S. Ya. Startsev, “Cascade Method of Laplace Integration for Linear Hyperbolic Systems of Equations”, Mat. Zametki, 83:1 (2008), 107–118; Math. Notes, 83:1 (2008), 97

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Matematicheskie Zametki, 2008, Volume 83, Issue 1, Pages 107–118
DOI: https://doi.org/10.4213/mzm4338 (Mi mzm4338)

This article is cited in 17 scientific papers (total in 17 papers)

Cascade Method of Laplace Integration for Linear Hyperbolic Systems of Equations

S. Ya. Startsev

Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences

Full-text PDF (467 kB) Citations (17)

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DOI: https://doi.org/10.4213/mzm4338

Abstract: We propose a generalization of the cascade method of Laplace integration to the case of linear hyperbolic systems of equations. On the basis of this generalization, we prove that the system of equations with vanishing product of Laplace invariants has a complete set of solutions depending on arbitrary functions.

Keywords: linear hyperbolic equation, cascade integration method, Laplace integration, Laplace invariant, differential substitution.

Received: 29.04.2005
Revised: 19.04.2007

English version:
Mathematical Notes, 2008, Volume 83, Issue 1, Pages 97–106
DOI: https://doi.org/10.1134/S0001434608010124

Bibliographic databases:

UDC: 517.956.32

Language: Russian

Citation: S. Ya. Startsev, “Cascade Method of Laplace Integration for Linear Hyperbolic Systems of Equations”, Mat. Zametki, 83:1 (2008), 107–118; Math. Notes, 83:1 (2008), 97–106

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Linking options:

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https://doi.org/10.4213/mzm4338

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This publication is cited in the following 17 articles:

I. V. Rakhmelevich, “Ob invariantakh Laplasa dvumernykh nelineinykh uravnenii vtorogo poryadka s odnorodnym polinomom”, Izv. vuzov. Matem., 2024, no. 8, 55–64
I.T. Habibullin, A.U. Sakieva, “On integrable reductions of two-dimensional Toda-type lattices”, Partial Differential Equations in Applied Mathematics, 11 (2024), 100854
I. V. Rakhmelevich, “On Laplace Invariants of Two-Dimensional Nonlinear Equations of the Second Order with Homogeneous Polynomial”, Russ Math., 68:8 (2024), 47
E. A. Sozontova, “Usloviya suschestvovaniya i edinstvennosti resheniya zadachi Gursa dlya sistemy uravnenii s dominiruyuschimi chastnymi proizvodnymi”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 27:2 (2023), 375–383
M. N. Kuznetsova, “On nonlinear hyperbolic systems related by Bäcklund transforms”, Ufa Math. J., 15:3 (2023), 80–87
S. Ya. Startsev, “Structure of set of symmetries for hyperbolic systems of Liouville type and generalized Laplace invariants”, Ufa Math. J., 10:4 (2018), 103–110
S. Ya. Startsev, “Symmetry Drivers and Formal Integrals of Hyperbolic Systems of Equations”, J. Math. Sci. (N. Y.), 252:2 (2021), 232–241
S. Ya. Startsev, “On differential substitutions for evolution systems”, Ufa Math. J., 9:4 (2017), 108–113
Shemyakova E., “Laplace transformations as the only degenerate Darboux transformations of first order”, Program. Comput. Softw., 38:2 (2012), 105–108
E. I. Ganzha, “Euler Integrals and Multi-Integrals of Linear Partial Differential Equations”, Math. Notes, 89:1 (2011), 37–50
Shemyakova E.S., “X- and Y-invariants of partial differential operators in the plane”, Program. Comput. Softw., 37:4 (2011), 192–196
Kaptsov O.V., “Ideals of differential operators and transformations of linear partial differential equations”, Program. Comput. Softw., 36:2 (2010), 97–102
Ekaterina Shemyakova, “Refinement of Two-Factor Factorizations of a Linear Partial Differential Operator of Arbitrary Order and Dimension”, Math.Comput.Sci., 4:2-3 (2010), 223
S. P. Tsarev, E. S. Shemyakova, “Differential Transformations of Parabolic Second-Order Operators in the Plane”, Proc. Steklov Inst. Math., 266 (2009), 219–227
A. V. Zhiber, Yu. G. Mikhailova, “Algoritm postroeniya obschego resheniya $n$-komponentnoi giperbolicheskoi sistemy uravnenii s nulevymi invariantami Laplasa i kraevye zadachi”, Ufimsk. matem. zhurn., 1:3 (2009), 28–45
Shemyakova E., “Multiple factorizations of bivariate linear partial differential operators”, Computer algebra in scientific computing, Lecture Notes in Computer Science, 5743, eds. Gerdt V., Mayr E., Vorozhtsov E., Springer-Verlag, Berlin, 2009, 299–309
Shemyakova E., “Invariant properties of third-order non-hyperbolic linear partial differential operators”, Intelligent computer mathematics, Lecture Notes in Computer Science, 5625, eds. Carette J., Dixon L., Coen C., Watt S., Springer-Verlag, Berlin, 2009, 154–169

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

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