Vestnik КRAUNC. Fiz.-Mat. Nauki. 2023. vol. 42. no. 1. P. 180-190. ISSN 2079-6641

Research Article
Full text in Russian
MSC 47G20

Contents of this issue

Read Russian Version

Elimination of the Integral Term in the Equations of One Hereditary System Related to the Hydromagnetic Dynamo

G.M Vodinchar, E. A. Kazakov^*

Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, Russia, 684034, Paratunka, Mirnaya st., 7.

Abstract. We study a two-dimensional system of integro-differential equations, which is the simplest hereditary model of a two-mode hydromagnetic dynamo. Accounting for the spatial and temporal nonlocality of interactions in dynamo systems is currently being actively studied. In the low-mode approximations of the dynamo equations, one can consider only temporal nonlocality, i.e. heredity (memory). Memory in the system under study is implemented in the form of feedback distributed over all past states of the system. The feedback is represented by a convolution-type integral term of a quadratic combination of phase variables with a fairly general kernel. This term models the quenching of the turbulent field generator (\alpha-effect) by a quadratic form in phase variables. In real dynamo systems, such quenchingn is provided by the Lorentz force. The main result of the work is a proof of the possibility of eliminating the integral term for one class of kernels. Such kernels are solutions of a homogeneous linear differential equation with constant coefficients. It is proved that the studed integro-differential system can be replaced by a higher-dimensional differential system with suitable initial conditions for additional phase variables. If the kernel is a solution to an n-order equation, then the dimension of the system can reach 3n−2 and depends on the initial conditions that the kernel satisfies. The work uses classical methods of the theory of differential equations. Examples are given of dynamical systems that arise for some kernels as a result of the elimination of the integral term. The results of the work can be used to verify computational algorithms and program codes developed for solving integro-differential equations.

Key words: hydromagnetic dynamo, memory, heredity, integro-differential equations.

Received: 13.03.2023; Revised: 20.03.2023; Accepted: 21.03.2023; First online: 22.03.2023

For citation. Vodinchar G. M., Kazakov E. A. Elimination of the integral term in the equations of one hereditary system related to the hydromagnetic dynamo. Vestnik KRAUNC. Fiz.-mat. nauki. 2023, 42: 1, 180-190. EDN: BRDBZK.

Funding. The work is supported by Russian Science Foundation, grant No. 22-11-00064.

Competing interests. There are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing
the final version of the article in print. The final version of the manuscript was approved by all authors.

^*Correspondence: E-mail:

The content is published under the terms of the Creative Commons Attribution 4.0 International License

© Vodinchar G. M., Kazakov E. A., 2023

© Institute of Cosmophysical Research and Radio Wave Propagation, 2023 (original layout, design, compilation


  1. Zeldovich Ya. B., Ruzmaikin A. A., Sokilov D. D. Magnitie polay v astrofizike [Magnetic fields in astrophysics], Moscow-Izhevsk, SIC «RHD», 2006.(In Russian).
  2. Krause F., R¨adler K.-H. Mean-field magnetohydrodynamics and dynamo theory. New York, PergamonPress, 1980.
  3. Vodinchar G., Kazakov E. The Lorenz system and its generalizations as dynamo models with memory. E3S Web of Conf., 2018, 62, 02011, DOI:10.1051/e3sconf/20186202011
  4. Vodinchar G.M. Hereditary Oscillator Associated with the Model of a Large-Scale \alpha\omega-Dynamo, Mathematics, 2020, 8(11), 2065, DOI: 10.3390/math8112065.
  5. Kazakov E. A. Hereditary low-mode dynamo model. Vestnik KRAUNC. Fiz.-mat. nauki, 2021, 35, 2, 40-47, DOI: 10.26117/2079-6641-2021-35-2-40-47, (In Russian).
  6. Kazakov E. A. Two-mode model of a hydromagnetic dynamo with memory. Computational Technologies, 2022, 27(6), 19–32. DOI:10.25743/ICT.2022.27.6.003, (In Russian.).
  7. Uchajkin V. V. Metod drobnyh proizvodnyh [Fractional derivative method]. Ulyanovsk, Artichoke, 2008, 510. (In Russian).
  8. Tarasov B. E. Modeli teoreticheskoi fiziki s integro-differencirovaniem drobnogo poraydka [Models of Theoretical Physics with Fractional Integro-Derivation], Moscow-Izhevsk, Izhevsk Institute of Computer Research, 2011. (In Russian).
  9. Herrmann R. Fractional Calculus: An Introduction for Physicists. Singapore, World Scientific, 2014.
  10. Vodinchar G., Feshchenko L. Fractal Properties of the Magnetic Polarity Scale in the Stochastic Hereditary \alpha\omega-Dynamo Model, Fractal Fract, 2022, 6(6), 328. DOI: 10.3390/math8112065.
  11. Korn G., Korn T. Spravochnik po matimatike dlay nauchnih rabotikov i ingenerov [Handbook of mathematics for scientists and engineers], Moscow, Nauka, 1968. (In Russian).

Information about authors

Vodinchar Gleb Mikhailovich – PhD (Math. & Phys.), Leading Researcher, Lab. for Simulation of Physical Processes,

Kazakov Evgeny Anatolevich – Lead programmer, Lab. of Electromagnetic Propogation,