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,
$$F(x) = -\frac1 4 e^{-2x} (2x + 1) \\<br /> \frac{dF}{dx} = x e^{-2x} \\<br /> \frac{d^2F}{dx^2} = e^{-2x} (1-2x)$$
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 $0 < x < 1$, 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.