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Conditional symmetries of nonlinear third-order ordinary differential equations
a. | International Institute for Symmetry Analysis and Mathematical Modeling, North-West University, Mafikeng Campus, P Bag X2046, Mafikeng, South Africa |
b. | School of Computer Science and Applied Mathematics DST-NRF Centre of Excellence in Mathematical and Statistical Sciences, University of the Witwatersrand, Johannesburg, Wits 2050, South Africa |
In this work, we take as our base scalar second-order ordinary differential equations (ODEs) which have seven equivalence classes with each class possessing three Lie point symmetries. We show how one can calculate the conditional symmetries of third-order non-linear ODEs subject to root second-order nonlinear ODEs which admit three point symmetries. Moreover, we show when scalar second-order ODEs taken as first integrals or conditional first integrals are inherited as Lie point symmetries and as conditional symmetries of the derived third-order ODE. Furthermore, the derived scalar nonlinear third-order ODEs without substitution are considered for their conditional symmetries subject to root second-order ODEs having three symmetries.
References:
[1] |
B. Abraham-Shrauner, K. S. Govinder and P. G. L. Leach,
Integration of second order ordinary differential equations not possessing Lie point symmetries, Phys. Lett. A, 203 (1995), 169-174.
doi: 10.1016/0375-9601(95)00426-4. |
[2] |
D. J. Arrigo and J. M. Hill,
Nonclassical symmetries for nonlinear diffusion and absorption, Stud. Appl. Math., 94 (1995), 21-39.
doi: 10.1002/sapm199594121. |
[3] |
G. W. Bluman and J. D. Cole,
The general similarity solution of the heat equation, J. Math. Mech., 18 (1969), 1025-1042.
|
[4] |
S. S. Chern, Sur la géométrie d'une équation différentielle du troiséme ordre, CR Acad Sci Paris, 1937. |
[5] |
S. S. Chern,
The geometry of the differential equation $ y''''=F(x, y, y'', y''')$, Sci. Rep. Nat. Tsing Hua Univ., 4 (1940), 97-111.
|
[6] |
R. Cherniha and M. Henkel,
On non-linear partial differential equations with an infinite-dimensional conditional symmetry, J. Math. Anal. Appl., 298 (2004), 487-500.
doi: 10.1016/j.jmaa.2004.05.038. |
[7] |
R. Cherniha and O. Pliukhin,
New conditional symmetries and exact solutions of reaction-diffusion-convection equations with exponential nonlinearities, J. Math. Anal. Appl., 403 (2013), 23-37.
doi: 10.1016/j.jmaa.2013.02.010. |
[8] |
P. A. Clarkson,
Nonclassical symmetry reductions of the Boussinesq equation, Chaos Solitons Fractals, 5 (1995), 2261-2301.
doi: 10.1016/0960-0779(94)E0099-B. |
[9] |
P. A. Clarkson,
Nonclassical symmetry reductions of nonlinear partial differential equations, Math. Comput. Model., 18 (1993), 45-68.
doi: 10.1016/0895-7177(93)90214-J. |
[10] |
P. L. Da Silva and I. L. Freire, Symmetry analysis of a class of autonomous even-order ordinary
differential equations, IMA J. Appl. Math., 80 (2015), 1739-1758, arXiv: 1311.0313v2 [mathph] 7 march 2014.
doi: 10.1093/imamat/hxv014. |
[11] |
A. Fatima and F. M. Mahomed,
Conditional symmetries for ordinary differential equations and applications, Int. J. Non-Linear Mech., 67 (2014), 95-105.
doi: 10.1016/j.ijnonlinmec.2014.08.013. |
[12] |
W. I. Fushchich,
Conditional symmetry of equations of nonlinear mathematical physics, Ukrain. Math. Zh., 43 (1991), 1456-1470.
doi: 10.1007/BF01067273. |
[13] |
G. Gaeta,
Conditional symmetries and conditional constants of motion for dynamical systems, Report of the Centre de Physique Theorique Ecole Polytechnique, Palaiseau France, 1 (1993), 1-24.
|
[14] |
A. Goriely,
Integrability and Nonintegrability of Dynamical Systems, Advanced Series in Nonlinear Dynamics, 19. World Scientific Publishing Co. , Inc. , River Edge, NJ, 2001.
doi: 10.1142/9789812811943. |
[15] |
G. Grebot,
The characterization of third order ordinary differential equations admitting a transitive fibre-preserving point symmetry group, J. Math. Anal. Appl., 206 (1997), 364-388.
doi: 10.1006/jmaa.1997.5219. |
[16] |
N. H. Ibragimov and S. V. Meleshko,
Linearization of third order ordinary differential equations by point and contact transformations, J. Math. Anal. Appl., 308 (2005), 266-289.
doi: 10.1016/j.jmaa.2005.01.025. |
[17] |
N. H. Ibragimov, S. V. Meleshko and S. Suksern, Linearization of fourth order ordinary differential equation by point transformations, J. Phys. A, 41 (2008), 235206, 19 pp.
doi: 10.1088/1751-8113/41/23/235206. |
[18] |
A. H. Kara and F. M. Mahomed,
A Basis of conservation laws for partial differential equations, J. Nonlinear Math. Phys., 9 (2002), 60-72.
doi: 10.2991/jnmp.2002.9.s2.6. |
[19] |
M. Kunzinger and R. O. Popovych,
Generalized conditional symmetries of evolution equations, J. Math. Anal. Appl., 379 (2011), 444-460.
doi: 10.1016/j.jmaa.2011.01.027. |
[20] |
P. G. L. Leach,
Equivalence classes of second-order ordinary differential equations with three-dimensional Lie algebras of point symmetries and linearisation, J. Math. Anal. Appl., 284 (2003), 31-48.
doi: 10.1016/S0022-247X(03)00147-1. |
[21] |
S. Lie,
Lectures on Differential Equations with Known Infinitesimal Transformations, Leipzig, Teubner, 1981 (in German Lie's Lectures by G. Sheffers). |
[22] |
F. M. Mahomed, I. Naeem and A. Qadir,
Conditional linearizability criteria for a system of third-order ordinary differential equations, Nonlinear Anal. B: Real World Appl., 10 (2009), 3404-3412.
doi: 10.1016/j.nonrwa.2008.09.021. |
[23] |
F. M. Mahomed,
Symmetry group classification of ordinary differential equations: Survey of some results, Math. Meth. Appl. Sci., 30 (2007), 1995-2012.
doi: 10.1002/mma.934. |
[24] |
F. M. Mahomed and A. Qadir,
Classification of ordinary differential equations by conditional linearizability and symmetry, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 573-584.
doi: 10.1016/j.cnsns.2011.06.012. |
[25] |
F. M. Mahomed and A. Qadir,
Conditional linearizability criteria for third order ordinary differential equations, J. Nonlinear Math. Phys., 15 (2008), 124-133.
doi: 10.2991/jnmp.2008.15.s1.11. |
[26] |
F. M. Mahomed and A. Qadir,
Conditional linearizability of fourth-order semilinear ordinary differential equations, J. Nonlinear Math. Phys., 16 (2009), 165-178.
doi: 10.1142/S140292510900039X. |
[27] |
F. M. Mahomed and P. G. L. Leach,
Symmetry Lie algebras of $n$ order ordinary differential equations, J. Math. Anal. Appl., 151 (1990), 80-107.
doi: 10.1016/0022-247X(90)90244-A. |
[28] |
S. V. Meleshko,
On linearization of third order ordinary differential equation, J. Phys. A, 39 (2006), 15135-15145.
doi: 10.1088/0305-4470/39/49/005. |
[29] |
S. Neut and M. Petitot,
La géométrie de l'équation $ y'''=f(x,y,y',y'')$, CR Acad. Sci. Paris Sér. I., 335 (2002), 515-518.
doi: 10.1016/S1631-073X(02)02507-4. |
[30] |
P. J. Olver and E. M. Vorob'ev, Nonclassical and conditional symmetries, in: N. H. Ibragiminov (Ed. ), CRC Handbook of Lie Group Analysis, vol. 3, CRC Press, Boca Raton, 1994. |
[31] |
E. Pucci and G. Saccomandi,
Evolution equations, invariant surface conditions and functional separation of variables, Physica D: Nonlinear Phenomena, 139 (2000), 28-47.
doi: 10.1016/S0167-2789(99)00224-9. |
[32] |
E. Pucci and G. Saccomandi,
On the weak symmetry groups of partial differential equations, J. Math. Anal. Appl., 163 (1992), 588-598.
doi: 10.1016/0022-247X(92)90269-J. |
[33] |
E. Pucci,
Similarity reductions of partial differential equations, J. Phys. A, 25 (1992), 2631-2640.
doi: 10.1088/0305-4470/25/9/032. |
[34] |
W. Sarlet, P. G. L. Leach and F. Cantrijn,
First integrals versus configurational invariants and a weak form of complete integrability, Physica D, 17 (1985), 87-98.
doi: 10.1016/0167-2789(85)90136-8. |
[35] |
S. Spichak and V. Stognii,
Conditional symmetry and exact solutions of the Kramers equation, Nonlinear Math. Phys., 2 (1997), 450-454.
|
[36] |
S. Suksern, N. H. Ibragimov and S. V. Meleshko,
Criteria for the fourth order ordinary differential equations to be linearizable by contact transformations, Common. Nonlinear Sci. Number. Simul., 14 (2009), 2619-2628.
doi: 10.1016/j.cnsns.2008.09.021. |
[37] |
C. Wafo Soh and F. M. Mahomed,
Linearization criteria for a system of second-order ordinary differential equations, Int. J. Non-Linear Mech., 36 (2001), 671-677.
doi: 10.1016/S0020-7462(00)00032-9. |
show all references
References:
[1] |
B. Abraham-Shrauner, K. S. Govinder and P. G. L. Leach,
Integration of second order ordinary differential equations not possessing Lie point symmetries, Phys. Lett. A, 203 (1995), 169-174.
doi: 10.1016/0375-9601(95)00426-4. |
[2] |
D. J. Arrigo and J. M. Hill,
Nonclassical symmetries for nonlinear diffusion and absorption, Stud. Appl. Math., 94 (1995), 21-39.
doi: 10.1002/sapm199594121. |
[3] |
G. W. Bluman and J. D. Cole,
The general similarity solution of the heat equation, J. Math. Mech., 18 (1969), 1025-1042.
|
[4] |
S. S. Chern, Sur la géométrie d'une équation différentielle du troiséme ordre, CR Acad Sci Paris, 1937. |
[5] |
S. S. Chern,
The geometry of the differential equation $ y''''=F(x, y, y'', y''')$, Sci. Rep. Nat. Tsing Hua Univ., 4 (1940), 97-111.
|
[6] |
R. Cherniha and M. Henkel,
On non-linear partial differential equations with an infinite-dimensional conditional symmetry, J. Math. Anal. Appl., 298 (2004), 487-500.
doi: 10.1016/j.jmaa.2004.05.038. |
[7] |
R. Cherniha and O. Pliukhin,
New conditional symmetries and exact solutions of reaction-diffusion-convection equations with exponential nonlinearities, J. Math. Anal. Appl., 403 (2013), 23-37.
doi: 10.1016/j.jmaa.2013.02.010. |
[8] |
P. A. Clarkson,
Nonclassical symmetry reductions of the Boussinesq equation, Chaos Solitons Fractals, 5 (1995), 2261-2301.
doi: 10.1016/0960-0779(94)E0099-B. |
[9] |
P. A. Clarkson,
Nonclassical symmetry reductions of nonlinear partial differential equations, Math. Comput. Model., 18 (1993), 45-68.
doi: 10.1016/0895-7177(93)90214-J. |
[10] |
P. L. Da Silva and I. L. Freire, Symmetry analysis of a class of autonomous even-order ordinary
differential equations, IMA J. Appl. Math., 80 (2015), 1739-1758, arXiv: 1311.0313v2 [mathph] 7 march 2014.
doi: 10.1093/imamat/hxv014. |
[11] |
A. Fatima and F. M. Mahomed,
Conditional symmetries for ordinary differential equations and applications, Int. J. Non-Linear Mech., 67 (2014), 95-105.
doi: 10.1016/j.ijnonlinmec.2014.08.013. |
[12] |
W. I. Fushchich,
Conditional symmetry of equations of nonlinear mathematical physics, Ukrain. Math. Zh., 43 (1991), 1456-1470.
doi: 10.1007/BF01067273. |
[13] |
G. Gaeta,
Conditional symmetries and conditional constants of motion for dynamical systems, Report of the Centre de Physique Theorique Ecole Polytechnique, Palaiseau France, 1 (1993), 1-24.
|
[14] |
A. Goriely,
Integrability and Nonintegrability of Dynamical Systems, Advanced Series in Nonlinear Dynamics, 19. World Scientific Publishing Co. , Inc. , River Edge, NJ, 2001.
doi: 10.1142/9789812811943. |
[15] |
G. Grebot,
The characterization of third order ordinary differential equations admitting a transitive fibre-preserving point symmetry group, J. Math. Anal. Appl., 206 (1997), 364-388.
doi: 10.1006/jmaa.1997.5219. |
[16] |
N. H. Ibragimov and S. V. Meleshko,
Linearization of third order ordinary differential equations by point and contact transformations, J. Math. Anal. Appl., 308 (2005), 266-289.
doi: 10.1016/j.jmaa.2005.01.025. |
[17] |
N. H. Ibragimov, S. V. Meleshko and S. Suksern, Linearization of fourth order ordinary differential equation by point transformations, J. Phys. A, 41 (2008), 235206, 19 pp.
doi: 10.1088/1751-8113/41/23/235206. |
[18] |
A. H. Kara and F. M. Mahomed,
A Basis of conservation laws for partial differential equations, J. Nonlinear Math. Phys., 9 (2002), 60-72.
doi: 10.2991/jnmp.2002.9.s2.6. |
[19] |
M. Kunzinger and R. O. Popovych,
Generalized conditional symmetries of evolution equations, J. Math. Anal. Appl., 379 (2011), 444-460.
doi: 10.1016/j.jmaa.2011.01.027. |
[20] |
P. G. L. Leach,
Equivalence classes of second-order ordinary differential equations with three-dimensional Lie algebras of point symmetries and linearisation, J. Math. Anal. Appl., 284 (2003), 31-48.
doi: 10.1016/S0022-247X(03)00147-1. |
[21] |
S. Lie,
Lectures on Differential Equations with Known Infinitesimal Transformations, Leipzig, Teubner, 1981 (in German Lie's Lectures by G. Sheffers). |
[22] |
F. M. Mahomed, I. Naeem and A. Qadir,
Conditional linearizability criteria for a system of third-order ordinary differential equations, Nonlinear Anal. B: Real World Appl., 10 (2009), 3404-3412.
doi: 10.1016/j.nonrwa.2008.09.021. |
[23] |
F. M. Mahomed,
Symmetry group classification of ordinary differential equations: Survey of some results, Math. Meth. Appl. Sci., 30 (2007), 1995-2012.
doi: 10.1002/mma.934. |
[24] |
F. M. Mahomed and A. Qadir,
Classification of ordinary differential equations by conditional linearizability and symmetry, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 573-584.
doi: 10.1016/j.cnsns.2011.06.012. |
[25] |
F. M. Mahomed and A. Qadir,
Conditional linearizability criteria for third order ordinary differential equations, J. Nonlinear Math. Phys., 15 (2008), 124-133.
doi: 10.2991/jnmp.2008.15.s1.11. |
[26] |
F. M. Mahomed and A. Qadir,
Conditional linearizability of fourth-order semilinear ordinary differential equations, J. Nonlinear Math. Phys., 16 (2009), 165-178.
doi: 10.1142/S140292510900039X. |
[27] |
F. M. Mahomed and P. G. L. Leach,
Symmetry Lie algebras of $n$ order ordinary differential equations, J. Math. Anal. Appl., 151 (1990), 80-107.
doi: 10.1016/0022-247X(90)90244-A. |
[28] |
S. V. Meleshko,
On linearization of third order ordinary differential equation, J. Phys. A, 39 (2006), 15135-15145.
doi: 10.1088/0305-4470/39/49/005. |
[29] |
S. Neut and M. Petitot,
La géométrie de l'équation $ y'''=f(x,y,y',y'')$, CR Acad. Sci. Paris Sér. I., 335 (2002), 515-518.
doi: 10.1016/S1631-073X(02)02507-4. |
[30] |
P. J. Olver and E. M. Vorob'ev, Nonclassical and conditional symmetries, in: N. H. Ibragiminov (Ed. ), CRC Handbook of Lie Group Analysis, vol. 3, CRC Press, Boca Raton, 1994. |
[31] |
E. Pucci and G. Saccomandi,
Evolution equations, invariant surface conditions and functional separation of variables, Physica D: Nonlinear Phenomena, 139 (2000), 28-47.
doi: 10.1016/S0167-2789(99)00224-9. |
[32] |
E. Pucci and G. Saccomandi,
On the weak symmetry groups of partial differential equations, J. Math. Anal. Appl., 163 (1992), 588-598.
doi: 10.1016/0022-247X(92)90269-J. |
[33] |
E. Pucci,
Similarity reductions of partial differential equations, J. Phys. A, 25 (1992), 2631-2640.
doi: 10.1088/0305-4470/25/9/032. |
[34] |
W. Sarlet, P. G. L. Leach and F. Cantrijn,
First integrals versus configurational invariants and a weak form of complete integrability, Physica D, 17 (1985), 87-98.
doi: 10.1016/0167-2789(85)90136-8. |
[35] |
S. Spichak and V. Stognii,
Conditional symmetry and exact solutions of the Kramers equation, Nonlinear Math. Phys., 2 (1997), 450-454.
|
[36] |
S. Suksern, N. H. Ibragimov and S. V. Meleshko,
Criteria for the fourth order ordinary differential equations to be linearizable by contact transformations, Common. Nonlinear Sci. Number. Simul., 14 (2009), 2619-2628.
doi: 10.1016/j.cnsns.2008.09.021. |
[37] |
C. Wafo Soh and F. M. Mahomed,
Linearization criteria for a system of second-order ordinary differential equations, Int. J. Non-Linear Mech., 36 (2001), 671-677.
doi: 10.1016/S0020-7462(00)00032-9. |
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Representative 2nd-order ODE |
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