How to treat point heat sources?

Questions
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1

Hi everyone!
I have to solve a 2D-PDE transient heat equation that contains point heat sources.
I’d like to know if I wrote the following equation correctly:

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q0 = 1;
Subscript[Q, 0] = 
  q0*(DiracDelta[x - x1]*DiracDelta[y - y1] + 
     DiracDelta[x - x1]*DiracDelta[y - y2]*DiracDelta[x - x2]*
      DiracDelta[y - y1] + DiracDelta[x - x2]*DiracDelta[y - y2]);
heqn1 = Subscript[k, 
    CLS]*(D[Subscript[T, D][x, y, t], x, x] + 
      D[Subscript[T, D][x, y, t], y, y]) + Subscript[Q, 0] == 
  Subscript[\[Rho], CLS]*Subscript[Cp, CLS]*
   D[Subscript[T, D][x, y, t], t]

I haven’t considered the heat source as time-dependant just yet to simplify just a bit.
Thank you in advance!

2

I typed

TeXForm[heqn1/.{DiracDelta[x_]:>\[Delta][x]}]

to hopefully get back something which matches your equation.

\begin{multline*} k_{\text{CLS}}\left(T _D^{(0,2,0)}(x,y,t)+T _D^{(2,0,0)}(x,y,t)\right)+\delta (x-\text{x1}) \delta (x-\text{x2}) \delta (y-\text{y1})\delta (y-\text{y2})\\ +\delta (x-\text{x1}) \delta(y-\text{y1})+\delta (x-\text{x2}) \delta(y-\text{y2})=\\ \text{Cp} _{\text{CLS}} \rho _{\text{CLS}}T _D^{(0,0,1)}(x,y,t) \end{multline*}

This looks right. If you want to keep these new variable names (y1 instead of b1…), but still be sure of equivalence, you could do more substitutions besides DiracDelta and end up with almost equivalent tex code. Converting the indexed Derivative to Newton notation is possible, but finicky – I think it reads just fine in this case.

3

I tried and the result seems fine. Your code line should just write the symbol instead of DiracDelta. I hope it will run properly then.
Thank you!