On the exact and approximate solutions of several differential equations with variational derivatives of the first and second orders
https://doi.org/10.29235/1561-2430-2020-56-1-51-71
Abstract
In this paper, we consider the problem of the exact and approximate solutions of certain differential equations with variational derivatives of the first and second orders. Some information about the variational derivatives and explicit formulas for the exact solutions of the simplest equations with the first variational derivatives are given. An interpolation method for solving ordinary differential equations with variational derivatives is demonstrated. The general scheme of an approximate solution of the Cauchy problem for nonlinear differential equations with variational derivatives of the first order, based on the use of the operator interpolation apparatus, is presented. The exact solution of the differential equation of the hyperbolic type with variational derivatives, similar to the classical Dalamber solution, is obtained. The Hermite interpolation problem with the conditions of coincidence in the nodes of the interpolated and interpolation functionals, as well as their variational derivatives of the first and second orders, is considered for functionals defined on the sets of differentiable functions. The found explicit representation of the solution of the given interpolation problem is based on an arbitrary Chebyshev system of functions. This solution is generalized for the case of interpolation of functionals on one out of two variables and applied to construct an approximate solution of the Cauchy problem for the differential equation of the hyperbolic type with variational derivatives. The description of the material is illustrated by numerous examples.
About the Authors
M. V. Ignatenko
Belarusian State University
Belarus
Belarus
Marina V. Ignatenko – Ph. D. (Physics and Mathematics), Associate Professor, Associate Professor of Web-Technologies and Computer Simulation Department
4, Nezavisimosti Ave., 220030, MinskL. A. Yanovich
Institute of Mathematics of the National Academy of Sciences of Belarus
Belarus
Belarus
Leonid A. Yanovich – Corresponding Member, Dr. Sc. (Physics and Mathematics), Professor, Chief Researcher
11, Surganov Str., 220072, Minsk
References
1. Lévy P. Concrete Problems of Functional Analysis. Moscow, Nauka Publ., 1967. 510 p. (in Russian).
2. Vainberg M. M. Variational Methods for Investigation of Non-linear Operators. Moscow, Gostekhizdat Publ., 1956. 345 p. (in Russian.).
3. Volterra V. Theory of Functionals and of Integral and Integro-Differential Equations. New York, Dover Publ., 2005. 288 p.
4. Daletsky Yu. L. Differential Equations with Functional Derivatives and Stochastic Equations for Generalized Random Processes. Doklady academii nauk SSSR = Doklady of the Academy of Sciences of USSR, 1966, vol. 166, no. 5, pp. 1035–1038 (in Russian).
5. Zadorozhniy V. G. Second-order Differential Equations with Variational Derivatives. Differentsial’nyye uravneniya = Differential equations, 1989, vol. 25, no. 10, pp. 1679–1683 (in Russian).
6. Danilovich V. P. Kovalchik I. M. Cauchy Formula for Linear Equations with Functional Derivatives. Differentsial’nyye uravneniya = Differential equations, 1977, vol. 13, no. 8, pp. 1509–1511 (in Russian).
7. Kovalchik I. M. Representation of Solutions of Certain Equations with Functional Derivatives Using Wiener Integrals. Doklady academii nauk Ukraїni. Seriya A. Fiziko-matematicheskiye i tekhnicheskiye nauki = Doklady of the Academy of Sciences of Ukraine. Series A. Physical, Mathematical, and Technical Sciences, 1978, vol. 12, pp. 1079–1083 (in Russian).
8. Kovalchik I. M. Linear Equations with Functional Derivatives. Doklady academii nauk SSSR = Doklady of the Academy of Sciences of USSR, 1970, vol. 194, no. 4, pp. 763–766 (in Russian).
9. Daletskii Yu. L. Infinite-dimensional Elliptic Operators and Parabolic Equations Connected with them. Russian Mathematical Surveys, 1967, vol. 22, no. 4, pp. 1–53. https://doi.org/10.1070/rm1967v022n04abeh003769
10. Averbukh V. I., Smolyanov O. G. The Theory of Differentiation in Linear Topological Spaces. Russian Mathematical Surveys, 1967, vol. 22, no. 6, pp. 201–258. https://doi.org/10.1070/rm1967v022n06abeh003761
11. Makarov V. L., Khlobystov V. V., Yanovich L. A. Methods of Operator Interpolation. Proceedings of the Institute of Mathematics of the National Academy of Sciences of Ukraine. Vol. 83: Mathematics and its Applications. Kiev, 2010. 516 p.
12. Yanovich L. A., Ignatenko M. V. Bases of the Theory of Interpolation of Functions of Matrix ariables. Minsk, Belaruskaya Navuka Publ., 2016. 281 р. (in Russian.).
13. Yanovich, L. A., Ignatenko M. V. Newton-Type Interpolation Functional Polynomials with nodes of the second multiplicity. Analiticheskiye metody analiza i differentsial’nykh uravneniy. Sbornik nauchnykh trudov = Analytical methods of analysis and differential equations. Collection of Sceintific Papers. Minsk, Belarussian State University, 2012, pp. 229–240 (in Russian).
14. Yanovich L. A., Ignatenko M. V. On a class of interpolation polynomials for nonlinear ordinary differential operators. Matematicheskoe Modelirovanie = Mathematical Models and Computer Simulations, 2014, vol. 26, no. 11, pp. 90–96 (in Russian).
15. Yanovich L. A., Ignatenko M. V. To the theory of Hermite – Birkhoff interpolation of nonlinear ordinary differential operators. Vestsі Natsyianal’nai akademіі navuk Belarusі. Seryia fіzіka-matematychnykh navuk = Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics series, 2017, no. 2, pp. 7–23 (in Russian).
16. Ignatenko M. V. To the interpolation theory of diferential operators of arbitrary order in partial derivatives. Trudy Instituta matematiki Natsional’noy akademii nauk Belarusi = Proceedings of the Institute of Mathematics of the National Academy of Sciences of Belarus, 2017, vol. 25, no. 2, pp. 11–20 (in Russian).
17. Ignatenko M. V., Yanovich L. A. Generalized interpolation Hermite – Birkhoff polynomials for arbitrary-order partial differential operators. Vestsі Natsyianal’nai akademіі navuk Belarusі. Seryia fіzіka-matematychnykh navuk = Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics series, 2018, vol. 54, no. 2, pp. 149–163 (in Russian). https://doi.org/10.29235/1561-2430-2018-54-2-149-163
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