I am currently reading a paper where the following PDE's are considered:
$a_t = a_{xx} -ab$ $,$ Eq. (1)
$b_t = b_{xx} -ab$ $,$ Eq. (2)
where the subscripts $x$ and $t$ denote partial derivatives with respect to space and time. The author transformed the above equations from $x$ and $t$ to $\eta$ and $t$ where
$\eta = \frac{x}{2\sqrt{t}} $.
He then obtained
$a_t -\frac{\eta a_{\eta}}{2t} = \frac{a_{\eta\eta}}{4t} -ab$ $,$ Eq. (5) and
$b_t -\frac{\eta b_{\eta}}{2t} = \frac{b_{\eta\eta}}{4t} -ab$ $,$ Eq. (6).
I have some difficulties to obtain Eq. (5) from Eq. (1) (or equivalently, Eq. (6) from Eq. (2)). My problem is with the term $\frac{\eta a_{\eta}}{2t}$. I do not see where this comes from. I believed at first that it came from changing the variable in $a_{t}$ but apparently no since $a_t$ still appears in Eq. (5). Can someone provide the details to obtain Eq. (5) from Eq. (1) ?
We have - $$ \eta = \frac{x}{2\sqrt{t}} $$ or $$x = 2\eta\sqrt{t}$$ So, \begin{align} a_x &= \frac{\partial a}{\partial (2\eta\sqrt{t})} \\ &= \frac{1}{2} \left( \frac{\partial a}{\partial \eta} \frac{\partial \eta}{\partial (\eta\sqrt{t})} + \frac{\partial a}{\partial t} \frac{\partial t}{\partial (\eta\sqrt{t})}\right)\\ &= \frac{1}{2} \left( \frac{\partial a}{\partial \eta} \frac{1}{\sqrt{t}} + \frac{\partial a}{\partial t} \frac{2\sqrt{t}}{\eta}\right)\\ &= \frac {a_\eta}{2\sqrt{t}} + \frac {a_t \sqrt{t}}{\eta} \\ \end{align}
By a similar process, we can get $a_{xx}$.