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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), (0t1), 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.
MSC:
65J10Equations with linear operators (numerical methods)
65M20Method of lines (IVP of PDE)
65L05Initial value problems for ODE (numerical methods)
65L12Finite difference methods for ODE (numerical methods)
35G10Initial value problems for linear higher-order PDE
35K25Higher order parabolic equations, general
34G10Linear ODE in abstract spaces