The positivity of the difference operator with periodic conditions and its application

Allaberen Ashyralyev, Fatih Sabahattin Tetikoğlu

2016

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DOI 10.1063/1.4959661
Data confidence 70 medium
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Abstract

In this study, the second order of approximation of the difference operator Ahx with domain which approximates the second order differential dependent operator Ax defined by the formula Axu=−a(x)uxx(x)+δu(x),δ≥0,a(x)=a(x+2π),x∈ℝ1 with domain D(Ax)={u(x):u(x),u′(x),u″(x)∈C(ℝ1),u(x)=u(x+2π),x∈ℝ1,∫02πu(x)dx=0} is presented. The positivity of operator Ahx with domain in the Banach space C(ℝ1h) is established. It is proved that for any α∈(0,12) the norms in spaces Ea=Ea(C(ℝ1h),Ahx) and C2a(ℝ1h) are equivalent uniformly with respect to h. The positivity of the operator Ahx with domain in Hölder spaces of C∘2a(ℝ1h), α∈(0,12) is proved. In application, theorems on well-posedness for difference schemes of the approximate solution of the boundary value problem for elliptic differential equations are presented.