On Second Order of Accuracy Difference Scheme of the Approximate Solution of Nonlocal Elliptic-Parabolic Problems

Abstract

A second order of accuracy difference scheme for the approximate solution of the abstract nonlocal boundary value problem $-d^{2}u\left(t\right)/d{t}^2+Au\left(t\right)=g\left(t\right)$, $\left(0\le t\le 1\right)$, $du\left(t\right)/dt-Au\left(t\right)=f\left(t\right)$, $\left(-1\le t\le 0\right)$, $u\left(1\right)=u\left(-1\right)+\mu$ for differential equations in a Hilbert space $H$ with a self-adjoint positive definite operator $A$ is considered. The well posedness of this difference scheme in Hölder spaces is established. In applications, coercivity inequalities for the solution of a difference scheme for elliptic-parabolic equations are obtained and a numerical example is presented.