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I'm trying to solve the Fokker-Planck equation

$$\frac{\partial p}{\partial L}(L, \eta)= \frac{\partial}{\partial \eta}\left[\left(\eta^{2}-1\right) \frac{\partial p}{\partial \eta}(L, \eta)\right] = (\eta^2-1)\frac{\partial^2p}{\partial\eta^2}(L,\eta)+2\eta\frac{\partial p}{\partial\eta}(L,\eta), \quad \eta\in(1,\infty).$$

with initial condition

$$p(L=0, \eta)=\delta(\eta-1).$$ I also enforce the initial conditions that

$$P(L=\max(L),\eta) = 0, \quad P(L,\eta=\max(\eta)) = 0.$$

The problem is the Dirac Delta function as the initial condition. I try and use a pulse of the form

f[eta_] = aa E^(-10000*(eta-1)^2)

Integrate[f[eta], {eta, 1, Infinity}] == 1;

aa = aa /. Solve[%, aa][[1]]

My code to solve the PDE is then

pde = D[P[LL, eta], LL] - D[(eta^2 - 1) D[P[LL, eta], eta], eta]

sol = NDSolve[{pde == 0, P[0, eta] == f[eta], P[100, eta] == 0, 
    w[LL, 100] == 0}, P[LL, eta], {LL, 0, 100}, {eta, 1, 100}, 
   MaxSteps -> {50000, Automatic}] // Flatten

Plotting gives

(*Plot hypersurface*)
Plot3D[
 Evaluate[P[LL, eta] /. s], {LL, 0, 100}, {eta, 1, 100},
 PlotPoints -> 100, PlotRange -> All, ImageSize -> 500, 
 AxesLabel -> {L, \[Eta], P}, AxesStyle -> 40, Boxed -> True, 
 TicksStyle -> 20, PlotTheme -> "Classic"]

enter image description here

The solution I get is no good I think, as I get error messages which I think come from the Dirac Delta implementation. Please someone help me incorporate it properly, thank you!

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  • $\begingroup$ The value of LLoc ? $\endgroup$ – Cesareo Feb 12 at 20:17
  • $\begingroup$ Sorry. It can just be set to 1. $\endgroup$ – rami_salazar Feb 12 at 20:18

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