Linearization of systems of four second-order ordinary differential equations

Safdar, Muhammad; Ali, Sajid; Mahomed, F. M.

doi:10.1007/s12043-011-0177-1

Cited by 9 publications

(11 citation statements)

References 7 publications

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“…Here the coefficients a j , b j and c j are the real and imaginary parts of the complex coefficients (10). The linearizable form of systems derived in this section by real method appears to be the same as one obtains by splitting the corresponding form of the scalar complex equation (9).…”

Section: 2 Use Of Real Symmetry Analysismentioning

confidence: 85%

Linearization of two dimensional complex-linearizable systems of second order ordinary differential equations

Dutt¹,

Safdar²

2015

ams

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Complex-linearization of a class of systems of second order ordinary differential equations (ODEs) has already been studied with complex symmetry analysis. Linearization of this class has been achieved earlier by complex method, however, linearization criteria and the most general linearizable form of such systems have not been derived yet. In this paper, it is shown that the general linearizable form of the complex-linearizable systems of two second order ODEs is (at most) quadratically semi-linear in the first order derivatives of the dependent variables. Further, linearization conditions are derived in terms of coefficients of system and their derivatives. These linearizable 2-dimensional complex-linearizable systems of second order ODEs are characterized here, by adopting both the real and complex procedures.issues are of secondary nature as one needs to first investigate linearizability of DEs. An explicit linearizable form and linearization criteria for the scalar second order ODEs have been derived by Sophus Lie (see, e.g., [1]). Similarly, linearization of higher order scalar ODEs and systems of these equations attracted a great deal of interest and studied comprehensively over the last decade (see, e.g., [2]-[7]).Complex symmetry analysis has been employed to solve certain classes of systems of nonlinear ODEs and linear PDEs. Of particular interest here, is linearization of systems of second order ODEs (see, e.g., [8]- [11]) that is achieved by complex methods. These classes are obtained from linearizable scalar and systems of ODEs by considering their dependent variables as complex functions of a real independent variable, which when split into the real and imaginary parts give two dependent variables. In this way, a scalar ODE produces a system of two coupled equations, with Cauchy-Riemann (CR) structure on both the equations. These CR-equations appear as constraint equations that restrict the emerging systems of ODEs to special subclasses of the general class of such systems. These subclasses of 2-dimensional systems of second order ODEs may trivially be studied with real symmetry analysis, however, they appear to be nontrivial when viewed from complex approach. Complex-linearizable (clinearizable) classes explored earlier [8]- [11] and studied in this paper provide us means to extend linearization procedure to m-dimensional systems (m ≥ 3), of n th order (n ≥ 2) ODEs. Though these classes are subcases of the general m-dimensional systems of n th order ODEs, their linearization has not been achieved yet, with real symmetry analysis. Presently symmetry classification and solvability of higher dimensional systems of higher order ODEs seems to be exploitable only with complex symmetry analysis.

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Section: 2 Use Of Real Symmetry Analysismentioning

confidence: 85%

Linearization of two dimensional complex-linearizable systems of second order ordinary differential equations

Dutt¹,

Safdar²

2015

ams

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“…Under this, transformation equation ( 36) is reduced to equation ( 6), with a � − k 2 2 and b � 3k 1 k 2 . e general solution of (36) now follows easily from equation (30) written in terms of x and y via point transformation (38), with a and b set to − k 2 2 and 3k 1 k 2 , respectively. We obtain…”

Section: Illustrative Examplesmentioning

confidence: 99%

“…When the simpler "target" equation is a linear equation, the problem is called the linearisation problem. e pioneering work on this, with regard to second-order ODEs, is attributed to Sophus Lie (see [2] and the references there in). Lie proved that, to be linearisable, a second-order ODE must be at most cubically semilinear, and the coefficients in it must satisfy an overdetermined system of conditions [3,4].…”

Section: Introductionmentioning

confidence: 99%

“…Lie proved that, to be linearisable, a second-order ODE must be at most cubically semilinear, and the coefficients in it must satisfy an overdetermined system of conditions [3,4]. A considerable amount of research has since been conducted on the linearisation problem [2,3,[5][6][7] leading, in particular, to a variety of ways of characterising linearisable second-order ODEs (see, e.g., eorem 8 of [3]). A scaled-down version of eorem 8 of [3] is presented here.…”

Section: Introductionmentioning

confidence: 99%

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Some Remarks on the Solution of Linearisable Second-Order Ordinary Differential Equations via Point Transformations

Sinkala

2020

Journal of Mathematics

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Transformations of differential equations to other equivalent equations play a central role in many routines for solving intricate equations. A class of differential equations that are particularly amenable to solution techniques based on such transformations is the class of linearisable second-order ordinary differential equations (ODEs). There are various characterisations of such ODEs. We exploit a particular characterisation and the expanded Lie group method to construct a generic solution for all linearisable second-order ODEs. The general solution of any given equation from this class is then easily obtainable from the generic solution through a point transformation constructed using only two suitably chosen symmetries of the equation. We illustrate the approach with three examples.

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“…Furthermore, by employing complex symmetry procedures: the energy stored in the field of a coupled harmonic oscillator was studied in [11] and linearizability of systems of two second order ODEs was addressed in [12,13]. The complex procedure, indeed, has been extended to higher dimensional systems of second order ODEs [14] and two-dimensional, systems of third order ODEs [15].…”

Section: Introductionmentioning

confidence: 99%

Comparison of Noether Symmetries and First Integrals of Two-Dimensional Systems of Second Order Ordinary Differential Equations by Real and Complex Methods

2019

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Noether symmetries and first integrals of a class of two-dimensional systems of second order ordinary differential equations (ODEs) are investigated using real and complex methods. We show that first integrals of systems of two second order ODEs derived by the complex Noether approach cannot be obtained by the real methods. Furthermore, it is proved that a complex method can be extended to larger systems and higher order.

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Linearization of systems of four second-order ordinary differential equations

Cited by 9 publications

References 7 publications

Linearization of two dimensional complex-linearizable systems of second order ordinary differential equations

Linearization of two dimensional complex-linearizable systems of second order ordinary differential equations

Some Remarks on the Solution of Linearisable Second-Order Ordinary Differential Equations via Point Transformations

Comparison of Noether Symmetries and First Integrals of Two-Dimensional Systems of Second Order Ordinary Differential Equations by Real and Complex Methods

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