Editing Heat equation (section)

=== Steady-state equation ===
A solution to the heat equation <math>\partial u/\partial t = \Delta u</math> is said to be a steady-state solution if it does not vary with respect to time:
: <math>0 = \frac{\partial u}{\partial t} = \Delta u.</math>
Flowing {{math|''u''}} via. the heat equation causes it to become closer and closer as time increases to a steady-state solution. For very large time, {{mvar|''u''}} is closely approximated by a steady-state solution. A steady state solution of the heat equation is equivalently a solution of [[Laplace's equation]].

Similarly, a solution to the nonhomogeneous heat equation <math>\partial u/\partial t = \Delta u + f</math> is said to be a steady-state solution if it does not vary with respect to time:
: <math>0 = \frac{\partial u}{\partial t} = \Delta u + f.</math>
This is equivalently a solution of [[Poisson's equation]].

In the steady-state case, a nonzero spatial thermal gradient <math>\nabla u</math> may (or may not) be present, but if it is, it does not change in time. The steady-state equation describes the end result in all thermal problems in which a source is switched on (for example, an engine started in an automobile), and enough time has passed for all permanent temperature gradients to establish themselves in space, after which these spatial gradients no longer change in time (as again, with an automobile in which the engine has been running for long enough). The other (trivial) solution is for all spatial temperature gradients to disappear as well, in which case the temperature become uniform in space, as well. The steady-state equations are simpler and can help to understand better the physics of the materials without focusing on the dynamics of heat transport. It is widely used for simple engineering problems assuming there is equilibrium of the temperature fields and heat transport, with time.