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I am looking for an explicit, closed form expression for the solution to a boundary value problem given by the linear PDE:

$$ u_{xx}+au_y-bu+c=0 $$ with the BCs $ u_x(-g,y)=ku $, $u_x(g,y)=-ku$, $ u(x,0)=u_0 $, $u(x,d)=u_d$.

$a,b,c,k,u_0$ and $u_d$ are all $>0$. What would be the best thing to do with this? Any help is much appreciated!

Kurt
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  • A linear equation with constant coefficients set on a domain of very simple geometry, well, this appeals to Fourier series. –  Mar 16 '15 at 15:24
  • Thank you, I'll try that road! – Kurt Mar 16 '15 at 17:39

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Except for the terms $-bu + c$, your pde is the heat equation, where $y$ plays the role of "time". It can be transformed to the heat equation by $u = e^{by/a} v + c/b$. But you have one boundary condition too many: you can't specify boundary conditions at both $y=0$ and $y=d$.

  • Let me put it this way. The boundary condition at $y=0$ determines $u(x,0)$ up to an additive constant. The value of $u(x,0)$ and the boundary conditions at $x = \pm g$ then determine $u(x,y)$ for all $y > 0$. It's very unlikely that this will satisfy your boundary condition at $y=d$. So if you can't drop the other boundary condition, your problem will have no solution. –  Mar 16 '15 at 18:17
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    That is an elliptic equation, second order in $y$. Your pde is parabolic, first order in $y$. That's not at all similar. –  Mar 16 '15 at 18:27