Abstract
This paper deals with two problems. The first of them is a boundary value problem for a nonlinear singular perturbed system of differential-algebraic equations having several solutions that can be applied to the simulation of the creep process of metal constructions. The second one is a practical problem of defining steady state stress in rotating solid disks at a constant temperature. The technique of neural network modeling is applied to solving these problems. The neural network approximate solutions agree well with the results of other authors obtained by the traditional methods.
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References
Hairer, E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems. Springer, Heidelberg (1987)
Hairer, E., Wanner, G.: Solving Ordinary Differential Equations. II: Stiff and Differential-Algebraic Problems. Springer, Heidelberg (1996)
Budkina, E.M., Kuznetsov, E.B.: Solving of boundary value problem for differential-algebraic equations. In: Proceedings of the XIX International conference on computational mechanics and modern applied software systems, pp. 44–46. MAI Publishing House, Moscow (2015)
Shalashilin, V.I., Kuznetsov, E.B.: Parametric Continuation and Optimal Parametrization in Applied Mathematics and Mechanics. Kluwer Academic Publishers, Dordrecht/Boston/London (2003)
Sosnin, O.V., Gorev, B.V.: Energy Variant of the Theory of Creep and Long-term Strength. Report 3. Creep and Long-Term Strength of Rotating Disks. Problems of Strength, vol. 3, pp. 3–7 (in Russian) (1974)
Vasilyev, A.N., Tarkhov, D.A.: Neural Network Modeling, Principles, Algorithms, Applications. SPbSPU Publishing House, Saint-Petersburg (2009). (in Russian)
Fletcher, C.A.J.: Computational Galerkin Methods. Springer, New York (1984)
Lazovskaya, T.V., Tarkhov, D.A.: Fresh approaches to the construction of parameterized neural network solutions of a stiff differential equation. St. Petersburg Polytechnical Univ. J. Physics and Mathematics 1, 192–198 (2015)
Kainov, N.U., Tarkhov, D.A., Shemyakina, T.A.: Application of neural network modeling to identification and prediction problems in ecology data analysis for metallurgy and welding industry. Nonlin. Phenom. Complex Syst. 17, 57–63 (2014)
Sosnin, O.V., Nikitenko, A.F., Gorev, B.V.: Justification of the energy variant of the theory of creep and long-term strength of metals. J. Appl. Mech. Tech. Phys. 51, 608–614 (2010)
Timoshenko, S.P., Goodier, J.N.: Theor. Elast. McGraw-Hill, New York (1970)
Polak, E.: Computational Methods in Optimization: A Unified Approach. Academic Press, New York (1971)
Achnowledgements
The work was supported by the Russian Foundation for Basic Research, project numbers 16-08-00943, 14-01-00660, and 14-01-00733.
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Budkina, E.M., Kuznetsov, E.B., Lazovskaya, T.V., Leonov, S.S., Tarkhov, D.A., Vasilyev, A.N. (2016). Neural Network Technique in Boundary Value Problems for Ordinary Differential Equations. In: Cheng, L., Liu, Q., Ronzhin, A. (eds) Advances in Neural Networks – ISNN 2016. ISNN 2016. Lecture Notes in Computer Science(), vol 9719. Springer, Cham. https://doi.org/10.1007/978-3-319-40663-3_32
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DOI: https://doi.org/10.1007/978-3-319-40663-3_32
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