A nonlocal diffusion problem that approximates the heat equation with Neumann boundary conditions

• Main Authors: Cesar A. Gómez, Julio D. Rossi
• Format: Article
• Language: English
• Published: Elsevier 2020-01-01
• Series: Journal of King Saud University: Science
• Online Access: http://www.sciencedirect.com/science/article/pii/S1018364717307887

In this paper we discuss a nonlocal approximation to the classical heat equation with Neumann boundary conditions. We considerwt∊(x,t)=1∊N+2∫ΩJx-y∊(w∊(y,t)-w∊(x,t))dy+C1∊N∫∂ΩJx-y∊g(y,t)dSy,(x,t)∈Ω‾×(0,T),w(x,0)=u0(x),x∈Ω‾,and we show that the corresponding solutions, w∊, converge to the classical solution of the local heat equation vt=Δv with Neumann boundary conditions, ∂v∂n(x,t)=g(x,t), and initial condition v(0)=u0, as the parameter ∊ goes to zero. The obtained convergence is in the weak star on L∞ topology. Keywords: Nonlocal diffusion, Neumann boundary conditions, Heat equation, 2010 Mathematics Subject Classification: 45A05, 45J05, 35K05