scholarly journals Asymptotic solutions of singularly perturbed integro-differential systems with rapidly oscillating coefficients in the case of a simple spectrum

2021 ◽  
Vol 6 (8) ◽  
pp. 8835-8853
Author(s):  
Abdukhafiz Bobodzhanov ◽  
◽  
Burkhan Kalimbetov ◽  
Valeriy Safonov ◽  
Keyword(s):  
Singularly Perturbed ◽  
Asymptotic Solutions ◽  
Simple Spectrum ◽  
Differential Systems ◽  
Oscillating Coefficients

scholarly journals Singularly Perturbed Integro-Differential Equations With Rapidly Oscillating Coefficients and With Rapidly Changing Kernel in the Case of a Multiple Spectrum

2021 ◽  
Vol 20 ◽  
pp. 84-96
Author(s):  
Burkhan Kalimbetov ◽  
Valery Safonov
Keyword(s):  
Integral Operator ◽  
Careful Analysis ◽  
Singularly Perturbed ◽  
Time Interval ◽  
Simple Spectrum ◽  
Integral Term ◽  
System A ◽  
Oscillating Coefficients ◽  
Oscillating Coefficient ◽  
Complex Cases

The paper investigates a system with rapidly oscillating coefficients and with a rapidly decreasing kernel of the integral operator. Previously, only differential problems of this type were studied in which the integral term was absent. The presence of an integral operator significantly affects the development of an algorithm for asymptotic solutions, for the implementation of which it is necessary to take into account essentially singularities generated by the rapidly decreasing spectral value of the kernel of the integral operator. In addition, resonances can arise in the problem under consideration (i.e., the case can be realized when an integer linear combination of the eigenvalues of the rapidly oscillating coefficient coincides with the points of the spectrum of the limiting operator over the entire considered time interval), as well as the case where the eigenvalue of the rapidly oscillating coefficient coincides with the points spectrum of the limiting operator. This case generates a multiple spectrum of the original singularly perturbed integro-differential system. A similar problem was previously considered in the case of a simple spectrum. More complex cases of resonance (for example, point resonance) require more careful analysis and are not considered in this article.

Efficient decomposition of singularly perturbed systems

2019 ◽  
Vol 14 (4) ◽  
pp. 410 ◽  
Author(s):  
Vladimir Sobolev
Keyword(s):  
Enzyme Kinetics ◽  
Singularly Perturbed ◽  
Singularly Perturbed Systems ◽  
Differential Systems ◽  
Integral Manifolds ◽  
Perturbed Systems ◽  
Method Of Integral Manifolds

The problem of the decomposition of singularly perturbed differential systems by the method of integral manifolds is studied and the application of the method to the problems of enzyme kinetics is considered.

scholarly journals REDUCTION OF MATHEMATICAL MODEL OF ROBOT WITH ELASTIC JOINTS

2017 ◽  
Vol 20 (3) ◽  
pp. 20-33
Author(s):  
O.V. Vidilina ◽  
N.V. Voropaeva
Keyword(s):  
Mathematical Model ◽  
Existence And Uniqueness ◽  
Integral Manifold ◽  
Singularly Perturbed ◽  
Slow Movement ◽  
Parameter System ◽  
Asymptotic Decomposition ◽  
Differential Systems ◽  
Small Dissipation ◽  
Elastic Joints

A model of n - joint manipulator with elastic joints with small dissipation is studied. Class of singularly perturbed differential systems that describe the dynamics of robot is singled out. For a given class of systems the existence and uniqueness of integral manifoldness of slow movement is established, its features are studied. It is proved that integral manifold may be constructed with any degree of accuracy as asymptotic decomposition inpowers of small parameter. System that is used to describe movement in manifolds may be used as a reduced model of initial system.

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