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P. G. L. Leach - One of the best experts on this subject based on the ideXlab platform.
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The method of Reduction of Order and linearization of the two-dimensional Ermakov system
Mathematical Methods in the Applied Sciences, 2007Co-Authors: A. Maharaj, P. G. L. LeachAbstract:We present the general form of the system of second-Order ordinary differential equations invariant under a representation of the Lie algebra sl(2, R) and show that a considerable simplification is achieved using a well-known Kummer–Liouville transformation. We show that the system can be reduced to a combination of linear second-Order ordinary differential equations and a conservation law. The Reduction makes the determination of the complete symmetry group of the standard Ermakov system an easier task than earlier reported (J. Nonlinear Math. Phys. 2005; 12:305–320). The reduced system is equivalent to the Reduction of the Kepler problem under a further constraint. Copyright © 2007 John Wiley & Sons, Ltd.
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jacobi s last multiplier and the complete symmetry group of the ermakov pinney equation
Journal of Nonlinear Mathematical Physics, 2005Co-Authors: M C Nucci, P. G. L. LeachAbstract:Abstract The Ermakov-Pinney equation possesses three Lie point symmetries with the algebra sl(2, R). This algebra does not provide a representation of the complete symmetry group of the Ermakov-Pinney equation. We show how the representation of the group can be obtained with the use of the method described in Nucci, J. Nonlin. Math. Phys. 12 (2005) (this issue), which is based on the properties of Jacobi’s last multiplier (Bianchi L, Lezioni sulla teoria dei gruppi continui finiti di trasformazioni, Enrico Spoerri, Pisa, 1918), the method of Reduction of Order (Nucci,J. Math. Phys 37 (1996), 1772–1775) and an interactive code for calculating symmetries (Nucci, Interactive REDUCE programs for calcuating classical, non-classical and Lie-Backlund symmetries for differential equations (preprint: Georgia Institute of Technology, Math 062090-051, 1990, and CRC Handbook of Lie Group Analysis of Differential Equations. Vol. 3: New Trends in Theoretical Developments and Computational Methods, Editor: Ibragimov N H...
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Exponential nonlocal symmetries and nonnormal Reduction of Order
Journal of Physics A: Mathematical and General, 2001Co-Authors: C Géronimi, M R Feix, P. G. L. LeachAbstract:The conventional approach to double Reduction of the Order of an ordinary differential equation using Lie symmetries is via the normal subgroups of point symmetries. We show that, provided that one is prepared to use nonlocal symmetries, initial Reduction by the nonnormal subgroup does not prevent the double Reduction. We further illustrate our results with the general third-Order equations invariant under the nonsolvable algebras, sl(2, R) (of which the Chazy equation is a noted example) and so(3).
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The Determination of Nonlocal Symmetries by the Technique of Reduction of Order
Journal of Mathematical Analysis and Applications, 2000Co-Authors: Maria Clara Nucci, P. G. L. LeachAbstract:Autonomous systems of ordinary differential equations can be rewritten as systems of first Order ordinary differential equations and one of the dependent variables chosen as a new independent variable. Some of the variables are eliminated to give a mixed system of first and second Order equations for which the determination of point symmetries can be automated without having to make an Ansatz on the detailed structure of the symmetry. Because the coefficient function for the original independent variable appears only as its derivative in the reduced system, symmetries which are nonlocal in this variable become local symmetries of the reduced system and can be computed algorithmically.
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Exceptional Properties of Second and Third Order Ordinary Differential Equations of Maximal Symmetry
Journal of Mathematical Analysis and Applications, 2000Co-Authors: Sibusiso Moyo, P. G. L. LeachAbstract:The Riccati transformation is used in the Reduction of Order of second and third Order ordinary differential equations of maximal symmetry. The sl(2, R) subalgebra is preserved under this transformation. The Riccati transformation is itself associated with the symmetry that is annihilated in the Reduction of Order. The solution symmetries and the intrinsically contact symmetries become nonlocal symmetries under the Riccati transformation. We investigate the fate and origins of the contact symmetries arising from the Riccati transformation. The exceptional properties of the second and third Order equations of maximal symmetry are indicated. In the context of generalised symmetries we express the solution symmetries, contact symmetries, and the sl(2, R) elements in terms of a Jacobian. We show that a basis for the solution set of equations of maximal symmetry is given in terms of the solution set of a second Order ordinary differential equation.
Milan Rajković - One of the best experts on this subject based on the ideXlab platform.
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Integration of third Order ordinary differential equations possessing two-parameter symmetry group by Lie’s method
Nonlinear Dynamics, 2007Co-Authors: Mladen Nikolić, Milan RajkovićAbstract:The solution of a class of third Order ordinary differential equations possessing two parameter Lie symmetry group is obtained by group theoretic means. It is shown that Reduction to quadratures is possible according to two scenarios: (1) if upon first Reduction of Order the obtained second Order ordinary differential equation besides the inherited point symmetry acquires at least one more new point symmetry (possibly a hidden symmetry of Type II). (2) First, Reduction paths of the fourth Order differential equations with four parameter symmetry group leading to the first Order equation possessing one known (inherited) symmetry are constructed. Then, Reduction paths along which a third Order equation possessing two-parameter symmetry group appears are singled out and followed until a first Order equation possessing one known (inherited) symmetry are obtained. The method uses conditions for preservation, disappearance and reappearance of point symmetries.
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The Role of Three-dimensional Subalgebra in the Analysis of Hidden Symmetries of Differential Equations
arXiv: Mathematical Physics, 2005Co-Authors: Mladen Nikolic, Milan RajkovićAbstract:Some new properties of symmetries that disappear as point symmetries after the first Reduction of Order of an ODE and reappear after the second are analyzed from the aspect of three-dimensional subalgebra of symmetries of differential equations. The form of a hidden symmetry is shown to consist of two parts, one of which always remains preserved as a point symmetry, and the second (fundamental) part which behaves as the complete hidden symmetry. Symmetry that disappears as point symmetry and remains hidden (non-local) during $n$ Reductions of Order before reappearing as a point symmetry is also introduced and termed convertible symmetry of Order $n-1.$ We discuss the necessity for such classification in Order to distinguish them from hidden symmetries of type I and type II, which are defined with respect to Reduction of Order by one.
Espen R Jakobsen - One of the best experts on this subject based on the ideXlab platform.
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on the rate of convergence of approximation schemes for bellman equations associated with optimal stopping time problems
Mathematical Models and Methods in Applied Sciences, 2003Co-Authors: Espen R JakobsenAbstract:We provide estimates on the rate of convergence for approximation schemes for Bellman equations associated with optimal stopping of controlled diffusion processes. These results extend (and slightly improve) the recent results by Barles & Jakobsen to the more difficult time-dependent case. The added difficulties are due to the presence of boundary conditions (initial conditions!) and the new structure of the equation which is now a parabolic variational inequality. The method presented is purely analytic and rather general and is based on earlier work by Krylov and Barles & Jakobsen. As applications we consider so-called control schemes based on the dynamic programming principle and finite difference methods (though not in the most general case). In the optimal stopping case these methods are similar to the Brennan & Schwartz scheme. A simple observation allows us to obtain the optimal rate 1/2 for the finite difference methods, and this is an improvement over previous results by Krylov and Barles & Jakobsen. Finally, we present an idea that allows us to improve all the above-mentioned results in the linear case. In particular, we are able to handle finite difference methods with variable diffusion coefficients without the Reduction of Order of convergence observed by Krylov in the nonlinear case.
Barbara Abraham-shrauner - One of the best experts on this subject based on the ideXlab platform.
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Hidden Symmetries, First Integrals and Reduction of Order of Nonlinear Ordinary Differential Equations
Journal of Nonlinear Mathematical Physics, 2002Co-Authors: Barbara Abraham-shraunerAbstract:Abstract The Reduction of nonlinear ordinary differential equations by a combination of first integrals and Lie group symmetries is investigated. The retention, loss or even gain in symmetries in the integration of a nonlinear ordinary differential equation to a first integral are studied for several examples. The differential equations and first integrals are expressed in terms of the invariants of Lie group symmetries. The first integral is treated as a differential equation where the special case of the first integral equal to zero is examined in addition to the nonzero first integral. The inverse problem for which the first integral is the fundamental quantity enables some predictions of the change in Lie group symmetries when the differential equation is integrated. New types of hidden symmetries are introduced.
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Symmetries of First Integrals and Their Associated Differential Equations
Journal of Mathematical Analysis and Applications, 1999Co-Authors: P. G. L. Leach, K. S. Govinder, Barbara Abraham-shraunerAbstract:The relationship between the Reduction of Order through point symmetries and integration is explored with particular emphasis on the loss and gain of point (contact for third Order) symmetries to and from nonlocal symmetries. It is seen that Reduction of Order can even lead to the loss of all point symmetries at the third Order level and their replacement at the second Order level from nonlocal symmetries. It is evident that nonlocal symmetries should be given more attention in applications.
Harry G. Kwatny - One of the best experts on this subject based on the ideXlab platform.
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Matrix analysis of some linear gyroscopic systems
Journal of the Franklin Institute, 1992Co-Authors: Leon Y. Bahar, Harry G. KwatnyAbstract:Abstract Explicit solutions for some gyroscopic linear dynamical systems are obtained by selecting a change of the dependent vector variable, which eliminates the velocity term in the transformed equation of motion. The transformation corresponds to the vector counterpart of the technique used for the Reduction of Order in ordinary scalar differential equations. The matrix coefficients of the equations considered obey a certain commutativity condition, which can be expressed in terms of the vanishing of their Lie product or commutator. For purely gyroscopic systems, the results obtained are compared to a generalization of a method originally proposed by Gantmacher.