\textbf{Problem 5}

Consider the continuously differentiable function $f ( x )$ of the real variable $x$. Let $f ( x ) \rightarrow 0$ as $| x | \rightarrow \infty$. $f ( x )$, its derivative $f ^ { \prime } ( x )$, and $x f ( x )$ are absolutely integrable. The Fourier transform of the function $f ( x )$ is denoted by $\mathcal { F } \{ f ( x ) \} ( u )$ or equivalently by $\hat { f } ( u )$, and defined by
$$\mathcal { F } \{ f ( x ) \} ( u ) = \hat { f } ( u ) \equiv \frac { 1 } { \sqrt { 2 \pi } } \int _ { - \infty } ^ { \infty } f ( x ) \exp ( - i u x ) \, d x \tag{1}$$
where $u$ is a real variable and $i$ is the imaginary unit. The Fourier transform is defined in the same way for other functions.

\textbf{I.} Express $\mathcal { F } \left\{ f ^ { \prime } ( x ) \right\} ( u )$ in terms of $\hat { f } ( u )$ and $u$.

\textbf{II.} Express $\frac { d \hat { f } ( u ) } { d u }$ in terms of $\mathcal { F } \{ x f ( x ) \} ( u )$.

\textbf{III.} Let the function $f ( x ) = \exp \left( - a x ^ { 2 } \right)$, where $a$ is a positive real constant $( a > 0 )$. The following relation holds for $f ( x )$:
$$f ^ { \prime } ( x ) = - 2 a x f ( x ) \tag{2}$$
Apply the Fourier transform on both sides of Eq. (2) to obtain a first-order ordinary differential equation in $\hat { f } ( u )$. Solve this ordinary differential equation to obtain $\hat { f } ( u )$. Note that the integration constant in the solution of this ordinary differential equation can be obtained by calculating $\hat { f } ( 0 )$ with the help of Eq. (1) and the value of the following improper integral:
$$\int _ { - \infty } ^ { \infty } \exp \left( - a x ^ { 2 } \right) d x = \sqrt { \frac { \pi } { a } } \tag{3}$$

\textbf{IV.} Consider the function $h ( x , t )$ of the real variables $x$ and $t$. Let $h ( x , t )$ be defined for $- \infty < x < \infty$ and $t \geq 0$, and satisfy the following partial differential equation:
$$\frac { \partial h ( x , t ) } { \partial t } = \frac { \partial ^ { 2 } h ( x , t ) } { \partial x ^ { 2 } } \quad ( t > 0 ) \tag{4}$$
given the initial condition
$$h ( x , 0 ) = \exp \left( - a x ^ { 2 } \right) \quad ( a > 0 ) \tag{5}$$
\begin{enumerate}
  \item Apply the Fourier transform with respect to the variable $x$ on both sides of the partial differential equation (4) to obtain an ordinary differential equation with $\hat { h } ( u , t ) \equiv \frac { 1 } { \sqrt { 2 \pi } } \int _ { - \infty } ^ { \infty } h ( x , t ) \exp ( - i u x ) \, d x$ and the independent variable $t$.
  \item By solving the ordinary differential equation found in Question IV.1, obtain $\hat { h } ( u , t )$.
  \item Use the inverse Fourier transform with respect to the variable $u$ to obtain a solution $h ( x , t )$ satisfying Eq. (4) and Eq. (5).
\end{enumerate}

\textbf{V.} Consider the continuous function $g ( x )$ and its Fourier transform $\hat { g } ( u )$. Let $g ( x ) \rightarrow 0$ as $| x | \rightarrow \infty$ and $g ( x )$ be absolutely integrable. The convolution of the functions $f ( x )$ and $g ( x )$ is defined by
$$( f * g ) ( x ) \equiv \int _ { - \infty } ^ { \infty } f ( y ) g ( x - y ) \, d y \tag{6}$$
\begin{enumerate}
  \item Express $\mathcal { F } \{ ( f * g ) ( x ) \} ( u )$ in terms of $\hat { f } ( u )$ and $\hat { g } ( u )$.
  \item Here, the function $h ( x , t )$ satisfies Eq. (4), given the initial condition $h ( x , 0 ) = g ( x )$. Use the result of Question V.1 to find an integral representation of a solution $h ( x , t )$, where $t > 0$.
\end{enumerate}