Well-Posedness of the Boundary Value Problem for Parabolic Equations in Difference Analogues of Spaces of Smooth Functions

A Ashyralyev

doi:10.1155/2007/90815

Abstract

The first and second orders of accuracy difference schemes for the approximate solutions of the nonlocal boundary value problemv′(t)+Av(t)=f(t)(0≤t≤1),v(0)=v(λ)+μ,0<λ≤1, for differential equation in an arbitrary Banach spaceEwith the strongly positive operatorAare considered. The well-posedness of these difference schemes in difference analogues of spaces of smooth functions is established. In applications, the coercive stability estimates for the solutions of difference schemes for the approximate solutions of the nonlocal boundary value problem for parabolic equation are obtained.

Highlights

For differential equation in an arbitrary Banach space E with the strongly positive operator A was established
Introduction: difference schemes It is known that many applied problems in fluid mechanics and other areas of physics and mathematical biology were formulated into nonlocal mathematical models
We consider the first order of accuracy implicit Rothe difference scheme uk

Summary

Introduction

For differential equation in an arbitrary Banach space E with the strongly positive operator A was established. This abstract result permits us to obtain the almost coercivity inequality and the coercive stability estimates for the solutions of difference schemes of the first and second orders of accuracy over time and of an arbitrary order of accuracy over space variables in the case of the nonlocal boundary value problem for the 2m-order multidimensional parabolic equation. The solutions of the difference schemes (1.3) and (1.4) in Cτβ,γ(E) (0 ≤ γ ≤ β, 0 < β < 1) obey the coercivity inequality τ−1 uk − uk−1

Results

Conclusion