Lie point symmetries and exact solutions of quasilinear differential equations with critical exponents

Bozhkov, Yuri; Martins, Antonio Carlos Gilli

doi:10.1016/j.na.2004.03.016

Cited by 31 publications

(30 citation statements)

References 16 publications

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“…In Goenner [19], the author uncovered symmetries of eqn (1.6) to explain integrability of (1.6) for certain values of the parameters considered in Goenner and Havas [18]. The reader is also referred to the works ( [23], [24], [6], [7]) for symmetries and solutions of Emden-type equations.…”

Section: Introductionmentioning

confidence: 99%

Lagrangian formulation of a generalized Lane-Emden equation and double reduction

Khalique¹,

Mahomed²,

Muatjetjeja³

2008

JNMP

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We classify the Noether point symmetries of the generalized Lane-Emden equation y ′′ + ny ′ /x + f (y) = 0 with respect to the standard Lagrangian L = x n y ′2 /2 − x n f (y)dy for various functions f (y). We obtain first integrals of the various cases which admit Noether point symmetry and find reduction to quadratures for these cases. Three new cases are found for the function f (y). One of them is f (y) = αy r , where r = 0, 1. The case r = 5 was considered previously and only a one-parameter family of solutions was presented. Here we provide a complete integration not only for r = 5 but for other r values. We also give the Lie point symmetries for each case. In two of the new cases, the single Noether symmetry is also the only Lie point symmetry.

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Section: Introductionmentioning

confidence: 99%

Lagrangian formulation of a generalized Lane-Emden equation and double reduction

Khalique¹,

Mahomed²,

Muatjetjeja³

2008

JNMP

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“…Bozkhov and Martins [7] have shown that for k = 0 (1.7) admits the generator of Lie point symmetries 16) where ∂ x = ∂/∂x, ∂ y = ∂/∂y. For the case k = 1 (1.7) admits the generator of Lie point symmetries X and…”

Section: First Integralsmentioning

confidence: 99%

“…For k = 1 Bozkhov and Martins [7] show that X from (2.16) is a Noether (variatonal) symmetry of (1.7). Equation (1.7) admits the Lagrangian…”

Section: K=1mentioning

confidence: 99%

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Alternate Derivation of the Critical Value of the Frank-Kamenetskii Parameter in Cylindrical Geometry

Harely¹,

Momoniat²

2008

JNMP

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Noether's theorem is used to determine first integrals admitted by a generalised LaneEmden equation of the second kind modelling a thermal explosion. These first integrals exist for rectangular and cylindrical geometry. For rectangular geometry the first integrals show the symmetry of the temperature gradients at the rectangular walls. For a cylindrical geometry the first integrals show the dependence of the critical parameter on the temperature gradient at the cylinder wall. The well known critical value for the Frank-Kamenetskii parameter, δ = 2, is obtained in a very natural way.

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“…(1) and obtained exact solutions for various cases which admitted Noether point symmetries. Some other works on symmetries and solutions of Lane-Emden-type equations can be found in [11][12][13][14][15][16][17]. For the applications of Lie group methods to differential equations the interested reader is referred to [18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

Noether, Partial Noether Operators and First Integrals for the Coupled Lane-Emden System

Muatjetjeja

Khalique

2010

MCA

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Abstract-Systems of Lane-Emden equations arise in the modelling of several physical phenomena, such as pattern formation, population evolution and chemical reactions. In this paper we construct Noether and partial Noether operators corresponding to a Lagrangian and a partial Lagrangian for a coupled Lane-Emden system. Then the first integrals with respect to Noether and partial Noether operators are obtained for the Lane-Emden system under consideration. We show that the first integrals for both the Noether and partial Noether operators are the same. However, the gauge function is different in certain cases.

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Lie point symmetries and exact solutions of quasilinear differential equations with critical exponents

Cited by 31 publications

References 16 publications

Lagrangian formulation of a generalized Lane-Emden equation and double reduction

Lagrangian formulation of a generalized Lane-Emden equation and double reduction

Alternate Derivation of the Critical Value of the Frank-Kamenetskii Parameter in Cylindrical Geometry

Noether, Partial Noether Operators and First Integrals for the Coupled Lane-Emden System

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