η=5.;xopt={η/2,η/(2Sqrt[3])};κ=.75;Tmax=5.;​​Ω=Polygon[Rationalize[{{0,0},{η,0},{η/2,(ηSqrt[3])/2}},0]];​​bC=Rationalize[DirichletCondition[P[x1,x2,t]==Exp[-10000t]1/((Sqrt[3]η^2)/4),True],0];​​iC=Rationalize[P[x1,x2,0]==Piecewise[{{1/((Sqrt[3]η^2)/4),RegionMember[Ω,{x1,x2}]}},0],0];​​fwrdKol=Rationalize[D[P[x1,x2,t],t]==-D[κ(xopt[[1]]-x1)*P[x1,x2,t],{x1,1}]-D[κ(xopt[[2]]-x2)*P[x1,x2,t],{x2,1}]+1/2D[P[x1,x2,t],{x1,2}]+1/2D[P[x1,x2,t],{x2,2}],0];​​Psol=NDSolveValue[{fwrdKol,iC,bC},P,{x1,x2}∈Ω,{t,0,Tmax}];​​domP[t_]:=NIntegrate[Rationalize[Psol[x1,x2,t],0],{x1,x2}∈Ω,AccuracyGoal->4];
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Plot[domP[t],{t,0,5},PlotTheme->"Scientific",PlotRange->All,FrameLabel->{"t","Prob. Mass Domain"}]
NIntegrate
:Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small.
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ContourPlot[Psol[x1,x2,Tmax/2],{x1,x2}∈Ω]
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