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EN
Classical solutions of initial boundary value problems are approximated in the paper by solutions of implicit difference schemes. A convergence analysis for methods is presented. It is shown by an example that the new methods are considerably better than the explicit schemes. The proof of the stability is based on a theorem on difference inequalities. Nonlinear estimates of the Perron type for given functions with respect to functional variables are used. The results obtained in the paper can be applied to differential integral problems and to equations with deviated variables.
EN
Classical solutions of mixed problems for first order partialfunctional differential systems in two independent variables areapproximated in the paper with solutions of a difference problemof the Euler type. The mesh for the approximate solutions isobtained 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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