Dmitry A. Lyakhov
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.
- Vladimir Arnold. Ordinary Differential Equations. Springer, Berlin, 1992.Google Scholar
- Muhammad Ayub, Masood Khan, and Fazal Mahomed. Algebraic linearization criteria for systems of ordinary differential equations. Nonlinear Dynamics 67, no. 3 (2012): 2053--2062.Google ScholarCross Ref
- Thomas Bachler, Vladimir Gerdt, Markus Lange-Hegermann, and Daniel Robertz. Algorithmic Thomas decomposition of algebraic and differential systems. Journal of Symbolic Computation 47, no. 10 (2012): 1233--1266. Google ScholarDigital Library
- Julia Bagderina. Linearization Criteria for a System of Two Second-Order Ordinary Differential Equations. Journal of Physics A: Mathematical and Theoretical 43, no. 46 (2010): 465201.Google ScholarCross Ref
- George Bluman and Stephen Anco. Symmetry and Integration Methods for Differential Equations. Springer, New York, 2001.Google Scholar
- John Carminati and Khai Vu. Symbolic Computation and Differential Equations:Lie Symmetries. Journal of Symbolic Computation 29, no. 1 (2000): 95--116. Google ScholarDigital Library
- Manuel Ceballos, Juan Núnez, and Angel Tenorio. Algorithmic method to obtain abelian subalgebras and ideals in Lie algebras. International Journal of Computer Mathematics 89, no. 10 (2012): 1388--1411. Google ScholarDigital Library
- Norbert Euler, Thomas Wolf, Peter Leach, and Marianna Euler. Linearizable Third-Order Ordinary DifferentialEquations and The Generalised Sundman Transformations: The Case X prime prime prime=0. Acta Applicandae Mathematicae 76, no. 1 (2003): 89--115.Google ScholarCross Ref
- Tarcisio Filho and Annibal Figueiredo. {SADE} a Maple package for the symmetryanalysis of differential equations. Computer Physics Communications 182, no. 2 (2011): 467--476.Google ScholarCross Ref
- Vladimir Gerdt. On decomposition of algebraic PDE systems into simple subsystems. Acta Applicandae Mathematicae 101, no. 1 (2008): 39--51.Google ScholarCross Ref
- Evelyne Hubert. Notes on triangular sets and triangulation-decomposition algorithms. II Differential systems. In: Franz Winkler, and Ulrich Langer (Editors), Symbolic and Numerical Scientific Computation, Hagenberg, 2001, Lecture Notes in Computer Science, Vol. 2630, Springer, Berlin, 2003, pp. 40--87. Google ScholarDigital Library
- Nail Ibragimov and Sergey Meleshko. Linearization of third-order ordinary differential equations by point and contact transformations. Journal of Mathematical Analysis and Applications 308, no. 1 (2005): 266--289.Google ScholarCross Ref
- Nail Ibragimov, Sergey Meleshko, and Supaporn Suksern. Linearization of fourth-order ordinary differential equations by point transformations. Journal of Physics A: Mathematical and Theoretical 41, no. 23 (2008): 235206.Google ScholarCross Ref
- Nail Ibragimov. A Practical Course in Differential Equations and Mathematical Modelling. Classical and New Methods. Nonlinear Mathematical Models. Symmetry and In variance Principles. Higher Education Press, World Scientific, New Jersey, 2009.Google Scholar
- Markus Lange-Hegermann. The Differential Dimension Polynomial for Characterizable Differential Ideals. arXiv:math-AC/1401.5959.Google Scholar
- Markus Lange-Hegermann. Differential Thomas: Thomas decomposition of differential systems. http://wwwb.math.rwth-aachen.de/thomasdecomposition/.Google Scholar
- Ziming Li and Dongming Wang. Coherent, Regular and Simple Systems in Zero Decompositions of Partial Differential Systems. Systems Science and Mathematical Sciences 12, no. 5 (1999): 43--60.Google Scholar
- Sophus Lie. Klassifikation und Integration von gewöhnlichen Differential gleichungenzwischen x, y, die eine Gruppe von Transformationen gestatten. III. Archivfor Matematik og Naturvidenskab, 8(4), 1883, 371--458. Reprinted in Lie's Gesammelte Abhandlungen, 5, paper XIY, 1924, 362--427.Google Scholar
- Sophus Lie. Vorlesungen über kontinuierliche Gruppen mit geometrischen und anderen Anwendungen. Bearbeitet und herausgegeben von Dr. G. Schefferes. Teubner, Leipzig, 1883.Google Scholar
- Fazal Mahomed and Peter Leach. Symmetry Lie Algebra of nth Order Ordinary Differential Equations. Journal of Mathematical Analysis and Applications 151, no. 1 (1990): 80--107.Google ScholarCross Ref
- Peter Olver. Applications of Lie Groups to Differential Equations, 2nd Edition.Graduate Texts in Mathematics, Vol. 107, Springer, New York, 1993.Google ScholarCross Ref
- Peter Olver. Equivalence, Invariance and Symmetry. Cambridge University Press, 1995.Google Scholar
- Lev Ovsyannikov. Group Analysis of Differential Equations. Academic Press, New York, 1992.Google Scholar
- Greg Reid. Algorithms for reducing a system of PDEs to standard form, determining the dimension of its solution space and calculating its Taylor series solution. European Journal of Applied Mathematics 2, no. 4 (1991): 293--318.Google ScholarCross Ref
- Greg Reid. Finding abstract Lie symmetry algebras of differential equations without integrating determining equations. European Journal of Applied Mathematics 2, no. 04 (1991): 319--340.Google ScholarCross Ref
- Daniel Robertz. Formal Algorithmic Elimination for PDEs. Lecture Notes in Mathematics, Vol. 2121, Springer, 2014.Google Scholar
- Oritz Schwarz. Algorithmic Lie Theory for solving ordinary Differential Equations.Chapman & HallCRS, Boca Raton, 2008.Google Scholar
- Werner Seiler. Involution: The Formal Theory of Differential Equations and its Applications in Computer Algebra. Algorithms and Computation in Mathematics 24. Springer, Heidelberg, 2010. Google ScholarDigital Library
- Celestin Soh and Fazal Mahomed. Canonical forms for systems of two second-order ordinary differential equations. Journal of Physics A: Mathematical and General 34 (2001): 2883--2991.Google ScholarCross Ref
- Sakka Sookmee and Sergey Meleshko. Linearization of Two Second-Order Ordinary Differential Equations via Fiber Preserving Point Transformations. ISRN Mathematical Analysis, Article ID: 452689, 2011.Google Scholar
- Joseph Thomas. Differential Systems. AMS Colloquium Publications, Vol. XXI, AMS,New York, 1937.Google Scholar
- Joseph Thomas. Systems and Roots. William Byrd Press, Richmond, 1962.Google Scholar
- Khai Vu, Grace Jefersson, and John Carminati. Finding higher symmetries of differential equations using the MAPLE package DESOLVII. Computer Physics Communications 183, no. 4 (2012): 1044--1054.Google ScholarCross Ref
- Dongming Wang. Elimination Methods. Springer, Wien, 2001.Google Scholar
Index Terms
- Algorithmic Verification of Linearizability for Ordinary Differential Equations
Recommendations
On the algorithmic linearizability of nonlinear ordinary differential equations
AbstractSolving nonlinear ordinary differential equations is one of the fundamental and practically important research challenges in mathematics. However, the problem of their algorithmic linearizability so far remained unsolved. In this ...
Linearization techniques for singular initial-value problems of ordinary differential equations
Linearization methods for singular initial-value problems in second-order ordinary differential equations are presented. These methods result in linear constant-coefficients ordinary differential equations which can be integrated analytically, thus ...
Approximate solution of linear ordinary differential equations with variable coefficients
In this paper, a novel, simple yet efficient method is proposed to approximately solve linear ordinary differential equations (ODEs). Emphasis is put on second-order linear ODEs with variable coefficients. First, the ODE to be solved is transformed to ...
Comments