The original question is:

Try and apply the Similarity solution method to the following boundary value problems for $\displaystyle u(x,t)$.

$\displaystyle u_t = k u_{xx}$ for all $\displaystyle x > 0$ with boundary conditions

$\displaystyle u_x(0,t) = 1$

$\displaystyle u(x,t) \to 0$ as $\displaystyle x \to \infty$

$\displaystyle u(x,0) = 0$ for $\displaystyle x > 0$.

I know from my tutorial that I should first find $\displaystyle u = \sqrt{kt} f(\eta)$ where $\displaystyle f$ is an unknown function of similarity variable $\displaystyle \displaystyle \eta = \frac{x}{\sqrt{kt}}$. What I don't know is how to find the similarity variable $\displaystyle \eta$ and the formula $\displaystyle u = \sqrt{kt} f(\eta)$.

Please tell me the procedure for finding similarity solution. Thank you in advance.