Дифференциальные уравнения, динамические системы и оптимальное управление

О приближении решения уравнения переноса-диффузии с непостоянным коэффициентом диффузии

А. Немдили, Х. Фуджита Яшима

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

Аннотация: 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.

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

УДК: 517.956.4

MSC: 35K58

Поступила 7 июня 2024 г., опубликована 8 ноября 2024 г.

DOI: 10.33048/semi.2024.21.070