International Journal of Applied Mechanics, Jun 1, 2016
A boundary value problem solution is presented to treatment the deformations of a closed flexible... more A boundary value problem solution is presented to treatment the deformations of a closed flexible elastic torso shell having perturbations at its axial edges. A so-called artificial parameter technique is applied to obtain a solution in the form of a double asymptotic series further summed using two-dimensional fractional rational approximations. Convergence of the approximations to the exact solution is proven.
The overwhelming majority of mathematical problems, describing realistic systems and processes, c... more The overwhelming majority of mathematical problems, describing realistic systems and processes, contain two parts: first, the problem needs to be characterized by an effective mathematical model and, second, the appropriate solutions are to be found.
There is no need to talk about the wide spreading of dynamic processes in nature, or about the in... more There is no need to talk about the wide spreading of dynamic processes in nature, or about the infrastructure created by man [...]
The idea of asymptotic approximation is one of the most important and profound in mathematics, es... more The idea of asymptotic approximation is one of the most important and profound in mathematics, especially in the parts of it those are in close contact with physics, mechanics, and engineering [...]
Asymptotic approaches for nonlinear dynamics of continual system are developed well for the infin... more Asymptotic approaches for nonlinear dynamics of continual system are developed well for the infinite in spatial variables. For the systems with finite sizes we have an infinite number of resonance, and Poincare-Lighthill-Go method does riot work. Using of averaging procedure or method of multiple scales leads to the infinite systems of nonlinear algebraic or ordinary differential equations systems and then using truncation method. which does not gives possibility to obtain all important properties of the solutions.
This short paper is devoted to analysis of contribution by Professor Pietraszkiewicz to the Theor... more This short paper is devoted to analysis of contribution by Professor Pietraszkiewicz to the Theory of Shells.
We consider the problem of analytic approximation of periodic Ateb functions, which are widely us... more We consider the problem of analytic approximation of periodic Ateb functions, which are widely used in nonlinear dynamics. Ateb functions are the result of the following procedure. Initial ordinary differential equation contains only the inertial and non-linear terms. Its integration leads to an implicit solution. To obtain explicit solutions one needs to invert incomplete Beta functions. As a result of this inversion we obtain the special Ateb functions. Their properties are well known, but the use of Ateb functions is difficult on practice. In this regard, the problem arises of approximation of Ateb functions with the help of smooth elementary functions. For this purpose in the present article the asymptotic method is used with a small parameter which is inverted to the exponent of nonlinearity. We also investigated the analytical approximation of Ateb functions’ period. Comparison of simulation results, obtained by the approximate expression, with the results of numerical solution of the corresponding ...
The main advantage of the homogenization method is simplicity of algorithms allowing for solving ... more The main advantage of the homogenization method is simplicity of algorithms allowing for solving the complicated problems. Besides the problems mentioned in our book, we can notice skewed [41], riveted [67], honeycomb [75,169], reticulated [76], multispan [82,88] and multilink [112,131,132], folded [151], fissued [136], grid [175], laminated [3,59,63,64,160,161] structures. Homogenization asymptotic applications bear new ideas for mathematicians. On the other hand, some well-developed branches of homogenization [108,193,195], are still waiting for applications.
International Journal of Applied Mechanics, Jun 1, 2016
A boundary value problem solution is presented to treatment the deformations of a closed flexible... more A boundary value problem solution is presented to treatment the deformations of a closed flexible elastic torso shell having perturbations at its axial edges. A so-called artificial parameter technique is applied to obtain a solution in the form of a double asymptotic series further summed using two-dimensional fractional rational approximations. Convergence of the approximations to the exact solution is proven.
The overwhelming majority of mathematical problems, describing realistic systems and processes, c... more The overwhelming majority of mathematical problems, describing realistic systems and processes, contain two parts: first, the problem needs to be characterized by an effective mathematical model and, second, the appropriate solutions are to be found.
There is no need to talk about the wide spreading of dynamic processes in nature, or about the in... more There is no need to talk about the wide spreading of dynamic processes in nature, or about the infrastructure created by man [...]
The idea of asymptotic approximation is one of the most important and profound in mathematics, es... more The idea of asymptotic approximation is one of the most important and profound in mathematics, especially in the parts of it those are in close contact with physics, mechanics, and engineering [...]
Asymptotic approaches for nonlinear dynamics of continual system are developed well for the infin... more Asymptotic approaches for nonlinear dynamics of continual system are developed well for the infinite in spatial variables. For the systems with finite sizes we have an infinite number of resonance, and Poincare-Lighthill-Go method does riot work. Using of averaging procedure or method of multiple scales leads to the infinite systems of nonlinear algebraic or ordinary differential equations systems and then using truncation method. which does not gives possibility to obtain all important properties of the solutions.
This short paper is devoted to analysis of contribution by Professor Pietraszkiewicz to the Theor... more This short paper is devoted to analysis of contribution by Professor Pietraszkiewicz to the Theory of Shells.
We consider the problem of analytic approximation of periodic Ateb functions, which are widely us... more We consider the problem of analytic approximation of periodic Ateb functions, which are widely used in nonlinear dynamics. Ateb functions are the result of the following procedure. Initial ordinary differential equation contains only the inertial and non-linear terms. Its integration leads to an implicit solution. To obtain explicit solutions one needs to invert incomplete Beta functions. As a result of this inversion we obtain the special Ateb functions. Their properties are well known, but the use of Ateb functions is difficult on practice. In this regard, the problem arises of approximation of Ateb functions with the help of smooth elementary functions. For this purpose in the present article the asymptotic method is used with a small parameter which is inverted to the exponent of nonlinearity. We also investigated the analytical approximation of Ateb functions’ period. Comparison of simulation results, obtained by the approximate expression, with the results of numerical solution of the corresponding ...
The main advantage of the homogenization method is simplicity of algorithms allowing for solving ... more The main advantage of the homogenization method is simplicity of algorithms allowing for solving the complicated problems. Besides the problems mentioned in our book, we can notice skewed [41], riveted [67], honeycomb [75,169], reticulated [76], multispan [82,88] and multilink [112,131,132], folded [151], fissued [136], grid [175], laminated [3,59,63,64,160,161] structures. Homogenization asymptotic applications bear new ideas for mathematicians. On the other hand, some well-developed branches of homogenization [108,193,195], are still waiting for applications.
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Papers by Igor Andrianov