Question

Evaluate \[\int\limits_0^x\text{erf}\left(\sqrt{x-t}\right)\text{erf}\left(\sqrt t\right)\text dt\]

Answer 1

\[\begin{align*}
y(t)&=\int\limits_0^x\text{erf}\left(\sqrt{x-t}\right)\text{erf}\left(\sqrt t\right)\text dt\\
Y(p)&=\mathcal L\left\{\int\limits_0^x\text{erf}\left(\sqrt{x-t}\right)\text{erf}\left(\sqrt t\right)\text dt\right\}\\
\\&=\mathcal L\left\{\text{erf}\left(\sqrt t\right)\ast\text{erf}\left(\sqrt t\right)\right\}\\
\\&=\frac {1}{p\sqrt{p+1}}\times\frac {1}{p\sqrt{p+1}}\\
\\&=\frac {1}{p^2(p+1)}\\
\end{align*}\]

Answer 2

is this right/?, how can i check?

Answer 3

This appears to be correct to me, so far. :)
I checked it here just by making sure each step was justified and matched up correctly to the definitions...
line 1-2: taking laplace transform of both sides \(\checkmark\)
line 2-3: definition of convolution. \(\checkmark\)
line 3-4: convolution theorem \(\checkmark\) etc.