Point source with Dirichlet Boundary

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Discussion Overview

The discussion revolves around the mathematical treatment of a point source with Dirichlet boundary conditions in a 2D domain, particularly in the context of heat transfer. Participants explore the implications of having a point source, which theoretically lacks a surface, and how this affects the definition and solvability of the boundary value problem (BVP).

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions whether the transient solution with a Dirichlet point source in a 2D domain is ill-defined due to the singularity at the source point or if it can be treated as a weak form.
  • Another participant inquires about the long-time solution for the problem posed.
  • A participant mentions obtaining a very high value at the source point at time t=0, suggesting that the point source may not be applicable for Dirichlet boundary conditions in practical scenarios.
  • One participant appears to challenge the necessity of further clarification, implying that the problem may have already been resolved.

Areas of Agreement / Disagreement

Participants express differing views on the applicability and definition of a point source under Dirichlet boundary conditions, indicating that multiple competing perspectives remain without a clear consensus.

Contextual Notes

The discussion highlights potential limitations regarding the treatment of singularities and the assumptions underlying the mathematical model, particularly in relation to practical applications.

Bruce
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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
 
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Do you know how to solve this for the long-time solution?
 
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')
242830
 
Last edited:
It looks like you solved it already. What is the problem?
 

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