In[]:=
ClearAll["`*"];Remove["`*"];
In[]:=
f[t_]=Piecewise[{{0,t<=-1},{2*Sqrt[1-t^2]/Pi,-1<t<1},{0,t>=1}}]
Out[]=
0
t≤-1
2
1-
2
t
π
-1<t<1
0
True
In[]:=
A=f[tau]*f[t-tau]//Simplify
Out[]=
0
t≥1+tau||1+t≤tau||tau≥1||tau≤-1
4
1-
2
(t-tau)
1-
2
tau
2
π
True
Integrate[2*A,{tau,-Infinity,Infinity},Assumptions->t∈Reals]//AbsoluteTiming(*Symbolicallycan'tcompute!!!*)
Out[]=
454.58,
0
t≥2||t≤-2
Integrate2
0
t≥1+tau||1+t≤tau||tau≥1||tau≤-1
4
1-
2
(t-tau)
1-
2
tau
2
π
True
,{tau,-∞,∞},Assumptionst∈&&-2<t<2
True

In[]:=
c[t_?NumericQ]:=2*NIntegrate
0
t≥1+tau||1+t≤tau||tau≥1||tau≤-1
4
1-
2
(t-tau)
1-
2
tau
2
π
True
,{tau,-Infinity,Infinity};
In[]:=
Plot[c[t],{t,-3,3}]
Out[]=
(*Anotherway:---------------------------------------------------*)
In[]:=
g[t_]=InverseLaplaceTransform[LaplaceTransform[f[t],t,s],s,t]
Out[]=
2
1-
2
t
UnitStep[1-t]
π
In[]:=
B=Integrate[2*g[tau]*g[t-tau],{tau,-Infinity,Infinity},Assumptions->t∈Reals]
Out[]=
4t(4+
2
t
)EllipticE1-
4
2
t
-8EllipticK1-
4
2
t

3
2
π
if 0<t<2
In[]:=
Plot{B,c[t]},{t,0,2},PlotStyle->{Red,{Dashed,Black}},PlotLegends"
4t(4+
2
t
)EllipticE1-
4
2
t
-8EllipticK1-
4
2
t

3
2
π
","Numerically"
Out[]=
4t(4+
2
t
)EllipticE1-
4
2
t
-8EllipticK1-
4
2
t

3
2
π
Numerically