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Numerical method of bicharacteristics for quasilinear hyperbolic functional differential systems

100%
Kropielnicka K.
Commentationes Mathematicae
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2005
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tom 45
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nr 1
EN
Classical solutions of mixed problems for first order partial functional differential systems in two independent variables are approximated in the paper with solutions of a difference problem of the Euler type. The mesh for the approximate solutions is obtained by a numerical solving of equations of bicharacteristics. The convergence of explicit difference schemes is proved by means of consistency and stability arguments. It is assumed that given functions satisfy nonlinear estimates of the Perron type. Differential systems with deviated variables and differential integral systems can be obtained from a general model by specializing given operators.
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Comparison of explicit and implicit difference schemes for parabolic functional differential equations

63%
Kamont Z., Kropielnicka K.
Annales Polonici Mathematici
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2012
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tom 103
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nr 2
135-160
EN
Initial-boundary value problems of Dirichlet type for parabolic functional differential equations are considered. Explicit difference schemes of Euler type and implicit difference methods are investigated. The following theoretical aspects of the methods are presented. Sufficient conditions for the convergence of approximate solutions are given and comparisons of the methods are presented. It is proved that the assumptions on the regularity of the given functions are the same for both methods. It is shown that the conditions on the mesh for explicit difference schemes are more restrictive than the suitable assumptions for implicit methods. There are implicit difference schemes which are convergent while the corresponding explicit difference methods are not convergent. Error estimates for both methods are constructed.
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