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.

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 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.

scholarly journals On the solvability of the modified Cauchy problem for linear systems of generalized ordinary differential equations with singularities

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Malkhaz Ashordia ◽  
Inga Gabisonia ◽  
Mzia Talakhadze
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Systems ◽  
Unique Solvability ◽  
Sufficient Conditions ◽  
The Cauchy Problem ◽  
Generalized Ordinary Differential Equations

AbstractEffective sufficient conditions are given for the unique solvability of the Cauchy problem for linear systems of generalized ordinary differential equations with singularities.

scholarly journals ASYMPTOTIC EXPANSION OF SOLUTION OF GENERAL BVP WITH INITIAL JUMPS FOR HIGHER-ORDER SINGULARLY PERTURBED INTEGRO-DIFFERENTIAL EQUATION

2018 ◽  
pp. 28-36
Author(s):  
Dauylbayev M. ◽  
Atakhan N. ◽  
Mirzakulova A.E.
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Higher Order ◽  
Singularly Perturbed ◽  
Jump Phenomenon ◽  
Integro Differential Equation ◽  
Asymptotic Expansion Of Solution ◽  
Initial Jump

In this article we constructed an asymptotic expansion of the solution undivided boundary value problem for singularly perturbed integro-differential equations with an initial jump phenomenon m – th order. We obtain the theorem about estimation of the remainder term’s asymptotic with any degree of accuracy in the smallparameter.

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