Compatibilty of the Dirichlet boundary condition

[bhatiaharsh]

Hi,

I am trying to solve a Poisson equation \nabla^2 \phi = f in \Omega, with Dirichlet boundary condition \phi = 0 on \partial \Omega. My problem is that I am trying to understand the condition under which a solution exists. All the text I consulted says that the problem is solvable.

However, I am working on contrived example for which I don't see how a solution is possible, yet I am unable to explain it. Consider a function and its first two derivatives,
<br /> F(x) = -\frac1 4 e^{-2x} (2x + 1) \\<br /> \frac{dF}{dx} = x e^{-2x} \\<br /> \frac{d^2F}{dx^2} = e^{-2x} (1-2x) <br />
Clearly, F(x) \neq 0 for x = 0,1. I am attaching the plots of these functions F(x) in black, \frac{dF}{dx} in red, and \frac{d^2F}{dx^2} in green.

Now, suppose, I solve the Poisson equation said above, with \nabla^2 \phi = e^{-2x} (1-2x) for 0 &lt; x &lt; 1, I hope to recover \phi = F uniquely upto a harmonic. However, the given that \phi = 0 for x = 0, 1, I don't see how this can produce a continuous \phi, which matches the black curve.

I think this is because the information I pass to the system is corrupt, however, no textbook tells me any requirement on the compatibility between the source function and the boundary condition. Any insights are appreciated.

 

[kai_sikorski]

You've found a particular solution to the problem, but I think you're forgetting about the two homogenous solutions that you can use to match your boundary conditions.

 

[JJacquelin]

Hi !
Solution in attachment :

 

[bhatiaharsh]

Thanks both of you.

JJacquelin, I think you used (1-2x) instead of (1+2x), and therefore calculation of c_1,c_2 are wrong. But I got the general idea. Thanks a lot.