Convergence of Approximate Solutions for the Transport‐Diffusion Equation With Time and Space‐Dependent Diffusion Coefficient

In this paper, we study the transport‐diffusion equation in with a diffusion coefficient depending on both the position and time . We construct a family of approximate solutions using convolutions with the heat kernel and translations associated with the velocity field over discrete time steps. Under suitable regularity and boundedness assumptions on the given functions, we establish the uniform convergence of these approximate solutions to the solution of the transport‐diffusion equation. Furthermore, we prove the uniform convergence of the spatial derivatives of the approximate solutions to the corresponding derivatives of the solution. This method generalizes previous results obtained for constant, space‐dependent, and time‐dependent diffusion coefficients.