Differentical equations, dynamical systems and optimal control

On the approximation of the solution of transport-diffusion equation with a non-constant coefficient of diffusion

A. Nemdili, H. Fujita Yashima

Lab. Mathématiques appliquées et didactiq, École Normale Supérieure de Constantine, Ali Mendjeli, 25000, Constantine, Algeria

Abstract: The transport-diffusion equation with a non-constant diffusion coefficient in the whole space $\mathbb{R}^d $ is considered and a family of approximate solutions is defined by using the fundamental solution of the heat equation (heat kernel) and the translation corresponding to transport on each step of time discretization. Under appropriate conditions on the regularity of the data, the uniform convergence of approximate solutions to a function which satisfies the transport-diffusion equation is proved. To estimate and to prove the convergence of approximate solutions, we first estimate and prove the convergence of the “positions” with respect to which we apply the integral operator with the heat kernel. We also improve the convergence of the time derivative of approximate solutions.

Keywords: transport-diffusion equation, non-constant diffusion coefficient, approximation by time discretization, heat kernel.

UDC: 517.956.4

MSC: 35K58

Received June 7, 2024, published November 8, 2024

DOI: 10.33048/semi.2024.21.070