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Information × Registration Number 0219U001808, 0113U007686 , R & D reports Title Development of numerical methods for solving the multidimensional nonlinear direct and inverse spectral problems and specific classes of integral equations arising in the wave field theory popup.stage_title Head Voytovych Mykola Mykolayovych, Savenko Petro Oleksandrovychної, Registration Date 05-03-2019 Organization Pidstryhach Institute of Applied Problems of Mechanics and Mathematics NAS of Ukraine popup.description2 Object of research: non-linear integral equations of Hammerstein type, direct and inverse multi-parameter spectral eigenvalues problems, direct and inverse problems of the wave fields theory. Goal of work: To expand a class of nonlinear integral equations, whose solutions are given by polynomials of finite degrees; to generalize the method of solving nonlinear two-parameter spectral problems for a wider class of nonlinear multi-parameter problems; to develop, substantiate and test the numerical methods and algorithms for solving special classes of inverse spectral problems; to generalize the previously proposed method for solving systems of specific two-dimensional integral equations for the case of bodies with azimuthal symmetry Main results: The polynomial approach to finding the solutions of non-linear integral equations of Hammerstein type is generalized to the case of equations with an arbitrary dependence on the module of an unknown function. Systems of integral-transcendental equations are obtained, and the existence of equivalent groups of solutions is established. The necessary conditions for branching points and the system of equations for their calculation are obtained. The conditions of convergence and correctness of methods for certain types of integral equations are established. Branching off the primary real solutions is investigated in the presence of singularities in the kernel of the integral operator. In the first approximation, analytical representations of complex solutions branched off a real solution were found, their main properties were determined. The method of implicit functions is generalized for solving the nonlinear multi-parameter problems, consisting of finding the spectrum of holomorphic operator-functions defined in Banach spaces. The application of the method of implicit functions to solving the nonlinear three-parameter spectral problems for integral holomorphic operator-functions is considered. The modification of Newton's method for finding the solutions to multi-parameter spectral linear and nonlinear problems is proposed. An algorithm that uses an algebraic (spectral) approach is developed, that is, eigenvalues are calculated as the roots of the corresponding characteristic equation of the problem without solving the initial characteristic equation. The analytical-numerical methods for solving the nonlinear integral equations that are used to solve the problem of electromagnetic scattering on small-sized bodies, inverse problems of reconstructing the form of the boundary of rotational body at resonant frequencies, determination of the body's shape according to its scattering diagram, and determination of dynamic stresses in the ring plates, are developed. Product Description popup.authors Андрійчук М.І. Булацик О.О. Войтович М.М. Заморська О.Ф. Кусий О.В. Подлевський Б.М. СінькевичО.О. Савенко П.О. Соляр Т.Я. Ткач М.Д. Тополюк Ю.М. popup.nrat_date 2020-04-02 Close