# PDE with variable boundary condition

1. Dec 21, 2012

### jafanasim

1. The problem statement, all variables and given/known data

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?

2. Relevant 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)]

3. 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

#### Attached Files:

• ###### Combination.pdf
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2. Dec 21, 2012

### pasmith

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

3. Dec 21, 2012

### jafanasim

Thank you. What method should I choose? Laplace?

4. Dec 21, 2012

### pasmith

That would be the obvious choice.