In[]:=
Q[xx_,tt_]:=Sqrt[t]*Exp[-x^2/(4*D*t)]-(Sqrt[π/D]*x/2)*Erfc[x/Sqrt[4*D*t]];​​G[t_,tau_,x_,ksi_]:=(Exp[-((x-ksi)^2)/(4*D*(t-tau))]+Exp[-((x+ksi)^2)/(4*D*(t-tau))])/Sqrt[4*π*D*(t-tau)];​​
In[]:=
Integrate[G[t,tau,x,ksi]*Q[ksi,tau],{tau,0,t},{ksi,0,∞},Assumptions->{t>0,x>0,D>0,tau∈Ν,ksi∈Ν}]
Out[]=
t2
-
2
x
4Dt

Dt
-
π
xErfc
x
2
Dt

2
D
In[]:=
Integrate[G[t,tau,x,ksi]*Q[ksi,tau],{tau,0,t},{ksi,0,∞},Assumptions->{t>0,x>0,D>0,tau>0,ksi>0}]
Out[]=
t2
-
2
x
4Dt

Dt
-
π
xErfc
x
2
Dt

2
D