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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.

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
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