The following equation is a solution of a convection-diffusion problem. I want to prove that the resultant solution is simplified to the initial condition when $\Delta t$ approaches zero. Consequently, I want to prove the following integral identity.
$$\lim_{\Delta t\ \rightarrow\ 0}\frac{1}{2 \Delta t} \int_{0}^\infty x\ F(x)\ e^ {-\frac{x^2 + r^2}{4 \Delta t}}\ I_0(\frac{x r}{2 \Delta t})\ {d}x = F(r)$$
where $F(r)$ is the initial solution of the problem whose solution is displayed by the LHS of the previous equation. $I_0$ is modified Bessel function of the first kind of zeroth order.