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Comparison of different approaches to construct first integrals for ordinary differential equations. (English) Zbl 1453.34021

Summary: Different approaches to construct first integrals for ordinary differential equations and systems of ordinary differential equations are studied here. These approaches can be grouped into three categories: direct methods, Lagrangian or partial Lagrangian formulations, and characteristic (multipliers) approaches. The direct method and symmetry conditions on the first integrals correspond to first category. The Lagrangian and partial Lagrangian include three approaches: Noether’s theorem, the partial Noether approach, and the Noether approach for the equation and its adjoint as a system. The characteristic method, the multiplier approaches, and the direct construction formula approach require the integrating factors or characteristics or multipliers. The Hamiltonian version of Noether’s theorem is presented to derive first integrals. We apply these different approaches to derive the first integrals of the harmonic oscillator equation. We also study first integrals for some physical models. The first integrals for nonlinear jerk equation and the free oscillations of a two-degree-of-freedom gyroscopic system with quadratic nonlinearities are derived. Moreover, solutions via first integrals are also constructed.

MSC:

34A34 Nonlinear ordinary differential equations and systems
34A05 Explicit solutions, first integrals of ordinary differential equations
34C14 Symmetries, invariants of ordinary differential equations

Software:

GeM
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Full Text: DOI

References:

[1] Naz, R.; Mahomed, F. M.; Mason, D. P., Comparison of different approaches to conservation laws for some partial differential equations in fluid mechanics, Applied Mathematics and Computation, 205, 1, 212-230 (2008) · Zbl 1153.76051 · doi:10.1016/j.amc.2008.06.042
[2] Laplace, P. S., English translation, Celestrial Mechanics, 1966 (1798), New York, NY, USA
[3] Kara, A. H.; Mahomed, F. M., Relationship between symmetries and conservation laws, International Journal of Theoretical Physics, 39, 1, 23-40 (2000) · Zbl 0962.35009 · doi:10.1023/A:1003686831523
[4] Noether, E., Invariant variation problems, Transport Theory and Statistical Physics, 1, 3, 186-207 (1971) · Zbl 0292.49008 · doi:10.1080/00411457108231446
[5] Kara, A. H.; Mahomed, F. M.; Naeem, I.; Wafo Soh, C., Partial Noether operators and first integrals via partial Lagrangians, Mathematical Methods in the Applied Sciences, 30, 16, 2079-2089 (2007) · Zbl 1130.70012 · doi:10.1002/mma.939
[6] Naeem, I.; Mahomed, F. M., First integrals for a general linear system of two second-order ODEs via a partial Lagrangian, Journal of Physics A. Mathematical and Theoretical, 41, 35 (2008) · Zbl 1160.34030 · doi:10.1088/1751-8113/41/35/355207
[7] Naeem, I.; Mahomed, F. M., Partial Noether operators and first integrals for a system with two degrees of freedom, Journal of Nonlinear Mathematical Physics, 15, 165-178 (2008) · Zbl 1362.70025 · doi:10.2991/jnmp.2008.15.s1.15
[8] Freire, I. L.; da Silva, P. L.; Torrisi, M., Lie and Noether symmetries for a class of fourth-order Emden-Fowler equations, Journal of Physics. A. Mathematical and Theoretical, 46, 24 (2013) · Zbl 1281.34046 · doi:10.1088/1751-8113/46/24/245206
[9] Naz, R.; Naeem, I.; Mahomed, F. M., First integrals for two linearly coupled nonlinear Duffing oscillators, Mathematical Problems in Engineering, 2011 (2011) · Zbl 1210.34055 · doi:10.1155/2011/831647
[10] Bozhkov, Y.; Freire, I. L., On the Lane-Emden system in dimension one, Applied Mathematics and Computation, 218, 21, 10762-10766 (2012) · Zbl 1272.34045 · doi:10.1016/j.amc.2012.04.033
[11] Naeem, I.; Mahomed, F. M., Noether, partial Noether operators and first integrals for a linear system, Journal of Mathematical Analysis and Applications, 342, 1, 70-82 (2008) · Zbl 1143.34023 · doi:10.1016/j.jmaa.2007.11.041
[12] Ibragimov, N. H., A new conservation theorem, Journal of Mathematical Analysis and Applications, 333, 1, 311-328 (2007) · Zbl 1160.35008 · doi:10.1016/j.jmaa.2006.10.078
[13] Atherton, R. W.; Homsy, G. M., On the existence and formulation of variational principles for nonlinear differential equations, Studies in Applied Math, 54, 1, 31-60 (1975) · Zbl 0322.49019
[14] Ibragimov, N. H., Integrating factors, adjoint equations and Lagrangians, Journal of Mathematical Analysis and Applications, 318, 2, 742-757 (2006) · Zbl 1102.34002 · doi:10.1016/j.jmaa.2005.11.012
[15] Ibragimov, N. H.; Torrisi, M.; Tracinà, R., Self-adjointness and conservation laws of a generalized Burgers equation, Journal of Physics. A. Mathematical and Theoretical, 44, 14 (2011) · Zbl 1216.35115 · doi:10.1088/1751-8113/44/14/145201
[16] Gandarias, M. L., Weak self-adjoint differential equations, Journal of Physics A. Mathematical and Theoretical, 44 (2011) · Zbl 1223.35203
[17] Ibragimov, N. H., Nonlinear self-adjointness and conservation laws, Journal of Physics A. Mathematical and Theoretical, 44 (2011) · Zbl 1270.35031
[18] Tracinà, R.; Bruzóna, M. S.; Gandarias, M. L.; Torrisi, M., Nonlinear selfadjointness, conservation laws, exact solutions of a system of dispersive evolution equations, Communications in Nonlinear Science and Numerical Simulation, 19, 3036-3043 (2014) · Zbl 1510.35277
[19] Freire, I. L.; Santos Sampaio, J. C., On the nonlinear self-adjointness and local conservation laws for a class of evolution equations unifying many models, Communications in Nonlinear Science and Numerical Simulation, 19, 2, 350-360 (2014) · Zbl 1344.35005 · doi:10.1016/j.cnsns.2013.06.010
[20] Freire, I. L., New classes of nonlinearly self-adjoint evolution equations of third- and fifth-order, Communications in Nonlinear Science and Numerical Simulation, 18, 3, 493-499 (2013) · Zbl 1286.35075 · doi:10.1016/j.cnsns.2012.08.022
[21] Torrisi, M.; Tracinà, R., Quasi self-adjointness of a class of third order nonlinear dispersive equations, Nonlinear Analysis. Real World Applications, 14, 3, 1496-1502 (2013) · Zbl 1261.35131 · doi:10.1016/j.nonrwa.2012.10.013
[22] Ibragimov, N. H.; Torrisi, M.; Tracinà, R., Quasi self-adjoint nonlinear wave equations, Journal of Physics A. Mathematical and Theoretical, 43, 44 (2010) · Zbl 1206.35174 · doi:10.1088/1751-8113/43/44/442001
[23] Steudel, H., Über die zuordnung zwischen invarianzeigenschaften und erhaltungssätzen, Zeitschrift für Naturforschung, 17, 129-132 (1962)
[24] Olver, P. J., Applications of Lie Groups to Differential Equations, 107, xxviii+513 (1993), New York, NY, USA: Springer, New York, NY, USA · Zbl 0785.58003 · doi:10.1007/978-1-4612-4350-2
[25] Naz, R.; Mason, D. P.; Mahomed, F. M., Conservation laws and conserved quantities for laminar two-dimensional and radial jets, Nonlinear Analysis. Real World Applications, 10, 5, 2641-2651 (2009) · Zbl 1177.35171 · doi:10.1016/j.nonrwa.2008.07.003
[26] Naz, R., Conservation laws for some compacton equations using the multiplier approach, Applied Mathematics Letters, 25, 3, 257-261 (2012) · Zbl 1387.35020 · doi:10.1016/j.aml.2011.08.019
[27] Naz, R., Conservation laws for a complexly coupled KdV system, coupled Burgers’ system and Drinfeld-Sokolov-Wilson system via multiplier approach, Communications in Nonlinear Science and Numerical Simulation, 15, 5, 1177-1182 (2010) · Zbl 1221.37119 · doi:10.1016/j.cnsns.2009.05.071
[28] Anco, S. C.; Bluman, G., Integrating factors and first integrals for ordinary differential equations, European Journal of Applied Mathematics, 9, 3, 245-259 (1998) · Zbl 0922.34006 · doi:10.1017/S0956792598003477
[29] Naz, R.; Naeem, I.; Khan, M. D., Conservation laws of some physical models via symbolic package GeM, Mathematical Problems in Engineering, 2013 (2013) · Zbl 1299.35001 · doi:10.1155/2013/897912
[30] Dorodnitsyn, V.; Kozlov, R., Invariance and first integrals of continuous and discrete Hamiltonian equations, Journal of Engineering Mathematics, 66, 253-270 (2010) · Zbl 1188.70054 · doi:10.1007/s10665-009-9312-0
[31] Prince, G. E.; Leach, P. G. L., The Lie theory of extended groups in Hamiltonian mechanics, The Journal of the Australian Mathematical Society B. Applied Mathematics, 3, 3, 941-961 (1980) · Zbl 0433.70005
[32] Ibragimov, N. H.; Nucci, M. C., Integration of third order ordinary differential equations by Lies method: equations admitting three-dimensional Lie algebras, Lie Groups and Their Applications, 1, 4964 (1994) · Zbl 0921.34015
[33] Mahomed, F. M., Symmetry group classification of ordinary differential equations: survey of some results, Mathematical Methods in the Applied Sciences, 30, 16, 1995-2012 (2007) · Zbl 1135.34029 · doi:10.1002/mma.934
[34] Kara, A. H.; Mahomed, F. M.; Adam, A. A., Reduction of differential equations using Lie and Noether symmetries associated with first integrals, Lie Groups and their Applications, 1, 1, 137-145 (1994) · Zbl 0938.35505
[35] Stephani, H., Differential Equations: Their Solutions Using Symmetries, xii+260 (1989), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0704.34001
[36] Sjöberg, A.; Mahomed, F. M., Non-local symmetries and conservation laws for one-dimensional gas dynamics equations, Applied Mathematics and Computation, 150, 2, 379-397 (2004) · Zbl 1102.76059 · doi:10.1016/S0096-3003(03)00259-5
[37] Sjöberg, A.; Mahomed, F. M., The association of non-local symmetries with conservation laws: applications to the heat and Burgers’ equations, Applied Mathematics and Computation, 168, 2, 1098-1108 (2005) · Zbl 1084.35075 · doi:10.1016/j.amc.2004.10.006
[38] Sjöberg, A., Double reduction of PDEs from the association of symmetries with conservation laws with applications, Applied Mathematics and Computation, 184, 2, 608-616 (2007) · Zbl 1116.35004 · doi:10.1016/j.amc.2006.06.059
[39] Sjöberg, A., On double reductions from symmetries and conservation laws, Nonlinear Analysis. Real World Applications, 10, 6, 3472-3477 (2009) · Zbl 1179.35038 · doi:10.1016/j.nonrwa.2008.09.029
[40] Bokhari, A. H.; Al-Dweik, A. Y.; Zaman, F. D.; Kara, A. H.; Mahomed, F. M., Generalization of the double reduction theory, Nonlinear Analysis. Real World Applications, 11, 5, 3763-3769 (2010) · Zbl 1201.35014 · doi:10.1016/j.nonrwa.2010.02.006
[41] Bokhari, A. H.; Al-Dweik, A. Y.; Kara, A. H.; Mahomed, F. M.; Zaman, F. D., Double reduction of a nonlinear (2 + 1) wave equation via conservation laws, Communications in Nonlinear Science and Numerical Simulation, 16, 3, 1244-1253 (2011) · Zbl 1221.35244 · doi:10.1016/j.cnsns.2010.07.007
[42] Naz, R.; Ali, Z.; Naeem, I., Reductions and new exact solutions of ZK, Gardner KP, and modified KP equations via generalized double reduction theorem, Abstract and Applied Analysis, 2013 (2013) · Zbl 1293.35066 · doi:10.1155/2013/340564
[43] Naz, R.; Khan, M. D.; Naeem, I., Conservation laws and exact solutions of a class of non linear regularized long wave equations via double reduction theory and Lie symmetries, Communications in Nonlinear Science and Numerical Simulation, 18, 4, 826-834 (2013) · Zbl 1255.35166 · doi:10.1016/j.cnsns.2012.09.011
[44] Ibragimov, N. H., CRC Handbook of Lie Group Analysis of Differential Equations, 1-3 (1996), Boca Raton, Fla, USA: Chemical Rubber Company, Boca Raton, Fla, USA · Zbl 0864.35003
[45] Ibragimov, N. H., Elementary Lie Group Analysis and Ordinary Differential Equations, 4, xviii+347 (1999), Chichester, UK: John Wiley & Sons, Chichester, UK · Zbl 1047.34001
[46] Leach, P. G. L., Lie symmetries and Noether symmetries, Applicable Analysis and Discrete Mathematics, 6, 2, 238-246 (2012) · Zbl 1289.70027 · doi:10.2298/AADM120625015L
[47] Ibragimov, N. H.; Kara, A. H.; Mahomed, F. M., Lie-Bäcklund and Noether symmetries with applications, Nonlinear Dynamics, 15, 2, 115-136 (1998) · Zbl 0912.35011 · doi:10.1023/A:1008240112483
[48] Gottlieb, H. P. W., Harmonic balance approach to periodic solutions of non-linear jerk equations, Journal of Sound and Vibration, 271, 671-683 (2004) · Zbl 1236.34049 · doi:10.1016/S0022-460X(03)00299-2
[49] Gottlieb, H. P. W., Harmonic balance approach to limit cycles for nonlinear jerk equations, Journal of Sound and Vibration, 297, 1-2, 243-250 (2006) · Zbl 1243.70020 · doi:10.1016/j.jsv.2006.03.047
[50] Ramos, J. I., Analytical and approximate solutions to autonomous, nonlinear, third-order ordinary differential equations, Nonlinear Analysis. Real World Applications, 11, 3, 1613-1626 (2010) · Zbl 1201.34014 · doi:10.1016/j.nonrwa.2009.03.023
[51] Nayfeh, A. H., Introduction to Perturbation Techniques, xiv+519 (1981), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0449.34001
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