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.

scholarly journals Asymptotics solutions of a singularly perturbed integro-differential fractional order derivative equation with rapidly oscillating coefficients

2021 ◽  
Vol 104 (4) ◽  
pp. 56-67
Author(s):  
M.A. Bobodzhanova ◽  
◽  
B.T. Kalimbetov ◽  
G.M. Bekmakhanbet ◽  
◽  
...  
Keyword(s):  
Fractional Order ◽  
Regularization Method ◽  
Careful Analysis ◽  
Singularly Perturbed ◽  
Order Derivative ◽  
Time Interval ◽  
Resonance Case ◽  
Fractional Order Derivative ◽  
Oscillating Coefficients ◽  
Complex Cases

In this paper, the regularization method of S.A.Lomov is generalized to the singularly perturbed integrodifferential fractional-order derivative equation with rapidly oscillating coefficients. The main goal of the work is to reveal the influence of the oscillating components on the structure of the asymptotics of the solution to this problem. The case of the absence of resonance is considered, i.e. the case when an integer linear combination of a rapidly oscillating inhomogeneity does not coincide with a point in the spectrum of the limiting operator at all points of the considered time interval. The case of coincidence of the frequency of a rapidly oscillating inhomogeneity with a point in the spectrum of the limiting operator is called the resonance case. This case is supposed to be studied in our subsequent works. More complex cases of resonance (for example, point resonance) require more careful analysis and are not considered in this work.

scholarly journals Regularization Method for Singularly Perturbed Integro-Differential Equations with Rapidly Oscillating Coefficients and Rapidly Changing Kernels

Axioms ◽  
2020 ◽  
Vol 9 (4) ◽  
pp. 131
Author(s):  
Burkhan Kalimbetov ◽  
Valeriy Safonov
Keyword(s):  
Asymptotic Behavior ◽  
Integral Operator ◽  
Differential Equations ◽  
Regularization Method ◽  
Original Problem ◽  
Singularly Perturbed ◽  
Integral Term ◽  
Oscillating Coefficients

In this paper, we consider a system with rapidly oscillating coefficients, which includes an integral operator with an exponentially varying kernel. The main goal of the work is to develop an algorithm for the regularization method for such systems and to identify the influence of the integral term on the asymptotic behavior of the solution of the original problem.

Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity

2021 ◽  
Vol 19 (1) ◽  
pp. 244-258
Author(s):  
Burkhan T. Kalimbetov ◽  
Olim D. Tuychiev
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Asymptotic Solution ◽  
Regularization Method ◽  
Singularly Perturbed ◽  
New Approach ◽  
The Cauchy Problem ◽  
Oscillating Coefficients ◽  
Oscillating Coefficient ◽  
The Relationship

Abstract In this paper, the regularization method of S. A. Lomov is generalized to integro-differential equations with rapidly oscillating coefficients and with a rapidly oscillating right-hand side. The main goal of the work is to reveal the influence of the oscillating components on the structure of the asymptotics of the solution of this problem. The case of coincidence of the frequencies of a rapidly oscillating coefficient and a rapidly oscillating inhomogeneity is considered. In this case, only the identical resonance is observed in the problem. Other cases of the relationship between frequencies can lead to so-called non-identical resonances, the study of which is nontrivial and requires the development of a new approach. It is supposed to study these cases in our further work.

scholarly journals Generalization of the Regularization Method to Singularly Perturbed Integro-Differential Systems of Equations with Rapidly Oscillating Inhomogeneity

Axioms ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 40
Author(s):  
Abdukhafiz Bobodzhanov ◽  
Burkhan Kalimbetov ◽  
Valeriy Safonov
Keyword(s):  
Integral Operator ◽  
Differential Equations ◽  
Unknown Function ◽  
Regularization Method ◽  
Original Problem ◽  
Singularly Perturbed ◽  
Systems Of Equations ◽  
Differential Systems ◽  
Right Hand ◽  
Oscillating Coefficient

In this paper, we consider systems of singularly perturbed integro-differential equations with a rapidly oscillating right-hand side, including an integral operator with a slowly varying kernel. Differential equations of this type and integro-differential equations with slowly varying inhomogeneity and with a rapidly oscillating coefficient at an unknown function are studied. The main goal of this work is to generalize the Lomov’s regularization method and to reveal the influence of the rapidly oscillating right-hand side on the asymptotics of the solution to the original problem.

scholarly journals Asymptotic Solutions of Scalar Integro-Differential Equations with Partial Derivatives and with Fast Oscillating Coefficients

2020 ◽  
Vol 13 (2) ◽  
pp. 287-302
Author(s):  
Burkhan Kalimbetov ◽  
Akisher Temirbekov ◽  
Abdimuhan Tolep
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Unique Solvability ◽  
Regularization Method ◽  
Singularly Perturbed ◽  
Integral Term ◽  
Integro Differential Equation ◽  
The Cauchy Problem ◽  
Oscillating Coefficients

In the paper, ideas of the Lomov regularization method are generalized to the Cauchy problem for a singularly perturbed partial integro-differential equation in the case when the integral term contains a rapidly varying kernel. Regularization of the problem is carried out, the normal and unique solvability of general iterative problems is proved.

scholarly journals Null Controllability of a Nonlinear Age Structured Model for a Two-Sex Population

2021 ◽  
Vol 2021 ◽  
pp. 1-20
Author(s):  
Amidou Traoré ◽  
Okana S. Sougué ◽  
Yacouba Simporé ◽  
Oumar Traoré
Keyword(s):  
Null Controllability ◽  
Auxiliary System ◽  
Time Interval ◽  
Observability Inequality ◽  
Controllability Problem ◽  
Structured Model ◽  
Male And Female ◽  
System A ◽  
Age Structured ◽  
Age Structured Model

This paper is devoted to study the null controllability properties of a nonlinear age and two-sex population dynamics structured model without spatial structure. Here, the nonlinearity and the couplage are at the birth level. In this work, we consider two cases of null controllability problem. The first problem is related to the extinction of male and female subpopulation density. The second case concerns the null controllability of male or female subpopulation individuals. In both cases, if A is the maximal age, a time interval of duration A after the extinction of males or females, one must get the total extinction of the population. Our method uses first an observability inequality related to the adjoint of an auxiliary system, a null controllability of the linear auxiliary system, and after Kakutani’s fixed-point theorem.

scholarly journals A New Accurate Numerical Method Based on Shifted Chebyshev Series for Nuclear Reactor Dynamical Systems

2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
Yasser Mohamed Hamada
Keyword(s):  
Nuclear Reactor ◽  
Approximate Solutions ◽  
Time Interval ◽  
Neutron Density ◽  
Chebyshev Series ◽  
System A ◽  
Temperature Feedback ◽  
Exponential Rate Of Convergence ◽  
Point Kinetics ◽  
The Given

A new method based on shifted Chebyshev series of the first kind is introduced to solve stiff linear/nonlinear systems of the point kinetics equations. The total time interval is divided into equal step sizes to provide approximate solutions. The approximate solutions require determination of the series coefficients at each step. These coefficients can be determined by equating the high derivatives of the Chebyshev series with those obtained by the given system. A new recurrence relation is introduced to determine the series coefficients. A special transformation is applied on the independent variable to map the classical range of the Chebyshev series from [-1,1] to [0,h]. The method deals with the Chebyshev series as a finite difference method not as a spectral method. Stability of the method is discussed and it has proved that the method has an exponential rate of convergence. The method is applied to solve different problems of the point kinetics equations including step, ramp, and sinusoidal reactivities. Also, when the reactivity is dependent on the neutron density and step insertion with Newtonian temperature feedback reactivity and thermal hydraulics feedback are tested. Comparisons with the analytical and numerical methods confirm the validity and accuracy of the method.

scholarly journals Numerical Solution of Singularly Perturbed Delay Differential Equations with Layer Behavior

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
F. Ghomanjani ◽  
A. Kılıçman ◽  
F. Akhavan Ghassabzade
Keyword(s):  
Boundary Value Problems ◽  
Difference Equations ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Time Interval ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Approximation Process ◽  
Negative Shift ◽  
Control Functions

We present a numerical method to solve boundary value problems (BVPs) for singularly perturbed differential-difference equations with negative shift. In recent papers, the term negative shift has been used for delay. The Bezier curves method can solve boundary value problems for singularly perturbed differential-difference equations. The approximation process is done in two steps. First we divide the time interval, intoksubintervals; second we approximate the trajectory and control functions in each subinterval by Bezier curves. We have chosen the Bezier curves as piecewise polynomials of degreenand determined Bezier curves on any subinterval byn+1control points. The proposed method is simple and computationally advantageous. Several numerical examples are solved using the presented method; we compared the computed result with exact solution and plotted the graphs of the solution of the problems.

An Investigation of High Pressure Coolant Injection (HPCI) Line Dynamic Transient at Dresden Nuclear Generating Station

2002 ◽  
Author(s):  
Jeff Drowley ◽  
Kevin Ramsden
Keyword(s):  
High Pressure ◽  
Time Interval ◽  
Hydraulic Behavior ◽  
System A ◽  
Coolant Injection ◽  
Analytical Work ◽  
Injection Line ◽  
Valve Opening ◽  
Plant Data ◽  
High Pressure Coolant

The purpose of this paper is to discuss the thermal hydraulic behavior and related dynamic response of the Dresden Unit 3 HPCI (High Pressure Coolant Injection) Injection piping. A hydraulic transient occurred on the HPCI injection line piping, that resulted in damge to piping support, pulling the anchors out of the wall. The ensuing investigation following the discovery of the failed support suspected a possible waterhammer event, but walkdowns of the system did not provide much evidence of a propagating disturbance in the system. A review of plant data subsequent to the July 5, 2001 trip showed that during an automatic HPCI injection initiation, a pressure transient occurred, with the potential to produce loads on the piping and supports in excess of their designed capacity. The ensuing investigation determined that non-condensable was present at two locations within the system. It was also determined that the temperatures of the piping downstream of the HPCI injection valve were sufficiently high to introduce some amount of steam to the upper portions of the HPCI line during the time interval between the injection valve opening and the HPCI pump reaching pressure. This paper will discuss the analytical work performed to investigate this transient, along with the piping forces calculated for various scenarios considered.

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