On second order of accuracy difference scheme of the approximate solution of nonlocal elliptic-parabolic problems.

*(English)*Zbl 1198.65120Summary: A second order of accuracy difference scheme for the approximate solution of the abstract nonlocal boundary value problem \( - d^{2}u(t)/dt^{2}+Au(t)=g(t), (0\leq t\leq 1), du(t)/dt - Au(t)=f(t), ( - 1\leq t\leq 0), u(1)=u( - 1)+\mu \) 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.

##### MSC:

65L10 | Numerical solution of boundary value problems involving ordinary differential equations |

34G10 | Linear differential equations in abstract spaces |

35M13 | Initial-boundary value problems for PDEs of mixed type |

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

##### Keywords:

difference scheme; abstract nonlocal boundary value problem; Hilbert space; self-adjoint positive definite operator; well posedness; Hölder spaces; elliptic-parabolic equations; numerical example
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\textit{A. Ashyralyev} and \textit{O. Gercek}, Abstr. Appl. Anal. 2010, Article ID 705172, 17 p. (2010; Zbl 1198.65120)

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