×

On the algorithmic linearizability of nonlinear ordinary differential equations. (English) Zbl 1454.34063

Summary: Solving 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 contribution, we propose a solution of this problem for a wide class of nonlinear ordinary differential equation of arbitrary order. We develop two algorithms to check if a nonlinear differential equation can be reduced to a linear one by a point transformation of the dependent and independent variables. In this regard, we have restricted ourselves to quasi-linear equations with a rational dependence on the occurring variables and to point transformations. While 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 our algorithms is discussed and evaluated using several examples.

MSC:

34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34C14 Symmetries, invariants of ordinary differential equations
68W30 Symbolic computation and algebraic computation
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Arnold, Vladimir, Ordinary Differential Equations (1992), Springer: Springer Berlin · Zbl 0744.34001
[2] Ayub, Muhammad; Khan, Masood; Mahomed, Fazal, Algebraic linearization criteria for systems of ordinary differential equations, Nonlinear Dyn., 67, 3, 2053-2062 (2012) · Zbl 1251.34053
[3] Bächler, Thomas; Gerdt, Vladimir; Lange-Hegermann, Markus; Robertz, Daniel, Algorithmic Thomas decomposition of algebraic and differential systems, J. Symb. Comput., 47, 10, 1233-1266 (2012) · Zbl 1315.35013
[4] Bagderina, Yulia, Linearization criteria for a system of two second-order ordinary differential equations, J. Phys. A, Math. Theor., 43, 46, Article 465201 pp. (2010) · Zbl 1213.34056
[5] Bluman, George; Anco, Stephen, Symmetry and Integration Methods for Differential Equations (2001), Springer: Springer New York · Zbl 1013.34004
[6] Carminati, John; Vu, Khai, Symbolic computation and differential equations: Lie symmetries, J. Symb. Comput., 29, 1, 95-116 (2000) · Zbl 0958.68543
[7] Ceballos, Manuel; Núñez, Juan; Tenorio, Angel, Algorithmic method to obtain abelian subalgebras and ideals in Lie algebras, Int. J. Comput. Math., 89, 10, 1388-1411 (2012) · Zbl 1255.17002
[8] Euler, Norbert; Wolf, Thomas; Leach, Peter; Euler, Marianna, Linearizable third-order ordinary differential equations and the generalised Sundman transformations: the case \(X''' = 0\), Acta Appl. Math., 76, 1, 89-115 (2003) · Zbl 1054.34002
[9] Filho, Tarcisio; Figueiredo, Annibal, [SADE] a Maple package for the symmetry analysis of differential equations, Comput. Phys. Commun., 182, 2, 467-476 (2011) · Zbl 1217.65165
[10] Gerdt, Vladimir, On decomposition of algebraic PDE systems into simple subsystems, Acta Appl. Math., 101, 1, 39-51 (2008) · Zbl 1154.68111
[11] Gerdt, Vladimir; Lange-Hegermann, Markus; Robertz, Daniel, The MAPLE package TDDS for computing Thomas decompositions of systems of nonlinear PDEs, Comput. Phys. Commun., 234, 202-215 (2019) · Zbl 07682604
[12] Grigoriev, Dima; Chistov, Alexander, Complexity of a standard basis of a D-module, St. Petersburg Math. J., 20, 5, 709-736 (2009) · Zbl 1206.16050
[13] Gubbiotti, Giorgio; Nucci, Maria Clara, Are all classical superintegrable systems in two-dimensional space linearizable?, J. Math. Phys., 58, Article 012902 pp. (2017) · Zbl 1393.35007
[14] Gustavson, Richard; Ovchinnikov, Alexey; Pogudin, Gleb, New order bounds in differential elimination algorithms, J. Symb. Comput., 85, 128-147 (2018) · Zbl 1379.68366
[15] Hubert, Evelyne, Notes on triangular sets and triangulation-decomposition algorithms. II Differential systems, (Winkler, Franz; Langer, Ulrich, Symbolic and Numerical Scientific Computation. Symbolic and Numerical Scientific Computation, Hagenberg, 2001. Symbolic and Numerical Scientific Computation. Symbolic and Numerical Scientific Computation, Hagenberg, 2001, Lecture Notes in Computer Science, vol. 2630 (2003), Springer: Springer Berlin), 40-87 · Zbl 1022.12005
[16] Ibragimov, Nail, A Practical Course in Differential Equations and Mathematical Modelling. Classical and New Methods. Nonlinear Mathematical Models. Symmetry and Invariance Principles (2009), Higher Education Press, World Scientific: Higher Education Press, World Scientific New Jersey · Zbl 1227.00035
[17] Ibragimov, Nail; Meleshko, Sergey, Linearization of third-order ordinary differential equations by point and contact transformations, J. Math. Anal. Appl., 308, 1, 266-289 (2005) · Zbl 1082.34003
[18] Ibragimov, Nail; Meleshko, Sergey; Suksern, Supaporn, Linearization of fourth-order ordinary differential equations by point transformations, J. Phys. A, Math. Theor., 41, 23, Article 235206 pp. (2008) · Zbl 1151.34029
[19] Lange-Hegermann, Markus, DifferentialThomas: Thomas decomposition of differential systems · Zbl 1386.12007
[20] Lange-Hegermann, Markus, The differential dimension polynomial for characterizable differential ideals, (Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory (2017), Springer: Springer Cham), 443-453 · Zbl 1480.12007
[21] Li, Ziming; Wang, Dongming, Coherent, regular and simple systems in zero decompositions of partial differential systems, Syst. Sci. Math. Sci., 12, 5, 43-60 (1999) · Zbl 1094.13545
[22] Lie, Sophus, Klassifikation und Integration von gewöhnlichen Differentialgleichungen zwischen x, y, die eine Gruppe von Transformationen gestatten, III, Arch. Math. Naturvidensk., 8, 4, 371-458 (1883), Reprinted in Lie’s Gesammelte Abhandlungen, 5, paper XIY, 1924, 362-427 · JFM 15.0751.03
[23] Lie, Sophus, Vorlesungen über kontinuierliche Gruppen mit geometrischen und anderen Anwendungen. Bearbeitet und herausgegeben von Dr. G. Schefferes (1883), Teubner: Teubner Leipzig · Zbl 0248.22010
[24] Lisle, Ian; Huang, Tracy, Algorithmic calculus for Lie determining systems, J. Symb. Comput., 79, 482-498 (2017) · Zbl 1361.35012
[25] Lyakhov, Dmitry A.; Gerdt, Vladimir P.; Michels, Dominik L., Algorithmic verification of linearizability for ordinary differential equations, (Burr, Michael, Proceedings of the 42nd International Symposium on Symbolic and Algebraic Computation. Proceedings of the 42nd International Symposium on Symbolic and Algebraic Computation, ISSAC’17 (2017), ACM: ACM New York), 285-292 · Zbl 1454.34064
[26] Mahomed, Fazal; Leach, Peter, Symmetry Lie algebra of nth order ordinary differential equations, J. Math. Anal. Appl., 151, 1, 80-107 (1990) · Zbl 0719.34018
[27] Olver, Peter, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, vol. 107 (1993), Springer: Springer New York · Zbl 0785.58003
[28] Olver, Peter, Equivalence, Invariance and Symmetry (1995), Cambridge University Press · Zbl 0837.58001
[29] Ovsyannikov, Lev, Group Analysis of Differential Equations (1992), Academic Press: Academic Press New York
[30] Reid, Greg, Algorithms for reducing a system of PDEs to standard form, determining the dimension of its solution space and calculating its Taylor series solution, Eur. J. Appl. Math., 2, 4, 293-318 (1991) · Zbl 0768.35001
[31] Reid, Greg, Finding abstract Lie symmetry algebras of differential equations without integrating determining equations, Eur. J. Appl. Math., 2, 04, 319-340 (1991) · Zbl 0768.35002
[32] Robertz, Daniel, Formal Algorithmic Elimination for PDEs, Lecture Notes in Mathematics, vol. 2121 (2014), Springer · Zbl 1339.35007
[33] Schwarz, Fritz, Algorithmic Lie Theory for Solving Ordinary Differential Equations (2008), Chapman & Hall/CRS: Chapman & Hall/CRS Boca Raton · Zbl 1139.34003
[34] Seiler, Werner, Involution: The Formal Theory of Differential Equations and Its Applications in Computer Algebra, Algorithms and Computation in Mathematics, vol. 24 (2010), Springer: Springer Heidelberg · Zbl 1205.35003
[35] Soh, Celestin; Mahomed, Fazal, Canonical forms for systems of two second-order ordinary differential equations, J. Phys. A, Math. Gen., 34, 2883-2991 (2001) · Zbl 0985.34028
[36] Sookmee, Sakka; Meleshko, Sergey, Linearization of two second-order ordinary differential equations via fiber preserving point transformations, ISRN Math. Anal., 2011, Article 452689 pp. (2011) · Zbl 1241.34047
[37] Thomas, Joseph, Differential Systems, AMS Colloquium Publications, vol. XXI (1937), AMS: AMS New York · Zbl 0016.30404
[38] Thomas, Joseph, Systems and Roots (1962), William Byrd Press: William Byrd Press Richmond · Zbl 0191.38203
[39] Vu, Khai; Jefersson, Grace; Carminati, John, Finding higher symmetries of differential equations using the MAPLE package DESOLVII, Comput. Phys. Commun., 183, 4, 1044-1054 (2012) · Zbl 1308.35002
[40] Wang, Dongming, Elimination Methods (2001), Springer: Springer Wien · Zbl 0964.13014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.