I have the following similarity solutions problem and solution:
Problem
$u_t = ku_{xx}$ for all $x > 0$, with $u_x (0, t) = 1$, $u(x, t) \to 0$ as $x \to \infty$, and $u(x, 0) = 0$ for $x > 0$.
Solution
Introducing $u′ = \dfrac{u}{u_0}$, $t′ = \dfrac{t}{t_0}$ and $x′ = \dfrac{x}{x_0}$, we will reduce the original PDE to
$$\dfrac{\partial{u′}}{\partial{t′}} = \dfrac{\partial^2{u'}}{\partial{x'}^2}$$
when choosing $x^2_0 = k t_0$.
The boundary condition $u_x(0, t) = 1$ will make us choose $x_0 = u_0$ so that $u'_{x'} (0, t') = 1$.
Choosing $t_0 = t$ we have
$$\sqrt{kt}u' \left( \dfrac{x}{\sqrt{kt}}, 1 \right) = \sqrt{kt} f(\eta), \eta = \dfrac{x}{\sqrt{kt}}$$
The PDE will reduce to the ODE
$$f''(\eta) + \dfrac{\eta}{2}f'(\eta) − \dfrac{1}{2}f(\eta) = 0$$
subject to $f'(0) = 0$ and $f(\eta) \to 0$ as $\eta \to \infty$.
I was told that, despite how this problem was posed, my instructor would provide us with the change of variables $u′ = \dfrac{u}{u_0}$, $t′ = \dfrac{t}{t_0}$ and $x′ = \dfrac{x}{x_0}$.
However, I'm struggling to understand the reasoning behind each of the steps taken in the solution. For all of these steps, I find myself asking "Why was this step taken/How did they know to take this step?". Can someone please take the time to annotate this solution with explanations of each step, so that I can understand this type of problem and be able to do it myself in the future? In other words, please walk me through the process, explaining why each step was taken.
For instance, why will the boundary condition $u_x(0, t) = 1$ make us choose $x_0 = u_0$? Where did $u'_{x'} (0, t') = 1$ come from?
And so on for the rest of the solution ...
I would really appreciate it if someone could please take the time to do this.