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Multiperiodic Solution of the Initial-Boundary Value Problem for an Integro-Differential Equation of the Parabolic Type

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Abstract

This paper investigates the initial-boundary value problem for an integro-differential equation of parabolic type with a differentiation operator with respect to multidimensional time. A fundamental solution is constructed and some of its properties associated with multiperiodicity in time and diagonal periodicity in space variables are discovered. An estimate is obtained for the fundamental solution of an integro-differential equation. The existence and uniqueness of multiperiodic solutions of the boundary value problem with respect to all arguments are established and some basic properties of the desired solutions are found.

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REFERENCES

  1. G. C. Evans, “Sull’equazione integro-differenziale di tipo parabolico,” Atti R. Accad. Lincei, Ser. 5, Cl. Sci. Fis., Mat. Nat. Rend. 21 (2), 25–31 (1912).

    Google Scholar 

  2. A. A. Il’yushin and B. E. Pobedrya, Fundamentals of the Mathematical Theory of Thermoviscoelasticity (Nauka, Moscow, 1970) [in Russian].

    Google Scholar 

  3. V. V. Vlasov and N. A. Rautian, “Integrodifferential equations in viscoelasticity theory,” Russ. Math. 56 (6), 48–51 (2012). https://doi.org/10.3103/S1066369X12060060

    Article  MATH  Google Scholar 

  4. A. M. Samoilenko and O. D. Nurzhanov, “The Bubnov–Galerkin method of constructing periodic solutions of Voltaire-type integro-differential equations,” Differ. Uravn. 15 (8), 1503–1517 (1979).

    Google Scholar 

  5. A. I. Botashev, Periodic Solutions of Volterra Integro-Differential Equations (MFTI, Moscow, 1998) [in Russian]

    Google Scholar 

  6. V. K. Kharasakhal, Almost Periodic Solutions of Ordinary Differential Equations (Nauka, Alma-Ata, 1970) [in Russian].

  7. D. U. Umbetzhanov, Almost Multiperiodic Solutions of Partial Differential Equations (Nauka, Alma-Ata, 1979) [in Russian].

  8. D. U. Umbetzhanov, Almost Periodic Solutions of Evolution Equations (Nauka Kaz. SSR, Alma-Ata, 1990) [in Russian].

    MATH  Google Scholar 

  9. A. T. Asanova, “An almost periodic solution of a semilinear parabolic equation,” Differ. Equations 34 (12), 1705–1707 (1998).

    MathSciNet  MATH  Google Scholar 

  10. Zh. A. Sartabanov, “Multiperiodic solution of one system of integro-differential equations,” Izv. Akad. Nauk Kaz. SSR, Ser. Fiz.-Mat., No. 5 (138), 51–56 (1987).

  11. Zh. A. Sartabanov, “Pseudoperiodic solution of a system of integrodifferential equations,” Ukr. Math. J. 41 (1), 116–120 (1989). https://doi.org/10.1007/BF01060661

    Article  MathSciNet  MATH  Google Scholar 

  12. G. A. Abdikalikova, G. M. Aitenova, and Zh. A. Sartabanov, “Multiperiodic solution of a system of partial integro-differential equations,” Vestn. Kaz. NPU im. Abaja, Ser. Fiz. Mat., No. 3 (63), 7–13 (2018).

  13. G. M. Aitenova, Zh. A. Sartabanov, G. A. Abdikalikova, and A. Kerimbekov, “Bounded and multiperiodic solutions of the system of partial integro-differential equations,” Vestn. Karagand. Univ. Ser.: Mat., No. 3 (95), 8–18 (2019). https://doi.org/10.31489/2019M2/8-18

  14. Zh. A. Sartabanov and G. M. Aitenova, “Multiperiodic solutions of linear systems integro-differential equations with D c-operator and ε-period of hereditary,” Izv. Nats. Akad. Nauk Resp. Kaz., Ser. Fiz.-Mat., No. 6 (328), 106–122 (2019). https://doi.org/10.32014/2019.2518-1726.77

  15. V. Volterra, Leçons sur la théorie mathématique de la lutte pour la vie (Gauthier-Villars, Paris, 1931).

  16. V. Volterra, Theory of Functionals and of Integral and Integro-Differential Equations (Blackie & Son, London, 1930).

  17. Ya. V. Bykov, On Some Problems of the Theory of Integro-Differential Equations (Kirg. Gos. Univ., Frunze, 1957) [in Russian].

  18. M. I. Imanaliev, Oscillations and Stability of Solutions of Singularly Perturbed Integro-Differential Systems (Ilim, Frunze, 1974) [in Russian].

  19. A. N. Filatov, Averaging Methods in Differential and Integro-Differential Equations (Fan, Tashkent, 1971) [in Russian].

  20. A. M. Nakhushev, Equations of Mathematical Biology (Vyssh. Shk., Moscow, 1995) [in Russian].

    MATH  Google Scholar 

  21. V. Lakshmikantham and M. Rama Mohana Rao, Theory of Integro-Differential Equations (CRC Press, Boca Raton, FL, 1995).

    MATH  Google Scholar 

  22. X. Ros Oton, “Integro-differential equations: Regularity theory and Pohozaev identities,” PhD Thesis (Univ. Politècnica de Catalunya, Barcelona, 2014).

  23. T. A. Dzhangveladze, “On a nonlinear integro-differential equation of parabolic type,” Differ. Uravn. 21 (1), 41–46 (1985).

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Zhubanov Aktobe Regional University, 030000, Aktobe, Republic of Kazakhstan

    Zh. A. Sartabanov & G. A. Abdikalikova

  2. Utemisov West-Kazakhstan University, 090000, Uralsk, Republic of Kazakhstan

    G. M. Aitenova

Authors
  1. Zh. A. Sartabanov
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  2. G. M. Aitenova
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  3. G. A. Abdikalikova
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Corresponding authors

Correspondence to Zh. A. Sartabanov, G. M. Aitenova or G. A. Abdikalikova.

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Translated by L. Trubitsyna

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Sartabanov, Z.A., Aitenova, G.M. & Abdikalikova, G.A. Multiperiodic Solution of the Initial-Boundary Value Problem for an Integro-Differential Equation of the Parabolic Type. Russ Math. 66, 46–55 (2022). https://doi.org/10.3103/S1066369X22080084

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