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TMF, 1995, Volume 104, Number 2, Pages 195–213 (Mi tmf1332)  

This article is cited in 25 scientific papers (total in 25 papers)

Noncommutative integration of linear differential equations

A. V. Shapovalova, I. V. Shirokovb

a Tomsk State University
b Omsk State University

Abstract: A method of noncommutative integration of linear partial differential equations that is analogous to noncommutative integration of finite-dimensional Hamiltonian systems is proposed. The method is based on the concept, introduced in the paper, of a $\lambda$ representation of Lie algebras. The method can be applied to the integration of the Klein–Gordon equation in Riemannian spaces of non-Stäckel type (i. e., in spaces that do not admit complete separation of the variables).

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English version:
Theoretical and Mathematical Physics, 1995, 104:2, 921–934

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Received: 17.05.1994

Citation: A. V. Shapovalov, I. V. Shirokov, “Noncommutative integration of linear differential equations”, TMF, 104:2 (1995), 195–213; Theoret. and Math. Phys., 104:2 (1995), 921–934

Citation in format AMSBIB
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\paper Noncommutative integration of linear differential equations
\jour TMF
\yr 1995
\vol 104
\issue 2
\pages 195--213
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\transl
\jour Theoret. and Math. Phys.
\yr 1995
\vol 104
\issue 2
\pages 921--934
\crossref{https://doi.org/10.1007/BF02065973}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. V. Shapovalov, I. V. Shirokov, “Noncommutative integration method for linear partial differential equations. Functional algebras and dimensional reduction”, Theoret. and Math. Phys., 106:1 (1996), 1–10  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. I. V. Shirokov, “Darboux coordinates on $K$-orbits and the spectra of Casimir operators on Lie groups”, Theoret. and Math. Phys., 123:3 (2000), 754–767  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. Ya. V. Lisitsyn, A. V. Shapovalov, “Integrable $N$-dimensional systems on the Hopf algebra and $q$-deformations”, Theoret. and Math. Phys., 124:3 (2000), 1172–1186  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. I. V. Shirokov, “Identities and Invariant Operators on Homogeneous Spaces”, Theoret. and Math. Phys., 126:3 (2001), 326–338  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. Klishevich, VV, “Exact solution of Dirac and Klein-Gordon-Fock equations in a curved space admitting a second Dirac operator”, Classical and Quantum Gravity, 18:17 (2001), 3735  crossref  mathscinet  zmath  adsnasa  isi
    6. S. P. Baranovskii, I. V. Shirokov, “Prolongations of Vector Fields on Lie Groups and Homogeneous Spaces”, Theoret. and Math. Phys., 135:1 (2003), 510–519  mathnet  crossref  crossref  mathscinet  zmath  isi
    7. O. L. Kurnyavko, I. V. Shirokov, “Construction of invariant scalar particle wave equations on Riemannian manifolds with external gauge fields”, Theoret. and Math. Phys., 156:2 (2008), 1169–1179  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    8. M. M. Goncharovskiy, I. V. Shirokov, “An integrable class of differential equations with nonlocal nonlinearity on Lie groups”, Theoret. and Math. Phys., 161:3 (2009), 1604–1615  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    9. A. I. Breev, I. V. Shirokov, A. A. Magazev, “Vacuum polarization of a scalar field on Lie groups and homogeneous spaces”, Theoret. and Math. Phys., 167:1 (2011), 468–483  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    10. Goncharovskii M.M., Shirokov I.V., “Classification of Degenerate Solutions of Linear Differential Equations”, Russian Physics Journal, 54:5 (2011), 527–535  crossref  isi
    11. Goncharovskii M.M., Shirokov I.V., “Klassifikatsiya vyrozhdennykh reshenii lineinykh differentsialnykh uravnenii”, Izvestiya vysshikh uchebnykh zavedenii. Fizika, 54:5 (2011), 20–26  elib
    12. A. A. Magazev, “Integrating Klein–Gordon–Fock equations in an external electromagnetic field on Lie groups”, Theoret. and Math. Phys., 173:3 (2012), 1654–1667  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    13. A. S. Popov, I. V. Shirokov, “Zvezdnoe proizvedenie na koalgebre Li i ego primenenie dlya vychisleniya kvantovykh integralov dvizheniya”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 1(30) (2013), 379–386  mathnet  crossref
    14. V. V. Mikheev, “Vysokotemperaturnoe razlozhenie matritsy plotnosti i ego prilozheniya”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 1(30) (2013), 369–378  mathnet  crossref
    15. Breev A.I. Goncharovskii M.M. Shirokov I.V., “Klein-Gordon Equation with a Special Type of Nonlocal Nonlinearity in Commutative Homogeneous Spaces with Invariant Metric”, Russ. Phys. J., 56:7 (2013), 731–739  crossref  isi
    16. A. I. Breev, “Scalar field vacuum polarization on homogeneous spaces with an invariant metric”, Theoret. and Math. Phys., 178:1 (2014), 59–75  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    17. Breev A.I., “Schrodinger Equation With Convolution Nonlinearity on Lie Groups and Commutative Homogeneous Spaces”, Russ. Phys. J., 57:8 (2014), 1050–1058  crossref  isi
    18. Breev A.I. Shapovalov A.V., “Yang-Mills Gauge Fields Conserving the Symmetry Algebra of the Dirac Equation in a Homogeneous Space”, XXII International Conference on Integrable Systems and Quantum Symmetries, Journal of Physics Conference Series, 563, ed. Burdik C. Navratil O. Posta S., IOP Publishing Ltd, 2014, 012004  crossref  isi
    19. Magazev A.A., “Algebra of Symmetry Operators and Integration of the Klein-Gordon Equation in An External Electromagnetic Field”, Russ. Phys. J., 57:6 (2014), 809–818  crossref  isi
    20. Alexey A. Magazev, Vitaly V. Mikheyev, Igor V. Shirokov, “Computation of Composition Functions and Invariant Vector Fields in Terms of Structure Constants of Associated Lie Algebras”, SIGMA, 11 (2015), 066, 17 pp.  mathnet  crossref
    21. Breev A.I., Kozlov A.V., “Vacuum Averages of the Energy-Momentum Tensor of a Scalar Field in Homogeneous Spaces With a Conformal Metric”, Russ. Phys. J., 58:9 (2016), 1248–1257  crossref  isi
    22. A. I. Breev, A. V. Shapovalov, A. V. Kozlov, “Integrirovanie relyativistskikh volnovykh uravnenii v kosmologicheskoi modeli B'yanki IX”, Kompyuternye issledovaniya i modelirovanie, 8:3 (2016), 433–443  mathnet  elib
    23. Breev A.I., Mosman E.A., “Noncommutative Integration and Symmetry Algebra of the Dirac Equation on the Lie Groups”, Russ. Phys. J., 59:8 (2016), 1153–1163  crossref  isi
    24. Boldyreva M.N., Magazev A.A., “On the Lie Symmetry Algebras of the Stationary Schr?dinger and Pauli Equations”, Russ. Phys. J., 59:10 (2017), 1671–1680  crossref  isi  scopus
    25. Breev A.I., Magazev A.A., “Integration of the Dirac Equation on Lie Groups in An External Electromagnetic Field Admitting a Noncommutative Symmetry Algebra”, Russ. Phys. J., 59:12 (2017), 2048–2058  crossref  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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