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Zbl 0818.65047
Ashyralyev, Allaberen; Sobolevskij, Pavel Evseyevich
(Ashyralev, A.)
Well-posed solvability of the Cauchy problem for difference equations of parabolic type.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 24, No.2, 257-264 (1995). ISSN 0362-546X

Various boundary value problems for evolutionary partial differential equations can be reduced to the abstract Cauchy problem (*) $v'(t) + Av(t) = f(t)$, $(0 \leq t \leq 1)$, $v(0) = v\sb 0$, 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.
[P.Talpalaru (Iaşi)]
MSC 2000:
*65J10 Equations with linear operators (numerical methods)
65M20 Method of lines (IVP of PDE)
65L05 Initial value problems for ODE (numerical methods)
65L12 Finite difference methods for ODE
35G10 Initial value problems for linear higher-order PDE
35K25 Higher order parabolic equations, general
34G10 Linear ODE 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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