Triple integral with dependent parameters

Question

The form of my problem is as follows:

$\Psi=C\int_{-\infty}^\infty\int_{-\infty}^\infty\left(f(x,y)\times \left(\int_0^t g(x,t)dt\right)dxdy\right)$

I have already attempted numerical solutions using Matlab's integrate3, without success. However integrate3 is meant for problems of the form:

$\Psi=\int_a^b\int_c^d\int_e^f f(x,y,z)dxdydz$

Similarly, attempts made with scipy's integration toolkit have also not borne fruit.

I have attempted to first calculate $\int_0^tg(x,t)$ at discrete $x$ values and then place it in $\Psi$ but for rather obvious reasons that does not work either.

Additionally, $g(x,t)$ cannot be factored in the form of $h(x)\times i(t)$, which might have allowed for a by parts solution which might be integrated symbolically (for x) and numerically for t.

Also $\int_0^t\int_{-\infty}^{\infty} g(x,t)$ has singularities at multiple points.

Is there a mathematica method to solve this?

From the documentation I see mathematica seems to handle only region level integrations with integrate, so perhaps a linear solution in $t$ with a corresponding region integration over $x$ and $y$?

UPDATE

More specifically, I'm trying to reproduce the work of Shen et. al. so my equations are

shenPhiG[_g,_t]:= (1/(1+2(t/tc)))(1-Exp[(-2m*g/(1+2(t/tc)))])
shenU[_g]:=Exp[-(1+I*v)g]
NIntegrate[shenPhiG[g]*Exp[-I*(theta/tc)*NIntegrate[shenPhiG[g,t],{t,0,tf}]],{x, -Infinity, Infinity}, {y, -Infinity, Infinity}]

I've tried with tf as 5 or 10000 but I always get

NIntegrate::inumr : The integrand shenPhiG[g,t] has evaluated to non-numerical values for all sampling points in the region with boundaries {{0,1000}}.

UPDATE 2

The equations being modeled are given in the paper (in spherical coordinates, hence the reduction of integrals) as:

$\Theta=\frac{\theta}{t_c}\int_0^t\frac{1}{1+\frac{2t^\prime}{t_c}}\left[1-\mathrm{exp}\left(\frac{-2mg}{1+2\frac{t^\prime}{t_c}}\right)dt^\prime\right]$

and

$U_p=C\int_0^\infty \mathrm{exp}[-(1+\mathrm{i}V)g]\mathrm{exp}(-\mathrm{i}\Theta) dg$

With the approximation $\mathrm{exp}(-\mathrm{i}\Theta)\approx1-\mathrm{i}\Theta$

$U=C\int_0^{\infty}(1-\mathrm{i}\Theta)\mathrm{exp}[-(1+\mathrm{i}V)g]dg$

The form I've explained in the original question is obtained on converting to Cartesian coordinates.

Upon using

shenPhi[g_,t_]:=(1/(1+2(t/tc)))(1-Exp[(-2m*g/(1+2(t/tc)))])

and

shenU[g_]:=Exp[-(1+I*v)*g]NIntegrate[shenPhiG[g]*Exp[-I*(theta/tc)*
Integrate[shenPhiG[g,t], {t, 0, tf}]],
{g, -Infinity, Infinity}]

I get no output at all..

With

shenU[_g,t_]=Exp[-(1+I*v)*g]NIntegrate[shenPhiG[g]
*Exp[-I*NIntegrate[shenPhiG[g,t],{t,0,tf}]],{g,-Infinity,Infinity},{t,-Infinity,Infinity}

and

shenPhi[g_,t_]:=(theta/tc)(1/(1+2(t/tc)))(1-Exp[(-2m*g/(1+2(t/tc)))])

I get

NIntegrate::inumr: The integrand $\frac{1-\mathrm{exp}\left(-\frac{2g}{1+Times[<<2>>]} \right)}{1+\frac{t}{2542}}$ has evaluated to non-numerical values for all sampling points in the refion with boundaries {{0,50000}}.

Throw::nocatch: Uncaught Throw[-HolonomicDifferentialRootReduceDumpy[NIntegrateLevinRuleDumpx\$373597]+HolonomicDifferentialRootReduceDumpy'[NIntegrateLevinRuleDumpx\$373597],NIntegrateLevinRuleDumpFastLookupHolonomicDifferentialEquation] returned to top level.

Answer 1

You said you have definitions for $f(x,y)$ and $g(x,t)$ already, so the first step to trying to solve this problem with Mathematica would be to attempt the numerical integral directly after setting tf to the upper bound of the inner integral:

NIntegrate[f[x,y] NIntegrate[g[x,t], {t, 0, tf}],
   {x, -Infinity, Infinity}, {y, -Infinity, Infinity}]

Due to the nested integrals, this may be a fairly slow calculation. It may be better to see if Integrate can find a reasonably closed form of the inner integral first:

NIntegrate[f[x,y] Evaluate[Integrate[g[x,t], {t, 0, tf}]],
   {x, -Infinity, Infinity}, {y, -Infinity, Infinity}]

Though from the sounds of it, $g(x,t)$ probably won't allow that to be effective.

If tf is meant to be left as a variable, then numerical integration methods aren't going to be appropriate for solving this problem.

To provide further advice, examples of $f(x,y)$ and $g(x,t)$ of equivalent complexity to the problem you're trying to solve would be necessary.

---

shenPhi[_g,_t]:=(1/(1+2(t/tc)))(1-Exp[(-2m*g/(1+2(t/tc)))])

Is actually a nonsensical declaration, since function definitions are divided into name_pattern, and thus the first and second arguments are actually nameless as it stands. Since this is going to be a numerical integration problem, ensure that tc and m have numerical values declared elsewhere and change that line to:

shenPhi[g_,t_]:=(1/(1+2(t/tc)))(1-Exp[(-2m*g/(1+2(t/tc)))])

Then based on your additional information, ensure that all constants also have numerical values and evaluate the integral:

shenU[g_]:=Exp[-(1+I*v)*g]NIntegrate[shenPhiG[g]*Exp[-I*(theta/tc)*
    Integrate[shenPhiG[g,t], {t, 0, tf}]],
    {g, -Infinity, Infinity}, {y, -Infinity, Infinity}]

Clearly x needed to be changed to g to match the problem definition, but I'm not clear on which variable y is supposed to match -- so that will need to be changed as well.

It also appears from a preliminary inspection that the inner integral will need to be Integrate rather than NIntegrate, as the inner NIntegrate won't generally have a value for g to plug in.