Approximation by the heat kernel of the solution to the transport-diffusion equation with the time-dependent diffusion coefficient

In this paper, we examined the transport-diffusion equation in ${\mathbb{R}}^d$, where the diffusion is represented by the Laplace operator multiplied by a function $\kappa (t)$ dependent on time. We transformed the equation using the inverse function of $s(t)={\int }_0^t\kappa (t^{\prime })d{t}^{\prime }$. This transformation allowed us to construct a family of approximate solutions by using the heat kernel and translation corresponding to the transport in each step of time discretization. We proved the uniform convergence of these approximate solutions and their first and second derivatives with respect to the spatial variables. We also showed that the limit function satisfies the transport-diffusion equation in the space ${\mathbb{R}}^d$.