Conditional Linearizability Criteria for Third Order Ordinary Differential Equations

Mahomed, F. M.; Naeem, Imran; Qadir, Asghar

doi:10.2991/jnmp.2008.15.s1.11

Cited by 16 publications

(20 citation statements)

References 14 publications

(41 reference statements)

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“…One should be able to extend the complex linearization procedure to third-and fourthorder systems by using the results for the corresponding scalar ODEs [19,20] straightforwardly. Also the extension to conditional linearizability of systems could be obtained [27,28,29]. However, in the former case the power of the geometric approach would be lost.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Linearizability criteria for systems of two second-order differential equations by complex methods

Ali¹,

Mahomed²,

Qadir³

2011

Nonlinear Dyn

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Lie's linearizability criteria for scalar second-order ordinary differential equations had been extended to systems of second-order ordinary differential equations by using geometric methods. These methods not only yield the linearizing transformations but also the solutions of the nonlinear equations. Here, complex methods for a scalar ordinary differential equation are used for linearizing systems of two second-order ordinary and partial differential equations, which can use the power of the geometric method for writing the solutions. Illustrative examples of mechanical systems including the LaneEmden type equations which have roots in the study of stellar structures are presented and discussed.

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Section: Conclusion and Discussionmentioning

confidence: 99%

Linearizability criteria for systems of two second-order differential equations by complex methods

Ali¹,

Mahomed²,

Qadir³

2011

Nonlinear Dyn

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“…Again, it can be seen that it would be of great interest to find methods to extend the results for higher-order root equations. This has been done in [17].…”

Section: Discussionmentioning

confidence: 99%

“…Taking the general class of the scalar third-order ODE one gets linearizability criteria for scalar third-order ODEs [17]. This class is not included in the Neut and Petitot [21] and Ibragimov and Meleshko classes [9].…”

Section: Introductionmentioning

confidence: 99%

Conditional Linearizability of Fourth-Order Semi-Linear Ordinary Differential Equations

Mahomed¹,

Qadir²

2021

JNMP

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By the use of geometric methods for linearizing systems of second-order cubically semi-linear ordinary differential equations and the conditional linearizability of third-order quintically semi-linear ordinary differential equations, we extend to the fourth-order by differentiating the third-order conditionally linearizable equation. This yields criteria for conditional linearizability of a class of fourth-order semi-linear ordinary differential equations, which have not been discussed in the literature previously.

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“…These include the works of Chern [18], Grebot [19], Neut and Petitot [20], Ibragimov and Meleshko [21] all for scalar third-order ODEs and Ibragimov et al [22] for scalar fourth-order ODEs. Conditional invariant linearization criteria for third-order ODEs have also been pursued by Mahomed and Qadir [23]. Moreover, invariant linearization criteria for quadratic and cubic nonlinear systems of second-order ODEs have been researched as well (Mahomed and Qadir [24,25]).…”

Section: Introductionmentioning

confidence: 98%

Algebraic linearization criteria for systems of ordinary differential equations

2011

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Algebraic linearization criteria by means of general point transformations for systems of two second-order nonlinear ordinary differential equations (ODEs) are revisited. In previous work due to Wafo Soh and Mahomed (Int. J. Non-Linear Mech. 36:671, 2001) two four-dimensional Lie algebras that result in linearizability in terms of arbitrary point transformation for such systems were studied. Here we consider three more algebras of dimension four that result in linearization. Therefore our results supplement those of Wafo Soh and Mahomed (Int. J. Non-Linear Mech. 36:671, 2001). Moreover, it is shown that these are the only other possibilities for dimension four. Hence we provide the complete algebraic linearization criteria for dimension four algebras. Necessary and sufficient conditions for linearization via invertible maps of a nonlinear to a linear system are given. These are shown to be built up from the Lie algebraic criteria for linearization of scalar second-order ODEs. These results together with very recent work (Bagderina in J. Phys. A, Math. Theor. 43:465201, 2010) give a complete picture on linearizability properties via general point transformations for systems of two second-order ODEs. Furthermore, we provide natural extensions of these algebraic criteria for linearizing arbitrary systems of nonlinear second-order ODEs by means of point transformations. We also obtain algebraic criteria for the reduction of a linear system to the simplest system. Examples from Newtonian mechanics and geodesic equations are presented to illustrate our results.

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Conditional Linearizability Criteria for Third Order Ordinary Differential Equations

Cited by 16 publications

References 14 publications

Linearizability criteria for systems of two second-order differential equations by complex methods

Linearizability criteria for systems of two second-order differential equations by complex methods

Conditional Linearizability of Fourth-Order Semi-Linear Ordinary Differential Equations

Algebraic linearization criteria for systems of ordinary differential equations

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