Algorithmic verification of linearizability for ordinary differential equations

Dmitry A. Lyakhov; Vladimir P. Gerdt; Dominik L. Michels

doi:10.1145/3087604.3087626

Algorithmic verification of linearizability for ordinary differential equations

Dmitry A. Lyakhov, Vladimir P. Gerdt, Dominik L. Michels

Computer Science

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

8 Scopus citations

Abstract

For a nonlinear ordinary differential equation solved with respect to the highest order derivative and rational in the other derivatives and in the independent variable, we devise two algorithms to check if the equation can be reduced to a linear one by a point transformation of the dependent and independent variables. The first algorithm is based on a construction of the Lie point symmetry algebra and on the computation of its derived algebra. The second algorithm exploits the differential Thomas decomposition and allows not only to test the linearizability, but also to generate a system of nonlinear partial differential equations that determines the point transformation and the coefficients of the linearized equation. The implementation of both algorithms is discussed and their application is illustrated using several examples.

Original language	English (US)
Title of host publication	ISSAC 2017 - Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation
Editors	Michael Burr
Publisher	Association for Computing Machinery
Pages	285-292
Number of pages	8
ISBN (Electronic)	9781450350648
DOIs	https://doi.org/10.1145/3087604.3087626
State	Published - Jul 23 2017
Event	42nd ACM International Symposium on Symbolic and Algebraic Computation, ISSAC 2017 - Kaiserslautern, Germany Duration: Jul 25 2017 → Jul 28 2017

Publication series

Name	Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC
Volume	Part F129312

Conference

Conference	42nd ACM International Symposium on Symbolic and Algebraic Computation, ISSAC 2017
Country/Territory	Germany
City	Kaiserslautern
Period	07/25/17 → 07/28/17

Keywords

Algorithmic linearization test
Determining equations
Differential Thomas decomposition
Lie symmetry algebra
Ordinary differential equations
Point transformation
Power series solutions

ASJC Scopus subject areas

General Mathematics

Access to Document

10.1145/3087604.3087626

Cite this

Lyakhov, D. A., Gerdt, V. P., & Michels, D. L. (2017). Algorithmic verification of linearizability for ordinary differential equations. In M. Burr (Ed.), ISSAC 2017 - Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation (pp. 285-292). (Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC; Vol. Part F129312). Association for Computing Machinery. https://doi.org/10.1145/3087604.3087626

Lyakhov, Dmitry A. ; Gerdt, Vladimir P. ; Michels, Dominik L. / Algorithmic verification of linearizability for ordinary differential equations. ISSAC 2017 - Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation. editor / Michael Burr. Association for Computing Machinery, 2017. pp. 285-292 (Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC).

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title = "Algorithmic verification of linearizability for ordinary differential equations",

abstract = "For a nonlinear ordinary differential equation solved with respect to the highest order derivative and rational in the other derivatives and in the independent variable, we devise two algorithms to check if the equation can be reduced to a linear one by a point transformation of the dependent and independent variables. The first algorithm is based on a construction of the Lie point symmetry algebra and on the computation of its derived algebra. The second algorithm exploits the differential Thomas decomposition and allows not only to test the linearizability, but also to generate a system of nonlinear partial differential equations that determines the point transformation and the coefficients of the linearized equation. The implementation of both algorithms is discussed and their application is illustrated using several examples.",

keywords = "Algorithmic linearization test, Determining equations, Differential Thomas decomposition, Lie symmetry algebra, Ordinary differential equations, Point transformation, Power series solutions",

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Lyakhov, DA, Gerdt, VP & Michels, DL 2017, Algorithmic verification of linearizability for ordinary differential equations. in M Burr (ed.), ISSAC 2017 - Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation. Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC, vol. Part F129312, Association for Computing Machinery, pp. 285-292, 42nd ACM International Symposium on Symbolic and Algebraic Computation, ISSAC 2017, Kaiserslautern, Germany, 07/25/17. https://doi.org/10.1145/3087604.3087626

Algorithmic verification of linearizability for ordinary differential equations. / Lyakhov, Dmitry A.; Gerdt, Vladimir P.; Michels, Dominik L.
ISSAC 2017 - Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation. ed. / Michael Burr. Association for Computing Machinery, 2017. p. 285-292 (Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC; Vol. Part F129312).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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N2 - For a nonlinear ordinary differential equation solved with respect to the highest order derivative and rational in the other derivatives and in the independent variable, we devise two algorithms to check if the equation can be reduced to a linear one by a point transformation of the dependent and independent variables. The first algorithm is based on a construction of the Lie point symmetry algebra and on the computation of its derived algebra. The second algorithm exploits the differential Thomas decomposition and allows not only to test the linearizability, but also to generate a system of nonlinear partial differential equations that determines the point transformation and the coefficients of the linearized equation. The implementation of both algorithms is discussed and their application is illustrated using several examples.

AB - For a nonlinear ordinary differential equation solved with respect to the highest order derivative and rational in the other derivatives and in the independent variable, we devise two algorithms to check if the equation can be reduced to a linear one by a point transformation of the dependent and independent variables. The first algorithm is based on a construction of the Lie point symmetry algebra and on the computation of its derived algebra. The second algorithm exploits the differential Thomas decomposition and allows not only to test the linearizability, but also to generate a system of nonlinear partial differential equations that determines the point transformation and the coefficients of the linearized equation. The implementation of both algorithms is discussed and their application is illustrated using several examples.

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KW - Determining equations

KW - Differential Thomas decomposition

KW - Lie symmetry algebra

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KW - Power series solutions

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Lyakhov DA, Gerdt VP, Michels DL. Algorithmic verification of linearizability for ordinary differential equations. In Burr M, editor, ISSAC 2017 - Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation. Association for Computing Machinery. 2017. p. 285-292. (Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC). doi: 10.1145/3087604.3087626

Algorithmic verification of linearizability for ordinary differential equations

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