Some variational principles associated with ODEs of maximal symmetry. Part 1: Equations in canonical form
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Jean-Claude Ndogmo
Abstract
Variational and divergence symmetries are studied in this paper for linear equations of maximal symmetry in canonical form, and the associated first integrals are given in explicit form. All the main results obtained are formulated as theorems or conjectures for equations of a general order. Some of these results apply to linear equations of a general form and of arbitrary orders or having a symmetry algebra of arbitrary dimension.
Funding statement: This research is supported by research grants from the University of Venda (grant number I538) and from the NRF of South Africa (grant number 97822).
References
[1] S. Anco, G. Bluman and T. Wolf, Invertible mappings of nonlinear PDEs to linear PDEs through admitted conservation laws, Acta Appl. Math. 101 (2008), no. 1–3, 21–38. 10.1007/s10440-008-9205-7Search in Google Scholar
[2] J. J. H. Bashingwa, A. H. Bokhari, A. H. Kara and F. D. Zaman, The geometry and invariance properties for certain classes of metrics with neutral signature, Int. J. Geom. Methods Mod. Phys. 13 (2016), no. 6, Article ID 1650080. 10.1142/S0219887816500808Search in Google Scholar
[3] G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Appl. Math. Sci. 81, Springer, New York, 1989. 10.1007/978-1-4757-4307-4Search in Google Scholar
[4] R. de la Rosa, M. L. Gandarias and M. S. Bruzón, On symmetries and conservation laws of a Gardner equation involving arbitrary functions, Appl. Math. Comput. 290 (2016), 125–134. 10.1016/j.amc.2016.05.050Search in Google Scholar
[5] G. P. Flessas, K. S. Govinder and P. G. L. Leach, Characterisation of the algebraic properties of first integrals of scalar ordinary differential equations of maximal symmetry, J. Math. Anal. Appl. 212 (1997), no. 2, 349–374. 10.1006/jmaa.1997.5506Search in Google Scholar
[6]
J. Krause and L. Michel,
Équations différentielles linéaires d’ordre
[7] S. Lie, Klassifikation und Integration von gewöhnlichen Differentialgleichungen zwischen xy, die eine Gruppe von Transformationen gestatten, Math. Ann. 32 (1888), no. 2, 213–281. 10.1007/BF01444068Search in Google Scholar
[8] A. B. Magan, D. P. Mason and F. M. Mahomed, Analytical solution in parametric form for the two-dimensional free jet of a power-law fluid, Int. J. Nonlinear Mech. 851 (2016), 94–108. 10.1016/j.ijnonlinmec.2016.06.005Search in Google Scholar
[9] F. M. Mahomed and P. G. L. Leach, Symmetry Lie algebras of nth order ordinary differential equations, J. Math. Anal. Appl. 151 (1990), no. 1, 80–107. 10.1016/0022-247X(90)90244-ASearch in Google Scholar
[10] J. C. Ndogmo, Generation and identification of ordinary differential equations of maximal symmetry algebra, Abstr. Appl. Anal. 2016 (2016), Article ID 1796316. 10.1155/2016/1796316Search in Google Scholar
[11] J. C. Ndogmo, Some variational principles associated with ODEs of maximal symmetry. Part 2: The general case, J. Appl. Anal. 24 (2018), to appear. 10.1515/jaa-2018-0017Search in Google Scholar
[12] J.-C. Ndogmo and F. M. Mahomed, On certain properties of linear iterative equations, Cent. Eur. J. Math. 12 (2014), no. 4, 648–657. 10.2478/s11533-013-0364-zSearch in Google Scholar
[13] P. J. Olver, Applications of Lie Groups to Differential Equations, Grad. Texts in Math. 107, Springer, New York, 1986. 10.1007/978-1-4684-0274-2Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Deferred weighted 𝒜-statistical convergence based upon the (p,q)-Lagrange polynomials and its applications to approximation theorems
- Some variational principles associated with ODEs of maximal symmetry. Part 1: Equations in canonical form
- On the solutions and conservation laws of a two-dimensional Korteweg de Vries model: Multiple exp-function method
- Existence of solutions for nonlinear Schrödinger systems with periodic data perturbations
- Higher-order conditions for strict local Pareto minima for problems with partial order introduced by a polyhedral cone
- On some nonlinear hyperbolic p(x,t)-Laplacian equations
- Analysis of the embedded cell method in 1D for the numerical homogenization of metal-ceramic composite materials
- Approximately linear recurrences
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- Deficit distributions at ruin in a regime-switching Sparre Andersen model
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Articles in the same Issue
- Frontmatter
- Deferred weighted 𝒜-statistical convergence based upon the (p,q)-Lagrange polynomials and its applications to approximation theorems
- Some variational principles associated with ODEs of maximal symmetry. Part 1: Equations in canonical form
- On the solutions and conservation laws of a two-dimensional Korteweg de Vries model: Multiple exp-function method
- Existence of solutions for nonlinear Schrödinger systems with periodic data perturbations
- Higher-order conditions for strict local Pareto minima for problems with partial order introduced by a polyhedral cone
- On some nonlinear hyperbolic p(x,t)-Laplacian equations
- Analysis of the embedded cell method in 1D for the numerical homogenization of metal-ceramic composite materials
- Approximately linear recurrences
- Existence and uniqueness of a problem in thermo-elasto-plasticity with phase transitions in TRIP steels under mixed boundary conditions
- Deficit distributions at ruin in a regime-switching Sparre Andersen model
- On some non-Gaussian wave packets
