Ukr. Mat. Zh. - 2013. - 65, № 10. - pp. 1363–1387
We give a theorem on the estimation of error for approximate solutions to ordinary functional differential equations. The error is estimated by a solution of an initial problem for a nonlinear functional differential equation. We apply this general result to the investigation of convergence of the numerical method of lines for evolution functional differential equations. The initial boundary-value problems for quasilinear equations are transformed (by means of discretization in spatial variables) into systems of ordinary functional differential equations. Nonlinear estimates of the Perron-type with respect to functional variables for given operators are assumed. Numerical examples are given.
Ukr. Mat. Zh. - 2003. - 55, № 12. - pp. 1678-1796
We consider initial-value problems for infinite systems of first-order partial functional differential equations. The unknown function is the functional argument in equations and the partial derivations appear in the classical sense. A theorem on the existence of a solution and its continuous dependence upon initial data is proved. The Cauchy problem is transformed into a system of functional integral equations. The existence of a solution of this system is proved by using integral inequalities and the iterative method. Infinite differential systems with deviated argument and differential integral systems can be derived from the general model by specializing given operators.
Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 985–996
We consider initial boundary-value problems of Dirichlet type for nonlinear equations. We give sufficient conditions for the convergence of a general class of one-step difference methods. We assume that the right-hand side of the equation satisfies an estimate of Perron type with respect to the functional argument.