Electronic Modeling

VOL 36, NO 3 (2014)

СОДЕРЖАНИЕ

Integral Equations in the Problems of Mathematical Modeling

3-6

Mathematical Modeling and Computation Methods

	APARTSIN A.S. Volterra Nonclassical Equations of the First Kind in the Integral Models of DevelopingSystems	7-18
	BULATOV M.V. Investigation of Integral Equations with a Identically Degenerate Matrix before the Principal Part	19-30
	MENSHIKOV Yu.L. Method for Ensuring Adequacy of Dynamic Models	31-40
	VERLAN D.A. Method of Degenerate Kernels under Numerical Implementation of Integral Dynamic Models	41-58

Computational Processes and System

	GAVRYSH V.I. Numerical-analytical Solution of a Nonlinear Stationary Heat Conduction Problem for the Infinite Temperature-sensitive Multilayer Plate	59-70
	FEDORCHUK V.A., MAKHOVICH A.I. Method for Studying the Dynamics of Nonstationary Thermal Processes in the Presence of Symmetric Boundary Conditions	71-80

Application of Modelling Methods and Facilities

	STARKOV V.N., SEMENOV A.A., GOMONAY E.V. The First Kind Operator Equation in the Problem of Statistics Reconstruction of the Number of Quantum Light Photons	81-94
	ABDIKARIMOV R.A. Modeling of Dynamic Stability of Visco-Elastic Orthotropic Rectangular Plates of Variable Rigidity	95-104

Short Notes

	VERLAN A.F., KHUDOYAROV B.A., FAIZIBOEV E.F. Modeling of a Flutter of Visco-Elastic Cylindrical Shell in the Gas Flow	105-112
	IVANYUK V.A., KOST’YAN N.L. Method for Construction of a Dynamic Model of a Linear Object by the Reaction to the Input Action of Arbitrary Shape	113-120

Volterra Nonclassical Equations of the First Kind in the Integral Models of DevelopingSystems

APARTSIN A.S.

ABSTRACT

The sufficient conditions for the existence and uniqueness of continuous solutions of the Volterra linear nonclassical equation arising in the integral models of developing systems have been obtained. Some Volterra equations of the first kind with piecewise smooth kernels have also been studied. An analysis of test problems allowing one to understand specificity of insufficiently studied integral equations is given.

KEYWORDS

nonclassical Volterra equations of the first kind , sufficient conditions, breaking kernel.

REFERENCES

1. Glushkov, V.M. (1977), “About one class of dynamic macroeconomic models”, Upravlyayushchie sistemy i mashiny, no. 2, pp. 3-6.
2. Glushkov, V.M., Ivanov, V.V. and Yanenko, V.M. (1983), Modelirovanie razvivayushchikhsya sistem [Simulation of developing systems], Nauka, Moscow, Russia.
3. Yatsenko, Yu.P. (1991), Integralnye uravneniya sistem s upravlyaemoy pamyatyu [Integral equations of systems with managed memory], Naukova Dumka, Kiev, Ukraine.
4. Hritonenko, N. and Yatcenko, Yu. (2003), Applied Mathematical Modeling of Engineering Problems, Kluwer Academic Publishers, Dortrecht.
5. Apartsyn, A.S. (1999), Neklassicheskie uravneniya Volterry I roda: teoriya i chislennye metody [Nonclassical Volterra equations of the first kind: theory and numerical methods], Nauka, Novosibirsk, Russia.
6. Apartsyn, A.S. (2003), Nonclassical linear Volterra equations of the first kind, VSP, Utrecht-Boston.
7. Messina, E., Russo, E. and Vecchio, A. (2008), “A stable numerical method for Volterra integral equations with discontinuous kernel”, J. Math. Anal. Appl., Vol. 337, pp. 1383-1393.
8. Sidorov, D.N. (2013), “About parameter family of solutions of Volterra integral equations of the first kind with piecewise smooth kernels”, Differentsialnye uravneniya, Vol. 49, no. 3, pp. 209-215.
9. Kantorovich, L.V. and Akilov, G.P. (1977), Funktsionalnyy analiz [Functional analysis], Nauka, Moscow, Russia.
10. Apartsyn, A.S. (2004), “About polylinear Volterra equations of the first kind”, Avtomatika i telemekhanika, no. 2, pp. 118-125.
11. Apartsyn, A.S. (2007), “Polylinear Volterra integral equation of the first kind: elements of theory and numerical methods”,Izv. Irkutskogo gos. un-ta. Seriya «Matematika», no. 1, pp. 13-41.
12. Apartsyn, A.S. (2008), “Polylinear Volterra equations and some management tasks”,Avtomatika i telemekhanika, no. 4, pp. 3-16.
13. Apartsyn, A.S. (2013), “Polynomial Volterra integral equations of the first kind and Lambert's function”, Proc. of the Steklov Institute of Mathematics, Vol. 280 (1), pp. 26-38.
14. Corless, R.M., Gonnet, G.H., Hare, D.E.G. and Jeffrey, D.J. (1993), “Lambert's W function in Maple”, The Maple Technical Newsletter, no. 9, pp. 12-22.
15. Corless, R.M., Gonne,t G.H., Hare, D.E.G. and et al. (1996), “On the Lambert W function”, Adv. Comput. Math., Vol. 5, no. 4, pp. 329-359.

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Investigation of Integral Equations with a Identically Degenerate Matrix before the Principal Part

BULATOV M.V.

ABSTRACT

The article deals with a system of Volterra integral equations with a singular matrix before the principal part. The fundamental differences of such systems from integral equations of the first and the second kinds have been established. The conditions of existence of unique continuous solution have been defined.

KEYWORDS

Volterra integral equations, index, matrix polynomials, integral-algebraic equations.

REFERENCES

1. Apartsyn, A.S. (1999), Neklassicheskie uravneniya Volterry I roda: teoriya i chislennye metody [Nonclassical Volterra equations of the first kind: theory and numerical methods], Nauka, Novosibirsk, Russia.
2. Verlan, A.F. and Sizikov, V.S. (1986), Integralnye uravneniya: metody, algoritmy, programmy [Integral equation methods, algorithms, programs], Naukova dumka, Kiev, Ukraine.
3. Krasnov, M.L. (1975), Integralnye uravneniya [Integral equations], Nauka, Moscow, Russia.
4. Samko, S.G., Kilbas, A.A. and Marichev, O.I. (1987), Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya [Integrals and derivatives of fractional order, and some their applications], Nauka i tekhnika, Minsk, Belorussia.
5. Brunner, H. (2004), Collocation Methods for Volterra Integral and Related Funktional Differential Equations, Cambridge University Press.
6. Brunner, H. and van der Houwen, P.J. (1986), The Numerical Solution of Volterra Equations. CWI Monographs 3, North-Holland, Amsterdam.
7. Linz, P. (1985), Analytical and Numerical Methods for Volterra Equations, Studies in Applied Mathematics, Philadelphia.
8. Brunner, H. (1997), “1896—1996: One Hundred Years of Volterra Integral Equations of the First Kind”, Applied Numerical Mathematics, Vol. 24, pp. 83-93.
9. Chistyakov, V.F. (1987), “Singular systems of ordinary differential equations and their integrated counterparts”, Funktsii Lyapunova i ikh primeneniya, Novosibirsk,pp. 231-239.
10. Bulatov, M.V. and Budnikova, O.S. (2013), “Research multi-step methods for the solution of integral-algebraic equations: the construction of the stability regions”, Zhurn. vychislitelnoy matematiki i matematicheskoy fiziki, Vol. 53, no. 9, pp. 1448-1459.
11. Bulatov, M.V. (2002), Regularization of singular systems of integral equations”, Ibid., Vol. 42, no. 3, pp. 58-63.
12. Chistyakov, V.F. (2006), “On some properties of Volterra integral equations of the 4th kind with kernel of convolution type”, Matematicheskie zametki, Vol. 80 (1), pp. 115-118.
13. Hadizadeh, M., Ghoreishi, F. and Pishbin, S. (2011), “Jacobi Spectral Solution for Integral Algebraic Equations of Index-2”, Applied Numerical Mathematics, Vol. 161 (1), pp. 131-148.
14. Boyarintsev, Yu.E. (1980), Regulyarnye i singulyarnye sistemy obyknovennykh differentsialnykh uravneniy [Regular and singular system of ordinary differential equations], Nauka, Novosibirsk, Russia.
15. Gantmakher, F.R. (1966), Teoriya matrits [Matrix theory], Nauka, Moscow, Russia.
16.Vaarman, O. (1988), Obobshchennye obratnye otobrazheniya [Generalized inverse display], Valgus, Tallinn.
17. Ten, Men Yan (1985), “An approximate solution of linear integral Volterra equations of the first kind”, Abstract of Cand. Sci. (Phys.-Math.) dissertation, Irkutsk.
18. Bulatov, M.V. and Chistyakov, V.F. (1997), “The properties of differential-algebraic systems and their integral analogs”, Preprint, Memorial University of Newfoundland, p. 35.
19. Apartsyn, A.S (1987), “Digitization methods of regularization of some integral equations of the first kind”, Sbornik trudov «Metody chislennogo analiza i optimizatsii» [Methods of numerical analysis and optimization], Nauka, pp. 263-297.

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Method for Ensuring Adequacy of Dynamic Models

MENSHIKOV Yu.L.

ABSTRACT

The problem of synthesis of adequate mathematical description of physical processes has been studied. It is shown that in the general case, this problem is reduced to solution of some first kind Volterra integral equations (ill-posed problem). Several possible statements of such problems were proposed. Modifications of the regularization method were used for obtaining stable results of synthesis.

KEYWORDS

mathematical simulation, adequacy, integralequations.

REFERENCES

1. Shanon, R. (1975), Systems Simulation — the Art and Science, Prentice-Hall, Inc., Englewood Cliffs, New Jersey.
2. Menshikov, Yu.L. (2008), “The adequacy of the results of mathematical modelling”, Trudy Mezhdunar. konf. “Modelirovanie-2008” [Proceedings of the Intern. Conf. "Modeling 2008"], Kiev IPME im. Pukhova, G.Ye. NAN Ukrainy, pp. 119-124.
3. Soeiro, N. and еt al. (2004), “Vibratory Working Modelling through Methods of Continuum Mechanics”, Proc. Eleventh International Congress on Sound and Vibration, St. Petersburg, Russia, pp. 2821-2828.
4. Sarmar, I. and Malik, A. (2008), “Modeling, Analysis and Simulation of a Pan Tilt Platform Based on Linear and Nonlinear Sуstems”, Proc. IEEE/ASME MESA, China, pp. 147-152.
5.Menshikov, Yu.L. (2013), “Synthesis of Adequate Mathematical Description as Solution of Special Inverse Problems”, European Journal of Mathematical Sciences, Vol. 2, no. 3, pp. 256-271.
6.Menshikov, Yu.L. and Polyakov, N.V. (2013), “Synthesis algorithm of sustainable mathematical description of evolutionary process”, Tezy XI Mizhnar. naukovo-praktychnoyi konf. “Matematychne ta programne zabezpechennya intelektualnykh system”, MPZIS-2013 [Proceedings XI Intern. Scientific-practical conference "Mathematical and software of intelligent systems”, MPZIS-2013], Dnepropetrovsk, November 20-22, 2013, pp. 114-115.
7. Stepashko, V.S. (2008), “The method of critical variances as an analytical apparatus of the theory of inductive modelling”,Problemy upravleniya i informatiki, no. 2, pp. 27-32.
8. Gubarev, V.F. (2008), “The method of iterative identification of multidimensional systems with inaccurate data. Part 1. Theoretical Foundations”,Ibid., no, 2, pp. 8-26.
9. Zhukov, O.A. (2008), “Iterative algorithms for identification of multidimensional systems”, Trudy 15 mezhdunar. konf. po avtomaticheskomu upravleniyu “Avtomatika-2008” [Proceedings of the 15th Intern. Conf. on Automatic Control "Automation 2008"], Odessa, INIA, pp. 774-777.
10. Gelfandbeyn, Yu.M. and Kolosov, L.S. (1972), Retrospektivnaya identifikatsiya vozmushcheniy i pomekh [The retrospective identification of disturbances and interference], Nauka, Moscow, Russia.
11. Bublik, B.N. and Kirichenko, N.F. (1975), Osnovy teorii upravleniya [Fundamentals of control theory], Vishcha shkola, Kiev, Ukraine.
12. Tikhonov, A.N. and Arsenin, V.Ya. (1979), Metody resheniya nekorrektnykh zadach [Methods of solving ill-posed problems], Nauka, Moscow, Russia.
13. Menshikov, Yu.L. and Polyakov, N.V. (2009), Identifikatsiya modeley vneshnikh vozdeystviy [Identification of models of external influences], Nauka ta Osvíta, Dnepropetrovsk, Ukraine.
14. Menshikov, Yu.L. (2013), “Identification of External Loads as Method of Adequate Mathematical Description Synthesis”, Identification and Control (MIC 2013), Proc. of 32nd IASTED International Conference on Modelling, Innsbruck, Austria, February 11-13, 2013, p. 8.

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Method of Degenerate Kernels under Numerical Implementation of Integral Dynamic Models

VERLAN D.A.

ABSTRACT

The possibilities of numerical implementation of integral dynamic models are considered. The models are non-parametric and are presented by Volterra integral equations of type II and type I, and also by integro-differential equations. Recurrent numerical algorithms for solving these equations are developed. The degenerate (separable) kernels method is used for constructing the algorithms. The optimization method of approximation for kernels of the Volterra integral operators is proposed.

KEYWORDS

integral equations, algorithms, approximation, resolvent, integral-differential equations.

REFERENCES

1. Butkovskiy, A.G. (1979), Kharakteristiki sistem s raspredelennymi parametrami : Spravochnoe posobie [Characteristics of systems with distributed parameters: Reference Guide], Nauka, Moscow, Russia.
2. Manzhirov, A.V. and Polyanin, A.D. (2000), Spravochnik po integralnym uravneniyam: Metody resheniya[Handbook of Integral Equations: Methods of solution], Factorial Press, Moscow, Russia.
3. Verlan, D.A. (2011), “Approximation of functions of two variables in control problems”, Zbirnyk prats Matematychne ta kompyuterne modelyuvannya. Ser. Tekhnichni nauky. Vyp. 5, Kamyanets-Podilskyy natsionalnyy un-t im. Ivana Ohienka [Mathematical and computer modelling], Vol. 5, pp. 62-70.
4. Nikolskiy, S.M. (1988), Kvadraturnye formuly. 4-e izd. dop. [Quadrature formulas. 4th ed.] Nauka, Moscow, Russia.
5. Verlan, D.A. (2009), “Iterative approximation algorithms function of two variables”, Matematychne ta kompyuterne modelyuvannya. Ser.Tekhnichni nauky, Vyp. 2, Kamyanets-Podilskyy natsionalnyy un-t im. Ivana Ohiyenka [Mathematical and computer modelling], Vol. 2, pp. 24-32.
6. Verlan, D.A. (2013), “Gradient algorithm of bilinear approximation of kernels when solving the Fredholm integral equations of the second kind ”, Elektronnoye modelirovaniye, Vol. 35, no. 1, pp. 73-80.
7. Bakhvalov, N.S., Zhidkov, N.P. and Kobelkov, G.M. (2003), Chislennye metody [Numerical methods], Binom, Moscow, Russia.
8. Stechkin, S.B. and Subbotin, Yu.N. (1976), Splayny v vychislitelnoy matematike [Splines in computational mathematics], Nauka, Moscow, Russia.

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