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The original question is:

Try and apply the Similarity solution method to the following boundary value problems for [tex]u(x,t)[/tex].

[tex]u_t = k u_{xx}[/tex] for all [tex]x > 0[/tex] with boundary conditions

[tex]u_x(0,t) = 1[/tex]

[tex]u(x,t) \to 0[/tex] as [tex]x \to \infty[/tex]

[tex]u(x,0) = 0[/tex] for [tex]x > 0[/tex].

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

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

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# Similarity Solutions BVP

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