Second-order delay ordinary differential equations, their symmetries and application to a traffic problem. (English) Zbl 1519.34074
Summary: This article is the third in a series, the aim of which is to use Lie group theory to obtain exact analytic solutions of delay ordinary differential systems (DODSs). Such a system consists of two equations involving one independent variable \(x\) and one dependent variable \(y\). As opposed to ordinary differential equations (ODEs) the variable \(x\) figures in more than one point (we consider the case of two points, \(x\) and \(x_-\)). The dependent variable \(y\) and its derivatives figure in both \(x\) and \(x_-\). Two previous articles were devoted to first-order DODSs, here we concentrate on a large class of second-order ones. We show that within this class the symmetry algebra can be of dimension \(n\) with \(0 \leqslant n \leqslant 6\) for nonlinear DODSs and must be infinite-dimensional for linear or linearizable ones. The symmetry algebras can be used to obtain exact particular group invariant solutions. As a specific application we present some exact solutions of a DODS model of traffic flow.
MSC:
| 34K05 | General theory of functional-differential equations |
| 22E05 | Local Lie groups |
| 34C14 | Symmetries, invariants of ordinary differential equations |
| 76A30 | Traffic and pedestrian flow models |
| 90B20 | Traffic problems in operations research |
Keywords:
delay ordinary differential equation; Lie group classification; particular solutions for traffic flow modelReferences:
| [1] | Bluman G W and Kumei S 1989 Symmetries and Differential Equations (Berlin: Springer) · Zbl 0698.35001 · doi:10.1007/978-1-4757-4307-4 |
| [2] | Boyko V M, Popovych R O and Shapoval N M 2015 Equivalence groupoids of classes of linear ordinary differential equations and their group classification J. Phys.: Conf. Ser.621 012002 · doi:10.1088/1742-6596/621/1/012002 |
| [3] | Chandler R E, Herman R and Montroll E W 1958 Traffic dynamics: studies in car following Oper. Res.6 165-84 · doi:10.1287/opre.6.2.165 |
| [4] | Dorodnitsyn V A 1991 Transformation groups in net spaces J. Math. Sci.55 1490-517 · Zbl 0727.65074 · doi:10.1007/bf01097535 |
| [5] | Dorodnitsyn V 2011 Applications of Lie Groups to Difference Equations(Differential and integral equations series) (London: Chapman and Hall) · Zbl 1236.39002 |
| [6] | Dorodnitsyn V, Kozlov R and Winternitz P 2000 Lie group classification of second-order ordinary difference equations J. Math. Phys.41 480-504 · Zbl 0981.39010 · doi:10.1063/1.533142 |
| [7] | Dorodnitsyn V, Kozlov R and Winternitz P 2004 Continuous symmetries of Lagrangians and exact solutions of discrete equations J. Math. Phys.45 336-59 · Zbl 1070.70010 · doi:10.1063/1.1625418 |
| [8] | Dorodnitsyn V A, Kozlov R, Meleshko S V and Winternitz P 2018 Lie group classification of first-order delay ordinary differential equations J. Phys. A: Math. Theor.51 205202 · Zbl 1397.34106 · doi:10.1088/1751-8121/aaba91 |
| [9] | Dorodnitsyn V A, Kozlov R, Meleshko S V and Winternitz P 2018 Linear or linearizable first-order delay ordinary differential equations and their Lie point symmetries J. Phys. A: Math. Theor.51 205203 · Zbl 1402.34070 · doi:10.1088/1751-8121/aab3e9 |
| [10] | Driver R D 1977 Ordinary and Delay Differential Equations (Berlin: Springer) · Zbl 0374.34001 · doi:10.1007/978-1-4684-9467-9 |
| [11] | Elsgolts L E 1955 Qualitative Methods in Mathematical Analysis (Moscow: GITTL) |
| [12] | Gazis D C, Herman R and Rothery R W 1961 Nonlinear follow-the-leader models of traffic flow Oper. Res.9 545-67 · Zbl 0096.14205 · doi:10.1287/opre.9.4.545 |
| [13] | González-López A, Kamran N and Olver P J 1992 Lie algebras of vector fields in the real plane Proc. London Math. Soc.64 339-68 · Zbl 0872.17022 · doi:10.1112/plms/s3-64.2.339 |
| [14] | Grigoriev Y N, Ibragimov N H, Kovalev V F and Meleshko S V 2010 Symmetries of Integro-Differential Equations and Their Applications in Mechanics and Plasma Physics(Lecture Notes in Physics) vol 806 (Berlin: Springer) · Zbl 1203.45006 · doi:10.1007/978-90-481-3797-8 |
| [15] | Hale J 1977 Functional Differential Equations (Berlin: Springer) · Zbl 0352.34001 · doi:10.1007/978-1-4612-9892-2 |
| [16] | Kumei S and Bluman G W 1982 When nonlinear differential equations are equivalent to linear differential equations SIAM J. Appl. Math.42 1157-73 · Zbl 0506.35003 · doi:10.1137/0142079 |
| [17] | Levi D, Tremblay S and Winternitz P 2000 Lie point symmetries of difference equations and lattices J. Phys. A: Math. Gen.33 8507-23 · Zbl 0966.39007 · doi:10.1088/0305-4470/33/47/313 |
| [18] | Levi D, Tremblay S and Winternitz P 2001 Lie symmetries of multidimensional difference equations J. Phys. A: Math. Gen.34 9507-24 · Zbl 0998.39008 · doi:10.1088/0305-4470/34/44/311 |
| [19] | Levi D, Vinet L and Winternitz P 1997 Lie group formalism for difference equations J. Phys. A: Math. Gen.30 633-49 · Zbl 0956.39013 · doi:10.1088/0305-4470/30/2/024 |
| [20] | Levi D and Winternitz P 1991 Continuous symmetries of discrete equations Phys. Lett. A 152 335-8 · doi:10.1016/0375-9601(91)90733-o |
| [21] | Levi D and Winternitz P 2006 Continuous symmetries of difference equations J. Phys. A: Math. Gen.39 R1-R63 · Zbl 1112.37053 · doi:10.1088/0305-4470/39/2/r01 |
| [22] | Lie S 1880 Theorie der Transformationsgruppen I Math. Ann.16 441-528 · JFM 12.0292.01 · doi:10.1007/bf01446218 |
| [23] | Lie S 1880 Gesammelte Abhandlungen vol 6 (Leipzig: BG Teubner) pp 1-94 |
| [24] | Lie S 1924 Gruppenregister Gesammelte Abhandlungen vol 5 (Leipzig: BG Teubner) pp 767-3 |
| [25] | Kolmanovskii V and Myshkis A 1992 Applied Theory of Functional Differential Equations (Dordrecht: Kluwer) · Zbl 0785.34005 · doi:10.1007/978-94-015-8084-7 |
| [26] | Lighthill M J and Whitham G B 1955 On kinematic waves 2. A theory of traffic flow on long crowded roads Proc. R. Soc. A 229 317-45 · Zbl 0064.20906 · doi:10.1098/rspa.1955.0089 |
| [27] | Long F S and Meleshko S V 2017 Symmetry analysis of the nonlinear two-dimensional Klein-Gordon equation with a time-varying delay Math. Methods Appl. Sci.40 4658-73 · Zbl 1368.76058 · doi:10.1002/mma.4332 |
| [28] | Muhsen L and Maan N 2014 New approach to classify second order linear delay differential equations with constant coefficients to solvable Lie algebra IJMA8 973-93 · doi:10.12988/ijma.2014.44101 |
| [29] | Myshkis A D 1989 Differential Equations, Ordinary with Distributed Arguments Encyclopaedia of Mathematics vol 3 (Dordrecht: Kluwer) pp 144-7 |
| [30] | Nagel K, Wagner P and Woesler R 2003 Still flowing: approaches to traffic flow and traffic jam modeling Oper. Res.51 681-710 · Zbl 1165.90375 · doi:10.1287/opre.51.5.681.16755 |
| [31] | Nass A M 2019 Lie symmetry analysis and exact solutions of fractional ordinary differential equations with neutral delay Appl. Math. Comput.347 370-80 · Zbl 1428.34117 · doi:10.1016/j.amc.2018.11.002 |
| [32] | Nass A M 2019 Symmetry analysis of space-time fractional Poisson equation with a delay Quaest. Math.42 1221-35 · Zbl 1428.35017 · doi:10.2989/16073606.2018.1513095 |
| [33] | Newell G F 1961 Nonlinear effects in the dynamics of car following Oper. Res.9 209-29 · Zbl 0211.52301 · doi:10.1287/opre.9.2.209 |
| [34] | Olver P J 1986 Applications of Lie Groups to Differential Equations (Berlin: Springer) · Zbl 0588.22001 · doi:10.1007/978-1-4684-0274-2 |
| [35] | Olver P J 1995 Equivalence, Invariants and Symmetry (Cambridge: Cambridge University Press) · Zbl 0837.58001 · doi:10.1017/CBO9780511609565 |
| [36] | Ovsiannikov L V 1982 Group Analysis of Differential Equations (New York: Academic) · Zbl 0485.58002 |
| [37] | Ozbenli E and Vedula P 2020 Construction of invariant compact finite-difference schemes Phys. Rev. E 101 023303 · doi:10.1103/physreve.101.023303 |
| [38] | Patera J and Winternitz P 1977 Subalgebras of real three‐ and four‐dimensional Lie algebras J. Math. Phys.18 1449-55 · Zbl 0412.17007 · doi:10.1063/1.523441 |
| [39] | Patera J, Sharp R T, Winternitz P and Zassenhaus H 1977 Continuous subgroups of the fundamental groups of physics. III. The de Sitter groups J. Math. Phys.18 2259-88 · Zbl 0372.22008 · doi:10.1063/1.523237 |
| [40] | Polyanin A D and Zhurov A I 2014 The functional constraints method: application to non-linear delay reaction-diffusion equations with varying transfer coefficients Int. J. Non-Linear Mech.67 267-77 · doi:10.1016/j.ijnonlinmec.2014.09.008 |
| [41] | Polyanin A D and Zhurov A I 2014 Generalized and functional separable solutions to nonlinear delay Klein-Gordon equations Commun. Nonlinear Sci. Numer. Simul.19 2676-89 · Zbl 1510.35361 · doi:10.1016/j.cnsns.2013.12.021 |
| [42] | Polyanin A D and Sorokin V G 2015 Nonlinear delay reaction-diffusion equations: traveling-wave solutions in elementary functions Appl. Math. Lett.46 38-43 · Zbl 1524.34161 · doi:10.1016/j.aml.2015.01.023 |
| [43] | Polyanin A D and Zhurov A I 2014 The generating equations method: constructing exact solutions to delay reaction-diffusion systems and other non-linear coupled delay PDEs Int. J. Non-Linear Mech.71 104-15 · doi:10.1016/j.ijnonlinmec.2015.01.002 |
| [44] | Popovych R O, Boyko V M, Nesterenko M O and Lutfullin M W 2003 Realizations of real low-dimensional Lie algebras J. Phys. A: Math. Gen.36 7337-60 · Zbl 1040.17021 · doi:10.1088/0305-4470/36/26/309 |
| [45] | Smith H 2011 An Introduction to Delay Differential Equations with Applications to the Life Sciences (Berlin: Springer) · Zbl 1227.34001 · doi:10.1007/978-1-4419-7646-8 |
| [46] | Šnobl L and Winternitz P 2014 Classification and Identification of Lie Algebras(CRM Monograph Series) vol 33 (Providence, RI: American Mathematical Society) · Zbl 1331.17001 · doi:10.1090/crmm/033 |
| [47] | Tanthanuch J 2012 Symmetry analysis of the nonhomogeneous inviscid Burgers equation with delay Commun. Nonlinear Sci. Numer. Simul.17 4978-87 · Zbl 1352.35154 · doi:10.1016/j.cnsns.2012.05.031 |
| [48] | Tyagi V, Darbha S and Rajagopal K R 2009 A review of the mathematical models for traffic flow Int. J. Adv. Eng. Sci. Appl. Math.1 53-68 · doi:10.1007/s12572-009-0005-8 |
| [49] | Whitham G B 1990 Exact solutions for a discrete system arising in traffic flow Proc. R. Soc. A 428 49-69 · Zbl 0698.90030 · doi:10.1098/rspa.1990.0025 |
| [50] | Wu J 1996 Theory and Applications of Partial Functional Differential Equations (Berlin: Springer) · Zbl 0870.35116 · doi:10.1007/978-1-4612-4050-1 |
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