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Information × Registration Number 0218U006179, 0112U007388 , R & D reports Title Mathematical modeling of autooscillatory and autowave processes on the basis of evolutionary systems with fractional derivatives. popup.stage_title Head Datsko Bohdan Josypovych, Registration Date 27-03-2018 Organization Pidstryhach Institute of Applied Problems of Mechanics and Mathematics NAS of Ukraine popup.description2 The object of research: natural systems with heriditary properties and anomalous diffusion, nonlinear mathematical models with differential operators of fractional order, auto-oscillation and auto-wave systems with fractional derivatives. The purpose of the work is to construct mathematical models and to develop methods for the study of auto-oscillation and auto-wave processes in dissipative non-equilibrium media with complex structure. In particular: developing of a theory of linear stability; investigation of the conditions for the emergence of different types of instabilities and solutions in auto-oscillation and auto-wave systems with fractional derivatives; developing of new approaches for analytical and numerical research of basic and applied mathematical models with differential fractional order operators. Research method: Construction and research of mathematical models using methods of mathematical physics, functional analysis, nonequilibrium thermodynamics, numerical methods of mathematical physics and fractional calculus. During the project implementation, a general theory of stability for auto-oscillation and auto-wave systems with derivatives of fractional order was constructed. The conditions of occurrence of different types of nonlinear dynamics and boundary cycles, depending on the system parameters and the relation between the orders of fractional derivatives, are investigated. It is established that the orders of fractional derivatives and the relation between them can significantly change the stability conditions and the dynamics of nonlinear systems of fractional order. New classes of nonlinear auto-wave solutions and new types of boundary cycles that do not exist in the corresponding systems of the integer order are found. On the basis of small parameter methods, spatially inhomogeneous solutions in the vicinity of the bifurcation point in reaction- diffusions systems with classical and fractional derivatives are investigated. The spatial-temporal evolution of nonlinear solutions in such systems is studied, depending on the type of nonlinearity and the state of nonequilibrium of the system. Complex space-time solutions in nonlinear reaction-diffusion systems with fractional time derivatives in the stability domain of homogeneous solutions are investigated. Under conditions of subcritical bifurcation, new types of auto-wave solutions are found. Qualitative new mathematical models of long-term memory and model of educational process are developed. It is shown that the mathematical description of the dynamics of human memory requires the formalism of derivatives of fractional order. The results of the research can be used for the further development of mathematical models in the scientific institutes of the National Academy of Sciences of Ukraine. Product Description popup.authors Васюник З.І. Дацко Б.Й. Мелешко В.В. Павлюк В.С. popup.nrat_date 2020-04-02 Close