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First, Convolve only works when the output variable is different: Convolve[Exp[x], Exp[-x^2/2], x, x] won't work, but Convolve[Exp[x], Exp[-x^2/2], x, y] does, resulting in E^(1/2 + y) Sqrt[2 Pi] Your second problem is that $G(x)$ is not actually dependant on $x$, it is merely a constant. The convolution of two constant functions is infinite or ...

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First, I'll break down the function you're convolving. It's a real mess: (-E^(-2*x) + E^(-x))*(7 - 3*Piecewise[{{1, -E^(-2*x) + E^(-x) >= 0}}, 0]) + (-E^(-2*x) + 2/E^x)*(2 - 2*Piecewise[{{1, -E^(-2*x) + 2/E^x >= 0}}, 0]) + (1/(4*E^(2*x)) - 2/E^x + (7 - 6*x + 2*x^2)/4)* (3 - 2*Piecewise[{{1, 1/(4*E^(2*x)) - 2/E^x + (7 - 6*x + 2*x^2)/4 >= 0}}, ...

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I had to do something similar here: Unable to fit function convolved with NIntegrate with FindFit even with NumericQ (See my answer). Basically, you need to tell Mathematica to not try to find a symbolic solution and just wait until it can be evaluated numerically using ?NumericQ (there's lots of questions addressing this). Also, I found that it's ...

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Your computation (even after fixing the superfluous [x] and introducing HeavisideTheta) involves amounts of memory I'd call "extreme": After 20mins, it is well beyond 11 GB, and 20 mins later the kernel crashes (in a way, see below) due to all my 32 GB of RAM (and all of pagefile space) having been consumed. Sidenote: In the given case, ...

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Here is a simple example using UnitStep to create a conditional function h = Exp[-x] UnitStep[x] + (1 - UnitStep[x]) Exp[-2 x]; f = Exp[-2 x]; Convolve[f, h, x, y] -E^(-2 y) - E^(-2 y) y

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