Hello, I am afraid I continue to have problems with resolving some integrals. Given the functions:

Q[xx_,tt_]:=Sqrt[tt]*Exp[-xx^2/(4*D*tt)]-(Sqrt[π/D]*xx/2)*Erfc[xx/Sqrt[4*D*tt]];
G[tt_,ttau_,xx_,kksi_]:=(Exp[-((xx-kksi)^2)/(4*D*(tt-ttau))]+Exp[-((xx+kksi)^2)/(4*D*(tt-ttau))])/Sqrt[4*π*D*(tt-ttau)];

I need to determine analytically the following integral:

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}]

I believe this integral exists, and from some alternative calculations I expect it to be:

(Sqrt[π/D]/D/12)*((2*Sqrt[D*t/π])*(2*D*t-x^2)*Exp[-(x^2)/(4*D*t)]+(x^3)*Erfc[x/Sqrt[4*D*t]])

(but I just want to verify if this is true). Unfortunately, MATHEMATICA does not seem to be able to find this result - it gets frozen and does not output anything. I tried to tackle the univariate integral along variable ksi, involving the second term of function Q, but then MATHEMATICA returns the input Integrate command, which suggests that the integral does not exist. However, if I assume that x=0:

Integrate[G[t,tau,0,ksi]*Q[ksi,tau],{tau,0,t},{ksi,0,∞},Assumptions->{t>0,D>0,tau>0,ksi>0}]

I quickly get the result consistent with the expected formula:

t^(3/2)/3

So, is there any way to force MATHEMATICA to produce the expected output also for x > 0?

By the way, can anybody suggest any useful transformation of variables that might simplify this sort of integrals? I also need to determine the integral involving Q squared:

Integrate[G[t,tau,0,ksi]*Q[ksi,tau]^2,{tau,0,t},{ksi,0,∞},Assumptions->{t>0,D>0,tau>0,ksi>0}]

I have reasons to expect that the latter integral is proportional to t^2, with the proportionality coefficient not depending on t. I want to evaluate the coefficient numerically, but I don't see how this might be done, given the intricate dependence of the integrand on t.

Leslaw

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