Convergence of approximate solutions by heat kernel for transport-diffusion equation in a half-plane

In this paper, by using the heat kernel and the transport operator on each step of time discretization, approximate solutions for the transport-diffusion equation on the half-plane $ \mathbb{R}^2_+ $ are constructed, and their convergence to a function which satisfies the transport-diffusion equation and the initial and boundary conditions is proved. These approximate solutions can be considered as a deterministic version of (the approximation of) the stochastic representation of the solution to parabolic equation, realized by the relationship between the heat kernel and the Brownian motion. But as they are defined only by an integral operator and transport, their properties and their convergence are proved without using probabilistic notions. The result of this paper generalizes that of recent papers about the convergence of analogous approximate solutions on the whole space $ \mathbb{R}^n $. In case of the half-plane, it is necessary to elaborate (not trivial) estimates of the smoothness of the approximate solutions influenced by boundary condition.