From anomalous to classical diffusion in a non-linear heat equation

2022/02/07 by Oscar Jarrín, Jarrin, Oscar, Geremy Loachamín +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Fractional Differential Equations Solutions #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2202.03503

openalex publication_date 2022/02/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we consider the heat equation with the natural polynomial non-linear term; and with two different cases in the diffusion term. The first case (anomalous diffusion) concerns the fractional Laplacian operator with parameter 1<α<2, while, the second case (classical diffusion) involves the classical Laplacian operator. When α→ 2, we prove the uniform convergence of the solutions of the anomalous diffusion case to a solution of the classical diffusion case. Moreover, we rigorous derive a convergence rate, which was experimentally exhibit in previous related works.

Related

• The mathematical theories of diffusion. Nonlinear and fractional diffusion
• Decay estimates in time for classical and anomalous diffusion
• Decay of mass for nonlinear equation with fractional Laplacian
• Rate of convergence in the large diffusion limit for the heat equation with a dynamical boundary condition
• Numerical simulations of anomalous diffusion