Construction of a regularized asymptotic solution of an integro-differential equation with a rapidly oscillating cosine

Volume 32, Issue 1, pp 74--85 http://dx.doi.org/10.22436/jmcs.032.01.07

Publication Date: July 21, 2023 Submission Date: April 14, 2023 Revision Date: May 02, 2023 Accteptance Date: May 10, 2023

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Authors

A. Bobodzhanov - The National Research University, MPEI, Krasnokazarmennaya 14, Moscow, Russia. B. Kalimbetov - M. Auezov South Kazakhstan University, Tauke-xan 5, Shymkent, Kazakhstan. N. Pardaeva - Almalyk branch of the NRTU MISA, Amir Temur 56, Almalyk, Uzbekistan.

Abstract

In this paper, we consider a singularly perturbed integro-differential equation with a rapidly oscillating right-hand side, which includes an integral operator with a slowly varying kernel. Earlier, singularly perturbed differential and integro-differential equations with rapidly oscillating coefficients were considered. The main goal of this work is to generalize the Lomov's regularization method and to identify the rapidly oscillating right-hand side to the asymptotics of the solution to the original problem.

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Keywords

• Singular perturbation
• integro-differential equation
• oscillating right-hand side
• solvability of iterative problems
• regularization problems

MSC

•  34E05
•  34E20
•  45J05

References

• [1] D. Bibulova, B. Kalimbetov, V. Safonov, Regularized asymptotic solutions of a singularly perturbed Fredholm equation with a rapidly varying kernel and a rapidly oscillating inhomogeneity, Axioms, 11 (2022), 1–14
  • View Article
  • Google Scholar


• [2] M. A. Bobodzhanova, B. T. Kalimbetov, G. M. Bekmakhanbet, Asymptotics solutions of a singularly perturbed integrodifferential fractional order derivative equation with rapidly oscillating coefficients, Bulletin of the Karaganda, University- Mathematics, 104 (2021), 56–67
  • View Article
  • Google Scholar


• [3] A. A. Bobodzhanov, B. T. Kalimbetov, V. F. Safonov, Integro-differential problem about parametric amplification and its asymptotical integration, Int. J. Appl. Math., 33 (2020), 331–353
  • View Article
  • Google Scholar


• [4] A. A. Bobodzhanov, B. T. Kalimbetov, V. F. Safonov, Nonlinear singularly perturbed integro-differential equations and regularization method, WSEAS Trans. Math., 19 (2020), 301–311
  • View Article
  • Google Scholar


• [5] A. A. Bobodzhanov, B. T. Kalimbetov, V. F. Safonov, Asymptotic solutions of singularly perturbed integro-differential systems with rapidly oscillating coefficients in the case of a simple spectrum, AIMS Math., 6 (2021), 8835–8853
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] A. A. Bobodzhanov, V. F. Safonov, Singularly perturbed integro-differential equations with diagonal degeneration of the kernel in reverse time, Differ. Equ., 40 (2004), 120–127
  • View Article
  • Google Scholar
  • MATH


• [7] A. A. Bobodzhanov, V. F. Safonov, Asymptotic analysis of integro-differential systems with an unstable spectral value of the integral operator’s kernel, Comput. Math. Math. Phys., 47 (2007), 65–79
  • View Article
  • Google Scholar


• [8] A. A. Bobodzhanov, V. F. Safonov, Asymptotic solutions of Fredholm integro-differential equations with rapidly changing kernels and irreversible limit operator, Russian Math., 59 (2015), 1–15
  • View Article
  • Google Scholar


• [9] A. A. Bobodzhanov, V. F. Safonov, Regularized asymptotic solutions of the initial problem of systems of integro-partial differential equations, Mat. Zametki, 102 (2017), 28–38
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] A. A. Bobodzhanov, V. F. Safonov, Regularized asymptotics of solutions to integro-differential partial differential equations with rapidly varying kernels, Ufa Math. J., 2 (2018), 3–13
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] A. A. Bobodzhanov, V. F. Safonov, A generalization of the regularization method to the singularly perturbed integrodifferential equations with partial derivatives, Russ. Math., 62 (2018), 6–17
  • View Article
  • Google Scholar


• [12] A. A. Bobodzhanov, V. F. Safonov, V. I. Kachalov, Asymptotic and pseudoholomorphic solutions of singularly perturbed differential and integral equations in the Lomov’s regularization method, Axioms, 8 (2019), 1–20
  • View Article
  • Google Scholar
  • MATH


• [13] A. A. Bobodzhanov, B. T. Kalimbetov, V. F. Safonov, Generalization of the regularization method to singularly perturbed integro-differential systems of equations with rapidly oscillating inhomogeneity, Axioms, 10 (2021), 1–14
  • View Article
  • Google Scholar


• [14] A. A. Bobodzhanov, B. T. Kalimbetov, V. F. Safonov, Algorithm of the Regularization Method for a Nonlinear Singularly Perturbed Integro-Differential Equation with Rapidly Oscillating Inhomogeneities, Differ. Equ., 58 (2022), 392–404
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] A. A. Bobodzhanov, B. T. Kalimbetov, V. F. Safonov, Algorithm of the regularization method for a singularly perturbed integro-differential equation with a rapidly decreasing kernel and rapidly oscillating inhomogeneity, Zh. Sib. Fed. Univ. Mat. Fiz., 15 (2022), 214–223
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] Y. L. Daletsky, The asymptotic method for some differential equations with oscillating coefficients, In: Doklady Akademii nauk SSSR Reports of the Academy of Science of the USSR, 143 (1962), 1026–1029
  • View Article
  • Google Scholar


• [17] Y. L. Daletskiy, M. G. Krein, Stability of solutions of differential equations in Banach space, American Mathematical Society, Providence, R.I. (1974)
  • View Article
  • Google Scholar


• [18] S. F. Feˇsˇcenko, M. ¯I. Shkil, L. D. N¯ıkolenko, Asymptotic methods in the theory of linear differential equations, ``Naukova Dumka'', Kiev (1966)
  • View Article
  • Google Scholar


• [19] P. Hartman, Ordinary differential equations, John Wiley & Sons, New York-London-Sydney (1964)
  • View Article
  • Google Scholar


• [20] B. T. Kalimbetov, Regularized asymptotics of solutions for systems of singularly perturbed differential equations of fractional order, Int. J. Fuzzy Math. Arch., 16 (2018), 67–74
  • View Article
  • Google Scholar


• [21] B. T. Kalimbetov, On the Question of asymptotic integration of singularly perturbed fractional order problems, Asian J. Fuzzy Appl. Math., 6 (2019), 44–49
  • View Article
  • Google Scholar


• [22] B. T. Kalimbetov, E. Abylkasymova, G. Beissenova, On the asymptotic solutions of singulary perturbed differential systems of fractional order, J. Math. Comput. Sci., 24 (2002), 165–172
  • View Article
  • Google Scholar


• [23] B. T. Kalimbetov, Kh. F. Etmishev, Asymptotic solutions of scalar integro-differential equations with partial derivatives and with rapidly oscillating coefficients, Bulletin of KarSU, series Mathematics, 97 (2020), 52–67
  • View Article
  • Google Scholar


• [24] B. T. Kalimbetov, I. M. Omarova, D. A. Sapakov, Regularization method for singularly perturbed integro-differential systems with rapidly oscillating coefficients in resonance case, Bulletin of KarSU, series Mathematics, 75 (2014), 96–102
  • View Article
  • Google Scholar


• [25] B. T. Kalimbetov, N. A. Pardaeva, L. D. Sharipova, Asymptotic solutions of integro-differential equations with partial derivatives and with rapidly varying kernel, Sib. ` Elektron. Mat. Izv., 16 (2019), 1623–1632
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [26] B. T. Kalimbetov, V. F. Safonov, Regularization method for singularly perturbed integro-differential equations with rapidly oscillating coefficients and rapidly changing kernels, Axioms, 9 (2020), 1–12
  • View Article
  • Google Scholar


• [27] B. T. Kalimbetov, V. F. Safonov, Singularly perturbed integro-differential equations with rapidly oscillating coefficients and with rapidly changing kernel in the case of a multiple spectrum, WSEAS Trans. Math, 20 (2021), 84–96
  • View Article
  • Google Scholar


• [28] B.T. Kalimbetov, V. F. Safonov, Integro-differentiated singularly perturbed equations with fast oscillating coefficients, Bull. KarSU. Ser. Math., 94 (2019), 33–47
  • View Article
  • Google Scholar


• [29] B. T. Kalimbetov, V. F. Safonov, E. Madikhan, Singularly perturbed integral equations with a rapidly oscillating inhomogeneity, Int. J. Appl. Math., 34 (2021), 653–668
  • View Article
  • Google Scholar


• [30] B. T. Kalimbetov, V. F. Safonov, O. D. Tuychiev, Singular perturbed integral equations with rapidly oscillation coefficients, Sib. Elektron. Mat. Izv., 17 (2020), 2068–2083
  • View Article
  • Google Scholar
  • MATH


• [31] B. Kalimbetov, L. Tashimov, N. Imanbaev, D. Sapakov, Regularized asymptotical solutions of integro-differential systems with spectral singularities, Adv. Difference Equ., 2013 (2013), 6 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [32] B. T. Kalimbetov, M. A. Temirbekov, Z. O. Khabibullaev, Asymptotic solution of singular perturbed problems with an instable spectrum of the limiting operator, Abstr. Appl. Anal., 2012 (2012), 16 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [33] B. T. Kalimbetov, M. A. Temirbekov, B. I. Yeskarayeva, Discrete boundary layer for systems of integro-differential equations with zero points of spectrum, Bulletin of Karaganda SU, series Mathematics, 75 (2014), 88–95
  • View Article
  • Google Scholar


• [34] B. T. Kalimbetov, M. A. Temirbekov, B. I. Yeskarayeva, Mathematical description of the internal boundary layer for nonlinear integro-differential system, Bulletin of Karaganda SU, series Mathematics, 75 (2014), 77–87
  • View Article
  • Google Scholar


• [35] B. T. Kalimbetov, A. N. Temirbekov, A. S. Tolep, Asymptotic solutions of scalar integro-differential equations with partial derivatives and with fast oscillating coefficients, Eur. J. Pure Appl. Math., 13 (2020), 287–302
  • View Article
  • MathSciNet
  • Google Scholar


• [36] B. T. Kalimbetov, O. D. Tuychiev, Asymptotic solution of the Cauchy problem for the singularly perturbed partial integrodifferential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity, Open Math., 19 (2021), 244–258
  • View Article
  • Google Scholar


• [37] B. T. Kalimbetov, B. I. Yeskarayeva, Contrast structure in equations with zero spectrum of limit operator and irreversible spectral value of the kernel, Bulletin of Karaganda SU, series Mathematics, 78 (2015), 56–64
  • View Article
  • Google Scholar


• [38] S. A. Lomov, Introduction to General Theory of Singular Perturbations, American Mathematical Society, Providence, RI (1992)
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [39] S. A. Lomov, I. S. Lomov, Foundations of Mathematical Theory of Boundary Layer, Mosk. Gos. Univ., Moscow (2011)
  • View Article
  • Google Scholar


• [40] A. D. Ryˇzih, Asymptotic solution of a linear differential equation with a rapidly oscillating coefficient, Trudy Moskov. Orden. Lenin. ` Energet. Inst., 357 (1978), 92–94
  • View Article
  • Google Scholar


• [41] V. F. Safonov, A. A. Bobodzhanov, Course of higher mathematics. Singularly perturbed equations and the regularization method, Publishing House MPEI, Moscow (2012)
  • View Article
  • Google Scholar


• [42] N. I. Shkil, Asymptotic methods in differential equations, Naukova Dumka, Kiev (1971)
  • View Article
  • Google Scholar