Point source with Dirichlet Boundary

In summary, the conversation discusses the confusion around using a point source with Dirichlet boundary in a 2D domain for heat transfer. It is mentioned that a point source does not have a surface in a 2D domain, which raises questions about its validity. The conversation also includes a heat transfer equation and boundary and initial conditions. The solution for the long-time solution is mentioned to give a very high value at the source point, but integrating the source point into the area in a 2D domain has also been found to work. The applicability of a point source with Dirichlet boundary in reality is also questioned.
  • #1
Bruce
4
0
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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  • #2
Do you know how to solve this for the long-time solution?
 
  • #3
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:
  • #4
It looks like you solved it already. What is the problem?
 

1. What is a point source with Dirichlet boundary?

A point source with Dirichlet boundary is a mathematical model used in physics and engineering to represent a single point or location where a specific physical quantity, such as temperature or concentration, is known. The Dirichlet boundary condition specifies the value of this quantity at the boundary of a given region.

2. How is a point source with Dirichlet boundary different from a point source without a boundary condition?

A point source without a boundary condition is a mathematical idealization that assumes the physical quantity being studied is constant at all points in space. In contrast, a point source with Dirichlet boundary takes into account the boundary of a given region and specifies the value of the physical quantity at that boundary.

3. What are some real-world applications of a point source with Dirichlet boundary?

A point source with Dirichlet boundary is commonly used in various fields such as heat transfer, fluid dynamics, and pollution dispersion modeling. For example, it can be used to model the temperature distribution in a room with a heater at a specific location, or the concentration of a pollutant released from a single point source.

4. How is a point source with Dirichlet boundary solved mathematically?

The mathematical solution for a point source with Dirichlet boundary depends on the specific physical quantity being studied and the boundary conditions of the given region. In general, it involves solving a differential equation, such as the heat equation or the diffusion equation, using techniques such as separation of variables or numerical methods.

5. What are the limitations of using a point source with Dirichlet boundary in modeling physical systems?

One limitation is that it assumes the physical quantity being studied is constant at all points in space. In reality, this may not always be the case, especially in complex systems. Additionally, the accuracy of the model depends on the accuracy of the boundary conditions and the assumptions made in the mathematical solution.

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