A Point source with Dirichlet Boundary

[Bruce]

hi, all great brains, I have a question about BVP, which confused me a while, maybe someone can help to clarify it.

For a point source with Dirichlet boundary in a 2D domain, the response at any coordinate except the source point is dependent on the surface of the source, for example, heat transfer. But a point source, in theory has no surface in a 2D domain. So the transient solution with Dirichlet point source in a 2D domain is regarded as ill-defined problem(the solution has a singularity at source)? or it can be regarded as a weak form, are there any reference about this? or something is else wrong.

Just use the heat transfer as an example:

d(dT/dx)+d(dT/dy)=dT/dt

BC: T'(0,y,t)=T'(L,y,t)=T'(x,0,t)=T'(x,M,t)=0 ; 0<x<L;0<y<M;

IC: T(x,y,0)=delta(x-x',y-y',t-t') * T0thanks in advance.

Bruce

 

[Chestermiller]

Do you know how to solve this for the long-time solution?

 

[Bruce]

hi, chestermiller,

Yes, I can get the solution which gives very high value (>10^6 for T0=100) at source point at t=0. In some practice, source point was integrated into area in a 2D domain, which also works in this case. Then I just have this doubt, point source is not applicable for dirichlet boundary condition in a reality .

ps: source point at (x',z')

 

[Chestermiller]

It looks like you solved it already. What is the problem?