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 TMF, 2017, Volume 190, Number 1, Pages 58–77 (Mi tmf9144)

Solving evolutionary-type differential equations and physical problems using the operator method

K. V. Zhukovsky

Faculty of Physics, Lomonosov Moscow State University, Moscow, Russia

Abstract: We present a general operator method based on the advanced technique of the inverse derivative operator for solving a wide range of problems described by some classes of differential equations. We construct and use inverse differential operators to solve several differential equations. We obtain operator identities involving an inverse derivative operator, integral transformations, and generalized forms of orthogonal polynomials and special functions. We present examples of using the operator method to construct solutions of equations containing linear and quadratic forms of a pair of operators satisfying Heisenberg-type relations and solutions of various modifications of partial differential equations of the Fourier heat conduction type, Fokker–Planck type, Black–Scholes type, etc. We demonstrate using the operator technique to solve several physical problems related to the charge motion in quantum mechanics, heat propagation, and the dynamics of the beams in accelerators.

Keywords: inverse operator, exponential operator, inverse derivative, differential equation, Laguerre polynomial, Hermite polynomial, special function

DOI: https://doi.org/10.4213/tmf9144

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English version:
Theoretical and Mathematical Physics, 2017, 190:1, 52–68

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Revised: 08.02.2016

Citation: K. V. Zhukovsky, “Solving evolutionary-type differential equations and physical problems using the operator method”, TMF, 190:1 (2017), 58–77; Theoret. and Math. Phys., 190:1 (2017), 52–68

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tmf9144
• https://doi.org/10.4213/tmf9144
• http://mi.mathnet.ru/eng/tmf/v190/i1/p58

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Zhukovsky K., “Exact Negative Solutions For Guyer-Krumhansl Type Equation and the Maximum Principle Violation”, Entropy, 19:9 (2017), 440
2. Denisov V.I. Sokolov V.A. Svertilov S.I., “Vacuum Nonlinear Electrodynamic Polarization Effects in Hard Emission of Pulsars and Magnetars”, J. Cosmol. Astropart. Phys., 2017, no. 9, 004
3. Van P., Berezovski A., Fulop T., Grof G., Kovacs R., Lovas A., Verhas J., “Guyer-Krumhansl-Type Heat Conduction At Room Temperature”, EPL, 118:5 (2017), 50005
4. K. Zhukovsky, “Exact harmonic solution to ballistic type heat propagation in thin films and wires”, Int. J. Heat Mass Transf., 120 (2018), 944–955
5. K. V. Zhukovsky, “A harmonic solution for the hyperbolic heat conduction equation and its relationship to the Guyer-Krumhansl equation”, Mosc. Univ. Phys. Bull., 73:1 (2018), 45–52
6. K. Zhukovsky, D. Oskolkov, N. Gubina, “Some exact solutions to non-Fourier heat equations with substantial derivative”, Axioms, 7:3 (2018), 48
7. M. E. Abishev, S. Toktarbay, N. A. Beissen, F. B. Belissarova, M. K. Khassanov, A. S. Kudussov, A. Zh. Abylayeva, “Effects of non-linear electrodynamics of vacuum in the magnetic quadrupole field of a pulsar”, Mon. Not. Roy. Astron. Soc., 481:1 (2018), 36–43
8. K. Zhukovsky, D. Oskolkov, “Exact harmonic solutions to Guyer-Krumhansl-type equation and application to heat transport in thin films”, Continuum Mech. Thermodyn., 30:6 (2018), 1207–1222
9. Sh. M. Nagiyev, A. I. Akhmedov, “Time evolution of quadratic quantum systems: Evolution operators, propagators, and invariants”, Theoret. and Math. Phys., 198:3 (2019), 392–411
10. Behr N., Dattoli G., Duchamp G.H.E., Licciardi S., Penson K.A., “Operational Methods in the Study of Sobolev-Jacobi Polynomials”, Mathematics, 7:2 (2019), 124
11. Zhukovsky K.V., “Exact Analytic Solution and Investigation of the Guyer-Krumhansl Heat Equation”, Russ. J. Math. Phys., 26:2 (2019), 237–254
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