21) becomes

a 2

u t = Δ u . (1.22)

It is called the three-dimensional heat-conduction equation . It is second-order, linear
and homogeneous.
Here, the thermal diffusivity a 2 is a state property of the medium and has a di-
mension of L 2 T . The physical significance of thermal diffusivity is associated with

the speed of propagation of heat into the solid during changes of temperature over
time. The propagation rate of heat in the medium is proportional to the thermal
diffusivity.

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Remark 1. For heat conduction in planes, we have the two-dimensional heat-
conduction equations u t = a 2 a 2 . For heat conduction

in rods, we have the one-dimensional heat-conduction equations u t =

Δ u and u t = Δ u + f ( x , y , t )

a 2 u xx and

a 2 u xx +

u t = f ( x , t ) . These are special cases of three-dimensional heat-conduction

equations.

Remark 2. Many other physical problems also lead to heat-conduction equations. In
general, the heat-conduction equation describes diffusion. The dependent variable u
is not necessarily the temperature, and can represent other physical quantities.

1.2.5 Potential Equations

We have used Approach 1 to derive wave equations and Approach 2 to develop heat-
conduction equations. Here we apply Approach 3 to derive potential equations by
considering the temperature in steady-state heat conduction and the electric potential
in an electrostatic field.

Temperature in Steady-State Heat Conduction

When the temperature is not dependent on the time any more, heat conduction be-
comes steady. The heat-conduction equations (1.21) and (1.22) reduce to

1
a 2 f

Δ u ( x , y , z )= − ( x , y , z ) , Δ u ( x , y , z )= 0 .

These are both called potential equations . The former is second-order, linear and
nonhomogeneous and is also called the three-dimensional Poisson equation . The lat-
ter is second-order, linear and homogeneous and is also called the three-dimensional