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UnkleRhaukus

  • 2 years ago

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

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

  2. UnkleRhaukus
    • 2 years ago
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    is this right/?, how can i check?

  3. AccessDenied
    • 2 years ago
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    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.

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