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On the Test Volterra Equations of the First Kind in the Integral Models of Developing Systems

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Abstract

The paper was devoted to analysis of the test Volterra equations of the first kind enabling one to study the specificity of important classes of integral equations in the mathematical models of developing system. Along with theoretical results, presented were numerical calculations for the model examples.

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References

  1. Glushkov, V.M., On a Class of Dynamic Macroeconomic Models, Upravl. Sist. Mash., 1977, no. 2, pp. 3–6.

    Google Scholar 

  2. Glushkov, V.M., Ivanov, V.V., and Yanenko, V.M., Modelirovanie razvivayushchikhsya sistem (Modeling of Developing Systems), Moscow: Nauka, 1983.

    MATH  Google Scholar 

  3. Yatsenko, Yu.P., Integral’nye modeli sistem s upravlyaemoi pamyat’yu (Integral Models of Controlledmemory Systems), Kiev: Naukova Dumka, 1991.

    MATH  Google Scholar 

  4. Hritonenko, N. and Yatsenko, Yu., Applied Mathematical Modelling of Engineering Problems, Dortrecht: Kluwer, 2003.

    Book  MATH  Google Scholar 

  5. Apartsin, A.S., Neklassicheskie uravneniya Volterra I roda: teoriya i chislennye metody (Nonclassical Volterra Equations of the First Kind: Theory and Numerical Methods), Novosibirsk: Nauka, 1999.

    Google Scholar 

  6. Messina, E., Russo, E., and Vecchio, A., A Stable Numerical Method for Volterra Integral Equations with Discontinuous Kernel, J. Math. Anal. Appl., 2008, no. 337, pp. 1383–1393.

    Article  MathSciNet  MATH  Google Scholar 

  7. Baker, C.T.H. and Derakhshan, M.S., Convergence and Stability of Quadrature Methods Applied to Volterra Equations with Delay, IMA J. Numer. Anal., 1993, no. 13, pp. 67–91.

    Article  MathSciNet  MATH  Google Scholar 

  8. Vermiglio, R., On the Stability of Runge-Kutta Methods for Delay Integral Equations, Numer. Math., 1992, no. 61, pp. 561–577.

    Article  MathSciNet  MATH  Google Scholar 

  9. Hu Qiya, Multilevel Correction for Discrete Collocation Solutions of Volterra Integral Equations with Delay Arguments, Appl. Numer. Math., 1999, no. 31, pp. 159–171.

    Article  MathSciNet  MATH  Google Scholar 

  10. El’sgol’ts, L.E. and Norkin, S.B., Vvedenie v teoriyu differentsial’nykh uravnenii s otklonyayushchimsya argumentom (Introduction to the Theory of Differential Equations with Divergent Argument), Moscow: Nauka, 1971.

    MATH  Google Scholar 

  11. Brunner, H., Collocation and Continuous Implicit Runge–Kutta Methods for a Class of Delay Volterra Integral Equations, J. Comput. Appl. Math., 1994, no. 53, pp. 61–72.

    Article  MathSciNet  MATH  Google Scholar 

  12. Hu Guang-Da, Hu Guang-Di, and Meguid, S., Stability of Runge-Kutta Methods for Delay Differential Systems with Multiple Delays, IMA J. Numer. Anal., 1999, no. 19, pp. 349–356.

    Article  MathSciNet  MATH  Google Scholar 

  13. Torelli, L. and Vermiglio, R., On the Stability of Continuous Quadrature Rules for Differential Equations with Several Constant Delays, IMA J. Numer. Anal., 1993, no. 13, pp. 291–302.

    Article  MathSciNet  MATH  Google Scholar 

  14. Apartsin, A.S. and Sidler, I.V., Using the Nonclassical Volterra Equations of the First Kind to Model the Developing Systems Control, Autom. Remote Control, 2013, vol. 74, no. 6, pp. 899–910.

    Article  MathSciNet  MATH  Google Scholar 

  15. Corless, R.M., Gonnet, G.H., Hare, D.E.G., and Jeffrey, D.J., Lambert’s W Function in Maple, Maple Technic. Newslett., 1993, no. 9, pp. 12–22.

    Google Scholar 

  16. Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J., and Knuth, D.E., On the LambertW Function, Adv. Comput. Math., 1996, vol. 5, no. 1, pp. 329–359.

    Article  MathSciNet  MATH  Google Scholar 

  17. Apartsyn, A.S., Multilinear Volterra Equations of the First Kind, Autom. Remote Control, 2004, vol. 65, no. 2, pp. 263–269.

    Article  MathSciNet  MATH  Google Scholar 

  18. Kantorovich, L.V. and Akilov, G.P., Funktsional’nyi analiz (Functional Analysis), Moscow, Nauka, 1977.

    MATH  Google Scholar 

  19. Vilenkin, N.Ya., Kombinatorika (Combinatorics), Moscow: Nauka, 1969.

    Google Scholar 

  20. Apartsin, A.S., Inversion Formulas and Their Finite-dimensional Counterparts for Some Classes of Volterra Equations of the I Kind with Discontinuous Kernels, in Proc. Int. Conf. “Topical Problems of Numerical and Appied Mathematics—2015” Devoted to the 90th Anniversary of Acad. Gurii Ivanovich Marchuk, (CD-Proceedings), Novosibirsk, 2015, pp. 62–69.

    Google Scholar 

  21. Apartsin, A.S., On the Theory of Volterra Integral Equations of the First Kind with Discontinuous Kernels, Comput. Math. Math. Phys., 2016, vol. 56, no. 5, pp. 810–825.

    Article  MathSciNet  MATH  Google Scholar 

  22. Sidorov, D.N., On Parametric Families of Solutions of Volterra Integral Equations of the First Kind with Piecewise Smooth Kernels, Differ. Equat., 2013, vol. 49, no. 2, pp. 210–216.

    Article  MathSciNet  MATH  Google Scholar 

  23. Apartsin, A.S. and Sidler, I.V., On Numerical Solution of the Nonclassical Volterra Equations of the First Kind, in Collected Papers of IXth Int. Conf. “Analytical and Numerical Methods to Model and Social Problems,” Penza, Russia, 2014, pp. 59–64.

    Google Scholar 

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Authors and Affiliations

  1. Melentiev Energy Systems Institute, Siberian Branch, Russian Academy of Sciences, Irkutsk, Russia

    A. S. Apartsin & I. V. Sidler

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  1. A. S. Apartsin
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  2. I. V. Sidler
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Correspondence to A. S. Apartsin.

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Original Russian Text © A.S. Apartsin, I.V. Sidler, 2018, published in Avtomatika i Telemekhanika, 2018, No. 4, pp. 31–45.

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Apartsin, A.S., Sidler, I.V. On the Test Volterra Equations of the First Kind in the Integral Models of Developing Systems. Autom Remote Control 79, 604–616 (2018). https://doi.org/10.1134/S0005117918040033

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