- Menshikov, Yu., Synthesis of Adequate Mathematical Description as Solution of Special Inverse Problems. EUROPEAN JOURNAL OF MATHEMATICAL SCIENCES, Vol. 2, No. 3, 2013, 256-271.
The problem of mathematical simulation of dynamic system characteristics behavior and their adequacy to real experimental data, which correspond to these characteristics, is considered in this paper. The specified problem is still poorly investigated and hardly adapted to formalization. The requirements of related to the adequate mathematical simulation of dynamic system are considered for the case when mathematical description is represented by system of the ordinary differential equations. The conditions are obtained which allow to reduce a problem of the adequate mathematical description to the solution of the several integral equations of the first type. The methods of obtaining of the steady solutions are suggested. The domains of application of the obtained solutions are specified. For a case, when the differential equations of dynamic system are given with errors in coefficients, several variants of synthesis of the adequate mathematical descriptions depending on final goals of this description use are suggested. The examples of the adequate descriptions of concrete dynamic systems are given.
- Menshikov, Yu., Inverse Problem of Astrodynamics. World Journal of Mechanics, 5, 249-256. DOI: 10.4236/wjm.2015.512023
We consider the problem of determining the center of mass of an unknown gravitational body, using the disturbances in the motion of observed celestial bodies. In this paper an universal approach to obtain the approximate and stable estimate of problem solution is suggested. This approach can be used in other fields of Science. For example, it can be applied for investigation of interactions between fields of forces and elementary particles using known trajectories of elementary particles motions.
- Menshikov, Yu.L., One approach to solutions of measurement’s inverse problems. J. Mathematical Inverse Problems, Vol.1, No.2, 2014, USA., р.71-85.
The investigation of approximate solutions of inverse problems is given in work. For obtaining of the useful information about the exact solution of an inverse problem of measurement a special hypothesis is offered. Two practical inverse problems of measurement are considered where the this hypothesis is used: inverse problem of Le Verrier and identification of unbalance characteristics of rotor. For obtaining of stable solutions of these problems a various statements have been considered. Numerical calculation of real problems with application of regularization method is performed.
- Menshikov, Yu., Features of Parameters Identification of Algebraic Mathematical Models. International Journal of Engineering and Innovative Technology (IJEIT) Volume 3, Issue 11, May 2014.
The conditions related to algorithms of identification of parameters of mathematical models of physical processes in algebraic form is discussed in the paper. The features of these algorithms were shown. One possible approach for solutions this problem was suggested in determinate statement. On real observations were obtained calculations economical characteristics of Ukraine as example.
- Menshikov, Yu., Inverse Problems in Non-classical Statements. Int. Journal of Pure and Applied Mathematics, v. 67, №1,2011, p.79-96.
The inverse problems which cannot be solved in the classical framework are investigated in this article. They are as follows: Krylov inverse problem, early diagnostics of a rotor unbalance, the most probable solution of inverse problem. For obtaining the steady solutions of these problems some algorithms based on the method of Tikhonov regularization are offered. Krylov inverse problem in various statements has been considered and numerical calculation based on real measurements has been executed. Nonclassical statements of inverse problems extend of regularization method possibilities.
- Menshikov, Yu., Uncontrollable Inaccuracy in Inverse Problems. World Academy of Science, Engineering and Technology Vol:9, 2015-09-02, International Science Index Vol:9, No:8, 2015 waset.org/Publication/10002513.
In this paper the influence of errors of function derivatives in initial time which have been obtained by experiment (uncontrollable inaccuracy) to the results of inverse problem solution was investigated. It was shown that these errors distort the inverse problem solution as a rule near the beginning of interval where the solutions are analyzed. Several methods for removing the influence of uncontrollable inaccuracy have been suggested.
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