Semidiscretization for a nonlocal parabolic problem

Abderrahmane El Hachimi,Moulay Rchid Sidi Ammi

doi:10.1155/ijmms.2005.1655

Abstract

A time discretization technique by Euler forward scheme is proposed to deal with a nonlocal parabolic problem. Existence and uniqueness of the approximate solution are proved.

Highlights

In this work, we study the time discretization by Euler forward scheme of the nonlocal initial boundary value problem ∂u ∂t −u=λ f (u) Ω f (u)dx 2 in Ω × ]0; T[, u = 0 on ∂Ω × ]0; T[, u(0) = u0 in Ω, (1.1)with Ω ⊂ Rd (d ≥ 1) a bounded regular domain and λ a positive parameter
We study the time discretization by Euler forward scheme of the nonlocal initial boundary value problem u=λ f (u) Ω f (u)dx 2 in Ω × ]0; T[, u = 0 on ∂Ω × ]0; T[, u(0) = u0 in Ω, (1.1)
Where u represents the temperature generated by the electric current flowing through a conductor, φ the electric potential, σ(u) and k(u) are, respectively, the electric and thermal conductivities

Summary

Introduction

We study the time discretization by Euler forward scheme of the nonlocal initial boundary value problem. We recall that the Euler forward method was used by several authors to treat semidiscretization of nonlinear parabolic problems, see [3, 4]. Concerning problem (1.1), results of existence and uniqueness of solutions are known under particular forms of f , we refer to [2] and the references therein. In [1], the authors derived an L2 and H1-norm error by requiring more regularity on the solution u, for instance u, ut in H2(Ω) ∩ W1,∞(Ω). Such smoothness is not always possible since the function f is nonlinear

The semidiscrete problem

Stability

Error estimates for solutions