$\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\Re}{\operatorname{Re}}$
Consider Black-Scholes equation in the financial mathematics with the reverse-time Cauchy problem: \begin{align} &V_t + \frac{\sigma^2}{2}S^2V_{SS} +r_Sv_S -rV=0,\quad 0< S<\infty, t<T , \label{eq-3.D.1}\\ &V(T, S )=K(S) .\label{eq-3.D.2} \end{align} Rewriting equation (\ref{eq-3.D.1}) as \begin{align*} &V_t + \frac{\sigma^2}{2}S(SV_S)_S +\bigl(r-\frac{\sigma^2}{2}\bigr) SV_S -rV=0,\ \end{align*} and changing variable $x=\ln (S)$ we get \begin{align*} &V_t + \frac{\sigma^2}{2}V_{xx} +\bigl(r-\frac{\sigma^2}{2}\bigr) V_x -rV=0, \end{align*} which can be rewritten as \begin{align*} &V_t + \frac{\sigma^2}{2}\bigl( \partial_x - a\bigr)^2 V -bV = 0, \end{align*} with constants $a,b$ and plugging $V=e^{bt} e^{ax} W$, we get \begin{align} &W_t + W_{xx} = 0, \qquad -\infty < x< \infty,\ t< T\\ &W(T, x)=L(x) \end{align} which is Cauchy problem for heat equation with the reverse time.
Exercise 1.
Using solution of the Cauchy problem for heat equation and using transformations described above, write a solutiob of (\ref{eq-3.D.1})-(\ref{eq-3.D.2}) in terms of $K(S)$.