On well-posedness of the nonlocal boundary value problem for parabolic difference equations

A Ashyralyev,P E Sobolevskii,I Karatay

doi:10.1155/s1026022604403033

Abstract

We consider the nonlocal boundary value problem for difference equations(uk−uk−1)/τ+Auk=φk,1≤k≤N,Nτ=1, andu0=u[λ/τ]+φ,0<λ≤1, in an arbitrary Banach spaceEwith the strongly positive operatorA. The well-posedness of this nonlocal boundary value problem for difference equations in various Banach spaces is studied. In applications, the stability and coercive stability estimates in Hölder norms for the solutions of the difference scheme of the mixed-type boundary value problems for the parabolic equations are obtained. Some results of numerical experiments are given.

Highlights

IntroductionIn [6], the coercive stability estimates in Holder norms for the solution of the nonlocal boundary value problem v (t) + Av(t) = f (t), 0 ≤ t ≤ 1, v(0) = v(λ) + φ, 0 < λ ≤ 1,
In [6], the coercive stability estimates in Holder norms for the solution of the nonlocal boundary value problem v (t) + Av(t) = f (t), 0 ≤ t ≤ 1, v(0) = v(λ) + φ, 0 < λ ≤ 1, (1.1)in arbitrary Banach space E with the strongly positive operator A were proved
In [5], the convergence estimates for the solution of first-order accuracy implicit Rothe difference scheme uk

Summary

Introduction

In [6], the coercive stability estimates in Holder norms for the solution of the nonlocal boundary value problem v (t) + Av(t) = f (t), 0 ≤ t ≤ 1, v(0) = v(λ) + φ, 0 < λ ≤ 1,. The exact Shauder’s estimates in Holder norms of solution of the boundary value problem on the range {0 ≤ t ≤ 1, x ∈ Rn} for 2m-order multidimensional parabolic equations were obtained. We are interested in studying the well-posedness of this difference nonlocal boundary value problem (1.2) in various Banach spaces. Applying the method of [9, 11, 14], the well-posedness of difference problem (1.2) in various Banach spaces is studied. The stability and coercive stability estimates in Holder norms for the solutions of the difference schemes of the mixed-type. Notice that the well-posedness of differential and difference equations of the parabolic type has been developed extensively, see, for instance, [4, 10, 12]

The estimate of stability

Ashyralyev et al 275

Numerical analysis We consider the nonlocal boundary value problem