Compatibilty of the Dirichlet boundary condition

[bhatiaharsh]

Hi,

I am trying to solve a Poisson equation [itex]\nabla^2 \phi = f[/itex] in [itex]\Omega[/itex], with Dirichlet boundary condition [itex]\phi = 0[/itex] on [itex]\partial \Omega[/itex]. 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,
[tex]
F(x) = -\frac1 4 e^{-2x} (2x + 1) \\
\frac{dF}{dx} = x e^{-2x} \\
\frac{d^2F}{dx^2} = e^{-2x} (1-2x)
[/tex]
Clearly, [itex]F(x) \neq 0[/itex] for [itex]x = 0,1[/itex]. I am attaching the plots of these functions [itex]F(x)[/itex] in black, [itex]\frac{dF}{dx}[/itex] in red, and [itex]\frac{d^2F}{dx^2}[/itex] in green.

Now, suppose, I solve the Poisson equation said above, with [itex]\nabla^2 \phi = e^{-2x} (1-2x)[/itex] for [itex]0 < x < 1[/itex], I hope to recover [itex]\phi = F[/itex] uniquely upto a harmonic. However, the given that [itex]\phi = 0[/itex] for [itex]x = 0, 1[/itex], I don't see how this can produce a continuous [itex]\phi[/itex], 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 [itex](1-2x)[/itex] instead of [itex](1+2x) [/itex], and therefore calculation of [itex]c_1,c_2[/itex] are wrong. But I got the general idea. Thanks a lot.

 

[blue_raver22]

Dear researcher,

Thank you for sharing your findings and concerns regarding the compatibility of the Dirichlet boundary condition in solving a Poisson equation. It is important to note that the existence and uniqueness of a solution to a boundary value problem, such as the one you are working on, depends on the compatibility of the boundary conditions and the source function.

In this case, it seems that the source function, e^{-2x} (1-2x), is not compatible with the given boundary condition of \phi = 0 on \partial \Omega. This can be seen from the plots you have provided, where the solution \phi should match the black curve, but it does not due to the mismatch between the source function and boundary condition.

To ensure compatibility, it is necessary for the source function to satisfy the boundary condition at all points on the boundary. In other words, the source function should also be zero at x = 0 and x = 1 in this case. This requirement is often mentioned in textbooks, but it may not be explicitly stated in the context of the Dirichlet boundary condition.

In addition, it is important to note that the uniqueness of the solution may not hold if the compatibility of the boundary conditions and the source function is not satisfied. In some cases, the solution may not exist at all.

I hope this explanation helps in understanding the issue you have encountered. It is always important to carefully check the compatibility of the boundary conditions and the source function in order to ensure the existence and uniqueness of the solution to a boundary value problem.

Best regards,