Finding Similarity Solutions for Boundary Value Problems

[math2011]

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The original question is:
Try and apply the Similarity solution method to the following boundary value problems for $$u(x,t)$$.

$$u_t = k u_{xx}$$ for all $$x > 0$$ with boundary conditions
$$u_x(0,t) = 1$$
$$u(x,t) \to 0$$ as $$x \to \infty$$
$$u(x,0) = 0$$ for $$x > 0$$.

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

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

 

[math2011]

I have posted the solution on the page below.
http://www.mathhelpforum.com/math-help/f59/similarity-solutions-185537.html