Linearizability of Systems of Ordinary Differential Equations Obtained by Complex Symmetry Analysis

Safdar, Muhammad; Qadir, Asghar; Ali, Sajid

doi:10.1155/2011/171834

Cited by 8 publications

(14 citation statements)

References 22 publications

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“…respectively. Equating the mixed derivatives (ψ 1,yy ) z = (ψ 1,yz ) y , (ψ 1,yy ) x = (ψ 1,xy ) y , (ψ 1,xx ) y = (ψ 1,xy ) x , (ψ 1,xx ) z = (ψ 1,xz ) x , (ψ 1,xy ) z = (ψ 1,xz ) y , (ψ 2,xx ) y = (ψ 2,xy ) y and (ψ 2,xx ) z = (ψ 2,xz ) x gives us These are the same conditions that are already obtained (18-21), by employing complex analysis, i.e., splitting the linearization conditions associated with the base scalar equation (9), into the real and imaginary parts.…”

Section: 21 Sufficient Conditions For the Linearization Of A C-lisupporting

confidence: 67%

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: 21 Sufficient Conditions For the Linearization Of A C-lisupporting

confidence: 67%

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

Dutt¹,

Safdar²

2015

ams

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“…We see that system (1) is linearizable at the origin if the coefficients X (jq+1,jp) , Y (jq,jp+1) of the normal form (5) are zero for all j ∈ N.…”

Section: Preliminariesmentioning

confidence: 99%

“…Lie proved that the necessary and sufficient conditions for a scalar nonlinear ODEs to be linearizable is that it must have eight Lie point symmetries [3,4]. In [5], the authors considered linearizability of systems of ODEs obtained by complex symmetry analysis. In this paper, for a certain type of systems, i.e., p:−q resonant systems, we use approach based on computation of linearizability quantities.…”

Section: Introductionmentioning

confidence: 99%

Linearizability of 2:−3 Resonant Systems with Quadratic Nonlinearities

2021

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In this paper, the linearizability of a 2:−3 resonant system with quadratic nonlinearities is studied. We provide a list of the conditions for this family of systems having a linearizable center. The conditions for linearizablity are obtained by computing the ideal generated by the linearizability quantities and its decomposition into associate primes. To successfully perform the calculations, we use an approach based on modular computations. The sufficiency of the obtained conditions is proven by several methods, mainly by the method of Darboux linearization.

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“…The explicit use of complex functions of complex or real variables is demonstrated in [7][8][9][10] where solvability of systems of DEs is achieved through Noether symmetries and corresponding first integrals. 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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Linearizability of Systems of Ordinary Differential Equations Obtained by Complex Symmetry Analysis

Cited by 8 publications

References 22 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

Linearizability of 2:−3 Resonant Systems with Quadratic Nonlinearities

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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