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QUESTION

Use the Fourier transform to solve the heat equation
$$f_t=f_{xx},\ t>0,\ -\infty<x<\infty$$ given that
$$f(x,0)=e^{-\frac{x^2}{2}},\ f,f_x\to 0 \textrm{ as }x\to \pm
\infty$$.

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ANSWER

$f_t=f_{xx},\ t>0,\ -\infty<x<\infty,\ f(x,0)=e^{-\frac{x^2}{2}},\
f,f_x\to 0$ as $|x| \to 0$.

Fourier transform with respect to x:

$F_t=-k^2F\\ F(t,k)=F(0,k)e^{-k^2t}\\ F(0,k)=e^{-\frac{k^2}{2}}\\
F(t,k)=e^{-\frac{k^2(1+2t)}{2}}$\\ Inverse Fourier transform\\
$\ds f(t,x)=\frac{1}{2\pi}\int_{-\infty}^\infty
e^{-\frac{k^2(1+2t)}{2}+ikx}\,dk$\\ Change of variable\\
$\overline{k}=\sqrt{1+2t}k$ and
$\overline{x}=\frac{x}{\sqrt{1+2t}}$
\begin{eqnarray*}
f(t,x)&=&\frac{1}{2\pi}\frac{1}{\sqrt{1+2t}}\int_{-\infty}^\infty
e^{-\frac{\overline{k}^2}{2}+i\overline{k}\overline{x}}\,d\overline{k}\\
&=& \frac{1}{\sqrt{1+2t}}e^{-\frac{\overline{x}^2}{2}}\textrm{ by
the result we have used before}\\
&=&\frac{1}{\sqrt{1+2t}}e^{-\frac{x^2}{2(1+2t)}}
\end{eqnarray*}

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