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The following problem was given to me by a friend, so I can't really guaranty that a solution exists, but if, I certainly can't find it myself...

Let us consider the following Integro-differential equation

eq = D[\[CapitalGamma][t, T], t] == 
   1/2 DiracDelta[T] Integrate[
      A Exp[-\[Beta] (u - C)^2] \[CapitalGamma][t, u], {u, 0, 
       Infinity}] - 
    1/(Exp[\[Sigma] - \[Alpha] T ] + 1) \[CapitalGamma][t, T] - 
    D[\[CapitalGamma][t, T], T] // TeXForm

$$\Gamma ^{(1,0)}(t,T)=\frac{1}{2} \delta (T) \int_{-\infty}^{\infty } A e^{\beta \left(-(u-C)^2\right)} \Gamma (t,u) \, du-\frac{\Gamma (t,T)}{e^{\sigma -\alpha T}+1}-\Gamma ^{(0,1)}(t,T),$$

where $$A\in [0,1]\quad \beta,C,\sigma,\alpha=\text{const.}>0.$$ My friend was a bit vague about the initial conditions, but I think the following two must certainly apply $$\Gamma(0,T)=e^{-(T-5)^2}\quad T\in [0,10], \quad \Gamma(t,T<0)=0.$$ I'm quite sure that this isn't enough (at least something in regards to $\partial \Gamma/\partial t$ should probably be known as well), so just assume the simplest case possible.


Now I tried finding methods to solve this here on Stack Exchange, but all of them seemed rather specific (applying the Laplace-Transformation to the equation, taking the derivative, etc.) and I'm not really sure if these methods work for my problem. Also the delta-function is something that I don't really know how to deal with in Mathematica... and of course the fact that I have function of two variables instead of the standard case with one...

Can someone suggest a method of solving this, or how one could approach a problem like this?

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    $\begingroup$ Even without the integration this PDE is troublesome enough for numeric method. Do you know if the solution involves discontinuity or non-smoothness? Some related posts: mathematica.stackexchange.com/a/145479/1871 scicomp.stackexchange.com/q/28744/5331 $\endgroup$ – xzczd Sep 1 '19 at 5:19
  • $\begingroup$ @xzczd Just looking at what $\Gamma$ represents, I would assume the solution to be at least continuos. Not sure if it is smooth or not.. $\endgroup$ – Sito Sep 1 '19 at 11:10
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    $\begingroup$ Since there's a A Exp[-\[Beta] (u - C)^2] inside the integral, the method mentioned in e.g. mathematica.stackexchange.com/a/73974/1871 might worth trying. (The DiracDelta needs to be replaced by a carefully chosen continuous approximation of course, the grid size must be dense enough, and {u, -Infinity, Infinity} will be approximated by {u, 0, 10}. ) $\endgroup$ – xzczd Sep 2 '19 at 10:54
  • $\begingroup$ @xzczd Thank you for the link and the advice. I will give it a try, but my experience with Mathematica is rather limited to easier problems, so we'll see how this goes.. $\endgroup$ – Sito Sep 2 '19 at 20:38
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    $\begingroup$ That's the node for Gauss quadrature. If you find it hard to understand, you may turn to trapezoid rule as shown here: mathematica.stackexchange.com/a/175784/1871 $\endgroup$ – xzczd Sep 4 '19 at 5:53

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