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Information × Registration Number 0220U000276, 0117U001228 , R & D reports Title Development of research methods correctness direct and inverse problems for differential operators popup.stage_title Head Bokalo Mykola Mikhailovych, Registration Date 27-01-2020 Organization Ivan Franko National University of Lviv popup.description2 Research object: inverse problems and free-boundary problems for parabolic and hyperbolic equations with unknown coefficients, problems for equations with fractional derivatives, problems for nonlinear parabolic and elliptical-parabolic equations and their systems, problems for nonlinear integro-differential degenerate parabolic equations, systems of equations and variational inequalities with variable nonlinearity indicators, the family of Schr?dinger operators with sufficiently smooth potentials. Purpose: development of research methods of inverse problems for two-dimensional parabolic equations, direct and inverse free boundary problems for hyperbolic equations and systems, inverse problems for equations with fractional derivatives in temporal and spatial variables, problems for nonlinear parabolic and elliptical parabolic equation and their systems, building accurate models of quantum mechanics. Research methods: Green's function method, integral equation method, a priori estimation method, Green's vector function and Fourier series, Gallorkin's method, monotonicity and compactness method and methods of differential equation theory and functional analysis; coupled Green operator method, linear abstract operator theory and self-adjoint extension theory, Vyshik-Mirror method, elliptic regularization method, Faedo-Gallorkin method, monotonicity and compactness method, characteristics method and method of their reduction to integro-operator equations, the principle of fixed point compression, the method of successive approximations. Results obtained: conditions for existence and uniqueness of solution of inverse problems for two-dimensional equations with unknown higher coefficients, dependent on both temporal and spatial variables, were found; the dependence of the uniquely solvability of the inverse free boundary problem for a one-dimensional parabolic equation with degenerating at the initial moment of time on the nature of degeneration of the equation and boundary is established; conditions for uniquely solvable inverse problems were found for linear and semilinear diffusion equations with fractional derivatives of temporal and spatial variables in spaces of smooth and generalized functions; the conditions for the existence of solutions of problems for parabolic equations and variational inequalities with variable nonlinearities in ordinary and generalized Lebesgue and Sobolev spaces are found; conditions of local and global solvability in the generalized and classical meanings of nonlinear hyperbolic problems are established; structured population models that are described by problems for hyperbolic equations and systems are investigated. Product Description popup.authors Іванчов Микола Іванович Ільницька Ольга Володимирівна Андрусяк Руслан Васильович Бокало Микола Михайлович Бугрій Олег Миколайович Головатий Юрій Данилович Гузик Надія Миколаївна Кирилич Володимир Михайлович Лопушанська Галина Петрівна Пелюшкевич Ольга Володимирівна Скіра Ірина Володимирівна Цебенко Андрій Миколайович popup.nrat_date 2020-04-02 Close