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**Well-posed solvability of the Cauchy problem for difference equations of parabolic type.**
*(English)*
Zbl 0818.65047

Various boundary value problems for evolutionary partial differential equations can be reduced to the abstract Cauchy problem (*) \(v'(t) + Av(t) = f(t)\), \((0 \leq t \leq 1)\), \(v(0) = v_ 0\), in a Banach space, where \(A\) is an unbounded strongly positive operator. In this paper the authors investigate solvability of the Padé difference scheme for approximately solving the abstract Cauchy problem (*). The study is based on coercive inequalities for the solutions of this difference scheme because the existence of these inequalities is equivalent to the natural well-posed solvability of the difference scheme.

Reviewer: P.Talpalaru (Iaşi)

### MSC:

65J10 | Numerical solutions to equations with linear operators |

65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |

65L05 | Numerical methods for initial value problems involving ordinary differential equations |

65L12 | Finite difference and finite volume methods for ordinary differential equations |

35G10 | Initial value problems for linear higher-order PDEs |

35K25 | Higher-order parabolic equations |

34G10 | Linear differential equations in abstract spaces |

### Keywords:

evolutionary partial differential equations; abstract Cauchy problem; Banach space; unbounded strongly positive operator; difference scheme; coercive inequalities; well-posed solvability
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\textit{A. Ashyralyev} and \textit{P. E. Sobolevskij}, Nonlinear Anal., Theory Methods Appl. 24, No. 2, 257--264 (1995; Zbl 0818.65047)

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### References:

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