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fluidistic
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Homework Statement
Problem 8-17 from Mathew's and Walker's book:
Use a cosine transform with respect to y to find the steady-state temperature distribution in a semi-infinite solid [itex]x>0[/itex] when the temperature on the surface [itex]x=0[/itex] is unity for [itex]-a<y<a[/itex] and zero outside this strip.
Homework Equations
Heat equation: [itex]\frac{\partial u}{ \partial t}+k \nabla u =0[/itex].
Cosine Fourier transform: [itex]f(x)=\frac {1} {\pi} \int _0 ^{\infty } g(y) \cos (xy )dy[/itex].
The Attempt at a Solution
I've made a sketch of the situation, I don't think I have any problem figuring out the situation.
Now I'm stuck. Should I perform "brainlessly" a cosine transform to the heat equation as it is, or should I set [itex]\frac{\partial u }{\partial t}=0[/itex] since it's a steady state distribution of temperature? This would make [itex]\nabla u =0 \Rightarrow \frac{\partial u }{\partial x }+\frac{\partial u }{\partial y }=0[/itex] (Laplace equation). Should I apply now the cosine transform?