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 solvabilityReferences:
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