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Solve the following initial value problem using Fourier transform

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(25 marks) Solve the following initial value problem using Fourier transform.

$$ \begin{array}{l} u_{t}=u_{x x}, \quad-\infty< x <\infty,  t= >0 \\ u(x, 0)=\left(1-2 x^{2}\right) e^{-4 x^{2}}, \quad-\infty< x <\infty \end{array} $$

with \(u(x, t) \rightarrow 0\) and \(u_{x}(x, t) \rightarrow 0\) as \(x \rightarrow \pm \infty\).

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