In[]:=
ClearAll["`*"];Remove["`*"];
In[]:=
p[x_,t_]:=1-Erf[x/Sqrt[4*t]];pa[x_?NumericQ,t_?NumericQ]:=1NIntegrate-+HypergeometricPFQ{},,,-*p[x,s],{s,0,10},Method{Automatic,"SymbolicProcessing"0},PrecisionGoal20//Quiet
1/4
t
HypergeometricPFQ{},,,-
1
2
3
4
4
s
256t
Gamma
3
4
sHypergeometricPFQ{},,,-
3
4
5
4
4
s
256t
π
1/4
t
2
s
5
4
3
2
4
s
256t
2
t
Gamma1
4
In[]:=
n=5;A=ListLinePlot[Table[{t,Log[10,1-pa[1/10,10^t]]},{t,-3,8,n^-1}],AxesLabel{Log[t],Log[1-P_0.5[x,t]]},PlotRange->{Automatic,{-1.5,0}},PlotStyle->Red]
Out[]=
In[]:=
pX[x_,t_]:=1-Erf[x/Sqrt[4*t]];paX[x_?NumericQ,t_?NumericQ,m_?NumberQ]:=1NIntegrate-+Sum,{j,0,m}*pX[x,s],{s,0,10},Method{Automatic,"SymbolicProcessing"0},PrecisionGoal20//Quiet(*Forsvalue10=Infinty!!!Form=30termsofseries*)
1/4
t
Sum,{j,0,m}
-j
256
j
-
4
s
t
j!Pochhammer,jPochhammer,j
1
2
3
4
Gamma
3
4
sSum,{j,0,m}
-j
256
j
-
4
s
t
j!Pochhammer,jPochhammer,j
3
4
5
4
π
1/4
t
2
s
-j
256
j
-
4
s
t
j!Pochhammer,jPochhammer,j
5
4
3
2
2
t
Gamma1
4
In[]:=
n=5;B=ListLinePlot[Table[{t,Log[10,1-paX[1/10,10^t,30]]},{t,-3,8,n^-1}],AxesLabel{Log[t],Log[1-P_0.5[x,t]]},PlotRange->{Automatic,{-1.5,0}},PlotStyle->{Black,Dashed}]
Out[]=
In[]:=
Show[{A,B}]
Out[]=
