PDE with variable boundary condition

[jafanasim]

Homework Statement

I am trying to solve this PDE with variable boundary condition, and I want to use combination method. But I have problem with the second boundary condition, which is not transformed to the new variable. Can you please give me some advise?

Homework Equations

(∂^2 T)/(∂x^2 )=1/∝ ∂T/∂t
IC: T(x,0) = Ti
BC1: T(x→∞, t) = Ti
BC2: -k ∂T/∂x|x=0 = h[T∞ - T(0,t)]

The Attempt at a Solution

The combination variable I chose is η= x/((4∝t)^(1⁄2))

My work is attached in a PDF file, please take a look at it. The highlighted boundary condition is the problem.

Thank you

 

[pasmith]

That method does not work with that sort of boundary condition: you can't express $\partial \eta/\partial x$ as a function of $\eta$.

 

[jafanasim]

That method does not work with that sort of boundary condition: you can't express $\partial \eta/\partial x$ as a function of $\eta$.

Thank you. What method should I choose? Laplace?

 

[pasmith]

Thank you. What method should I choose? Laplace?

That would be the obvious choice.