Complex Lie symmetries for scalar second-order ordinary differential equations

Ali, Sajid; Mahomed, F. M.; Qadir, Asghar

doi:10.1016/j.nonrwa.2008.07.011

Cited by 32 publications

(39 citation statements)

References 7 publications

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“…The CLS analysis has also been used for linearization of CODEs [2,3]. Lie's results on necessary and sufficient conditions for linearization of a real scalar second-order ODE are extended to the complex domain.…”

Section: Summary and Discussionmentioning

confidence: 99%

“…The CLSs admitted by the above CODE and the corresponding RLSs of the respective system of PDEs corresponding to the CODE have been discussed in [1]. Assume the above CODE arises from a c-Lagrangian L(z, u, u ′ ), i.e.…”

Section: Lagrangians and El Equations In The Complex Domainmentioning

confidence: 99%

“…The CODE (3.16) admits an 8−dimensional complex Lie algebra [1]. It has 5 CLSs which are also the NSs with respect to the c-Lagrangian (3.17) (for simplicity we do not write the first extension)…”

Section: Euler-lagrange-like Equationsmentioning

confidence: 99%

“…The CLS of this CODE have been studied in [1]. An obvious c-Lagrangian admitted by such an equation is…”

Section: Euler-lagrange-like Equationsmentioning

confidence: 99%

“…The use of complex Lie symmetries (CLSs) for complex ordinary differential equations (CODEs) and the system of PDEs was developed earlier [1]. A CLS can also be used to reduce the order of certain systems of PDEs that correspond to a CODE.…”

Section: Introductionmentioning

confidence: 99%

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Complex Lie Symmetries for Variational Problems

Ali¹,

Mahomed²,

Qadir³

2008

JNMP

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We present the use of complex Lie symmetries in variational problems by defining a complex Lagrangian and considering its Euler-Lagrange ordinary differential equation. This Lagrangian results in two real "Lagrangians" for the corresponding system of partial differential equations, which satisfy Euler-Lagrange like equations. Those complex Lie symmetries that are also Noether symmetries (i.e. symmetries of the complex Lagrangian) result in two real Noether symmetries of the real "Lagrangians". Also, a complex Noether symmetry of a second order complex ordinary differential equation results in a double reduction of the complex ordinary differential equation. This implies a double reduction in the corresponding system of partial differential equations.

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Section: Summary and Discussionmentioning

confidence: 99%

Section: Lagrangians and El Equations In The Complex Domainmentioning

confidence: 99%

Section: Euler-lagrange-like Equationsmentioning

confidence: 99%

“…The CLS of this CODE have been studied in [1]. An obvious c-Lagrangian admitted by such an equation is…”

Section: Euler-lagrange-like Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Complex Lie Symmetries for Variational Problems

Ali¹,

Mahomed²,

Qadir³

2008

JNMP

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Hypercomplex analysis and integration of systems of ordinary differential equations

Soh

Mahomed

2016

Math Methods in App Sciences

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We review the theory of hypercomplex numbers and hypercomplex analysis with the ultimate goal of applying them to issues related to the integration of systems of ordinary differential equations (ODEs). We introduce the notion of hypercomplexification, which allows the lifting of some results known for scalar ODEs to systems of ODEs. In particular, we provide another approach to the construction of superposition laws for some Riccati‐type systems, we obtain invariants of Abel‐type systems, we derive integrable Ermakov systems through hypercomplexification, we address the problem of linearization by hypercomplexification, and we provide a solution to the inverse problem of the calculus of variations for some systems of ODEs. Copyright © 2016 John Wiley & Sons, Ltd.

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A Note on the Multiplier Approach for Derivation of Conservation Laws for Partial Differential Equations in the Complex Domain

Naz

Mahomed

2018

Springer Proceedings in Mathematics &Amp; Statistics

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Complex Lie symmetries for scalar second-order ordinary differential equations

Cited by 32 publications

References 7 publications

Complex Lie Symmetries for Variational Problems

Complex Lie Symmetries for Variational Problems

Hypercomplex analysis and integration of systems of ordinary differential equations

A Note on the Multiplier Approach for Derivation of Conservation Laws for Partial Differential Equations in the Complex Domain

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