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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. I. Denisov, B. N. Shvilkin, V. A. Sokolov, and M. I. Vasili’ev, “Pulsar radiation in post-Maxwellian vacuum nonlinear electrodynamics,” Phys. Rev. D, 94, 045021 (2016).
V. I. Denisov, A. V. Kozar’, and V. F. Sharikhin, “Investigation of the trajectories of a magnetized particle in the equatorial plane of a magnetic dipole,” Moscow Univ. Phys. Bull., 65, 164–169 (2010).
V. M. Pastukhov, Y. V. Vladimirova, and V. N. Zadkov, “Photon-number statistics from resonance fluorescence of a two-level atom near a plasmonic nanoparticle,” Phys. Rev. A, 90, 063831 (2014).
G. Dattoli, V. V. Mikhailin, and K. V. Zhukovsky, “Undulator radiation in a periodic magnetic field with a constant component,” J. Appl. Phys., 104, 124507 (2008).
G. Dattoli, V. V. Mikhailin, and K. V. Zhukovsky, “Influence of a constant magnetic field on the radiation of a planar undulator,” Mosc. Univ. Phys. Bull., 65, 507–512 (2009).
K. V. Zhukovsky, “High harmonic generation in the undulators for free electron lasers,” Opt. Commun., 353, 35–41 (2015).
K. V. Zhukovsky, “Analytical account for a planar undulator performance in a constant magnetic field,” J. Electromagn. Waves Appl., 28, 1869–1887 (2014).
K. V. Zhukovsky, “Harmonic generation by ultrarelativistic electrons in a planar undulator and the emission-line broadening,” J. Electromagn. Waves Appl., 29, 132–142 (2015).
V. I. Denisov, V. A. Sokolov, and M. I. Vasili’ev, “Nonlinear vacuum electrodynamics birefringence effect in a pulsar’s strong magnetic field,” Phys. Rev. D, 90, 023011 (2014).
K. V. Zhukovsky, “Emission and tuning of harmonics in a planar two-frequency undulator with account for broadening,” Laser Part. Beams, 34, 447–456 (2016).
K. V. Zhukovsky, “Violation of the maximum principle and negative solutions with pulse propagation in Guyer–Krumhansl model,” Internat. J. Heat Mass Transfer, 98, 523–529 (2016).
K. V. Zhukovsky, “Exact solution of Guyer–Krumhansl type heat equation by operational method,” Internat. J. Heat Mass Transfer, 96, 132–144 (2016).
Y. Zhang and W. Ye, “Modified ballistic–diffusive equations for transient non-continuum heat conduction,” Internat. J. Heat Mass Transfer, 83, 51–63 (2015).
K. V. Zhukovsky and H. M. Srivastava, “Analytical solutions for heat diffusion beyond Fourier law,” Appl. Math. Comput., 293, 423–437 (2017).
B. Căruntu and C. Bota, “Analytical approximate solutions for a general class of nonlinear delay differential equations,” Sci. World J., 2014, 631416 (2014).
S. Hesam, A. R. Nazemi, and A. Haghbin, “Analytical solution for the Fokker–Planck equation by differential transform method,” Scientia Iranica, 19, 1140–1145 (2012).
K. V. Zhukovsky, “Solution of some types of differential equations: Operational calculus and inverse differential operators,” Sci. World J., 2014, 454865 (2014).
K. V. Zhukovsky, “The operational solution of fractional-order differential equations, as well as Black–Scholes and heat-conduction equations,” Moscow Univ. Phys. Bull., 71, 237–244 (2016).
G. Dattoli, H. M. Srivastava, and K. V. Zhukovsky, “Operational methods and differential equations with applications to initial-value problems,” Appl. Math. Comput., 184, 979–1001 (2007).
K. V. Zhukovsky and G. Dattoli, “Evolution of non-spreading Airy wavepackets in time dependent linear potentials,” Appl. Math. Comput., 217, 7966–7974 (2011).
K. V. Zhukovsky, “A method of inverse differential operators using ortogonal polynomials and special functions for solving some types of differential equations and physical problems,” Mosc. Univ. Phys. Bull., 70, 93–100 (2015).
G. Dattoli, H. M. Srivastava, and K. V. Zhukovsky, “A new family of integral transforms and their applications,” Integral Transform. Spec. Funct., 17, 31–37 (2006).
P. Appell and J. Kampé de Fériet, Fonctions Hypergéométriques et Hypersphériques: Polynômes d’Hermite, Gauthier-Villars, Paris (1926).
D. T. Haimo and C. Markett, “A representation theory for solutions of a higher-order heat equation: I,” J. Math. Anal. Appl., 168, 89–107 (1992).
G. Dattoli, H. M. Srivastava, and K. V. Zhukovsky, “Orthogonality properties of the Hermite and related polynomials,” J. Comput. Appl. Math., 182, 165–172 (2005).
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York (1953).
P. A. Vshivtseva, V. I. Denisov, and I. P. Denisova, “An integral relation for tensor polynomials,” Theor. Math. Phys., 166, 186–193 (2011).
K. B. Wolf, Integral Transforms in Science and Engineering (Math. Concepts Meth. Sci. Engin., Vol. 11), Plenum, New York (1979).
H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Wiley, New York (1984).
H. W. Gould and A. T. Hopper, “Operational formulas connected with two generalizations of Hermite polynomials,” Duke Math. J., 29, 51–63 (1962).
G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press, Cambridge (1944).
K. V. Zhukovsky, “Operational solution for some types of second order differential equations and for relevant physical problems,” J. Math. Anal. Appl., 446, 628–647 (2017).
K. V. Zhukovsky, “Operational method of solution of linear non-integer ordinary and partial differential equations,” Springer Plus, 5, 119 (2016).
A. A. Sokolov, I. M. Ternov, V. Ch. Zhukovsky, and A. V. Borisov, Gauge fields [in Russian], Moscow State University, Moscow (1986).
W. J. Parker, R. J. Jenkins, C. P. Butler, and G. L. Abbott, “Flash method of determining thermal diffusivity, heat capacity, and thermal conductivity,” J. Appl. Phys., 32, 1679–1684 (1961).
R. Kovács and P. Ván, “Generalized heat conduction in heat pulse experiments,” Internat. J. Heat Mass Transfer, 83, 613–620 (2015).
C. Cattaneo, “Sur une forme de l’équation de la chaleur éliminant le paradoxe d’une propagation instantanée,” C. R. Acad. Sci. Paris, 247, 431–433 (1958).
V. Peshkov, “‘Second sound’ in Helium II,” J. Phys., 8, 381 (1944).
C. C. Ackerman and W. C. Overton, “Second sound in solid Helium-3,” Phys. Rev. Lett., 22, 764–766 (1969).
Author information
Authors and Affiliations
Corresponding author
Additional information
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 190, No. 1, pp. 58–77, January, 2017.
Rights and permissions
About this article
Cite this article
Zhukovsky, K.V. Solving evolutionary-type differential equations and physical problems using the operator method. Theor Math Phys 190, 52–68 (2017). https://doi.org/10.1134/S0040577917010044
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577917010044