Editing Heat equation (section)

=== Diffusivity constant ===

The diffusivity constant {{math|''α''}} is often not present in mathematical studies of the heat equation, while its value can be very important in engineering. This is not a major difference, for the following reason. Let {{mvar|u}} be a function with
:<math>\frac{\partial u}{\partial t}=\alpha\Delta u.</math>
Define a new function <math>v(t,x)=u(t/\alpha,x) </math>. Then, according to the [[chain rule]], one has
{{NumBlk|:|<math>
 \frac{\partial}{\partial t} v(t,x)
= \frac{\partial}{\partial t} u(t/\alpha,x)
= \alpha^{-1}\frac{\partial u}{\partial t}(t/\alpha,x)
= \Delta u(t/\alpha,x)
= \Delta v(t,x)
</math>|{{EquationRef|⁎}}}}
Thus, there is a straightforward way of translating between solutions of the heat equation with a general value of {{math|α}} and solutions of the heat equation with {{math|1=''α'' = 1}}. As such, for the sake of mathematical analysis, it is often sufficient to only consider the case {{math|1=''α'' = 1}}. 

Since <math>\alpha>0</math> there is another option to define a <math>v</math> satisfying <math display="inline">\frac{\partial}{\partial t} v = \Delta v </math> as in ({{EquationNote|⁎}}) above by setting <math>v(t,x) = u(t, \alpha^{1/2} x) </math>. Note that the two possible means of defining the new function <math>v</math> discussed here amount, in physical terms, to changing the unit of measure of time or the unit of measure of length.