In[]:=
CompoundExpression[
]
​​deploy
Sun 7 May 2023 19:55:31
Check Ulrich solution from forum​
​
Top-level notebooks:
- forum-mean-field-equation.nb

Checking Ulrich’s FixedPoint solution

Ulrich’s fixed solution for p=2

In[]:=
ClearAll["Global`*"];​​h=1/(1+#)^2&;​​a=-2;​​b=1;​​inf=10000;(*~Infinity*)​​nl=NestList[Function[{fa},Block[{int,x,t},int[t_?NumericQ]:=Block[{z},NIntegrate[h[z]fa[z,t],{z,0,inf(*Infinity*)},Method->{Automatic,"SymbolicProcessing"->0}]];​​NDSolveValue[{Derivative[0,1][f][x,t]==ah[x]f[x,t]+bh[x]int[t],f[x,0]==1},f,{x,0,inf},{t,0,10},MaxStepSize->{100,Automatic}]]],1&(*Identity*),10];​​Plot3D[h[x]nl[[-1]][x,t],{x,0,10},{t,0,10},PlotRange->All,MeshFunctions->{#3&},AxesLabel->{x,t,f[x,t]}]
Out[]=

Check against manual solution by DinosaurEgg

Dinosaur egg math.SE post​
This is unnomalized version, may have edge effects
In[]:=
lossVal[t_]:=Module[{x},​​obj=h[x]nl[[-1]][x,t];​​NIntegrate[obj,{x,0,inf}]​​];​​​​tvals=Table[t,{t,1.,10}];​​ulrichVals=Table[lossVal[t],{t,tvals}];​​​​Clear[i,s];​​x=
2/s
;​​i=10;​​expr=
2
(i+1)
2+s
2
(i+1)
+
1
s
2
(i+1)
+2
ArcTan[x]
1
x
1+
1
x
ArcTan[x]
;​​gdVals=Table
Pi
4t
,{t,tvals};​​dinasaurEggVals=TableInverseLaplaceTransform
ArcTan[x]
1
x
1+
1
x
ArcTan[x]
,s,t,{t,tvals};​​ListPlot[{ulrichVals,gdVals,dinasaurEggVals},PlotLegends->{"ulrich","gd","dinasaurEgg"}]​​
Out[]=
ulrich
gd
dinasaurEgg

Older stuff
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