Symmetries, Conservation Laws and Multipliers via Partial Lagrangians and Noether’s Theorem for Classically Non-Variational Problems

Marwat, Dil Nawaz Khan; Kara, A. H.; Mahomed, F. M.

doi:10.1007/s10773-007-9417-z

Cited by 13 publications

(8 citation statements)

References 8 publications

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“…• If a 1 = 0, a 2 = 0, a 3 = 0, the coefficients X 4 , X 5 could be vanished by F s 3 3 , F s 4 4 , by setting s 3 = arctan( a 4 a 5 ), s 4 = arctan( a 5 a 3 ) respectively.By scaling X we can put a 3 = 1 so X is reduced to case (7).…”

Section: Classification Of Subalgebrasmentioning

confidence: 99%

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On the invariance properties of Vaidya-Bonner geodesics via symmetry operators

Farrokhi,

Bakhshandeh-Chamazkoti,

Nadjafikhah

2021

Preprint

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In the present paper, we try to investigate the Noether symmetries and Lie point symmetries of the Vaidya-Bonner geodesics. Classification of one-dimensional subalgebras of Lie point symmetries are considered. In fact, the collection of pairwise non-conjugate one-dimensional subalgebras that are called the optimal system of subalgebras is determined. Moreover, as illustrative examples, the symmetry analysis is implemented on two special cases of the system.

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Section: Classification Of Subalgebrasmentioning

confidence: 99%

“…Noether's theorem [11,12] provides a method for finding conservation laws of differential equations arising from a known Lagrangian and having a known Lie symmetry. This theorem relies on the availability of a Lagrangian and the corresponding Noether symmetries which leave invariant the action integral, [7,9].…”

Section: Introductionmentioning

confidence: 99%

On the invariance properties of Vaidya-Bonner geodesics via symmetry operators

Farrokhi,

Bakhshandeh-Chamazkoti,

Nadjafikhah

2021

Preprint

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“…[22][23][24] In 1918, Noether 25 proved that each conservation law is associated with an appropriate symmetry 26,27 and can be derived from Lagrangian function and invariance principle. [28][29][30] In recent years, several fractional generalizations of Noether's theorem were proved, 31 and some of the fractional conservation laws of different equations have been calculated. Since the order of the (3 + 1)-dimensional time-fractional modified Burgers equation is only two, the conservation laws of the (3 + 1)-dimensional time-fractional modified Burgers equation are constructed by using a simpler Noether method.…”

Section: Introductionmentioning

confidence: 99%

(3 + 1)‐Dimensional time‐fractional modified Burgers equation for dust ion‐acoustic waves as well as its exact and numerical solutions

Tian

Ren

et al. 2019

Math Methods in App Sciences

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In this paper, the theoretical model of dust ion-acoustic waves in collisionless, unmagnetized dust plasma is analyzed. According to the (3 + 1)-dimensional nonlinear dynamic propagation equations with the electron distribution in the presence of trapping particles and the kinetic equation of charge of dust particle, based on multiscale analysis and perturbation method, a new (3 + 1)-dimensional modified Burgers equation is constructed. Because of the space property of dust plasma, (3 + 1)-dimensional modified Burgers equation is more suitable than low-dimensional equation to describe the dust ion-acoustic waves. Furthermore, applying the semi-inverse method and the fractional variational principle, time-fractional modified Burgers equation is given, which owns potential value for comprehending the propagation feature of dust ion-acoustic waves in three-dimensional space. Following, the conservation laws of the time-fractional modified Burgers equation are discussed by using the Noether method. Finally, the shock wave solutions of the new model is obtained with the help of the G ′ ∕G-expansion method; meanwhile, the radial basis function method is also used to get the numerical solutions of the (3 + 1)-dimensional time-fractional modified Burgers equation. KEYWORDS(3 + 1)-dimensional time-fractional modified Burgers equation, conservation laws, dust ion-acoustic waves, radial basis function method, shock wave solution MSC CLASSIFICATION 35Q51; 35Q55After solving the equation Ac k =B k , we get the algebraic system at each time step, and we can obtain the solution by using Equation (50).

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“…In the absence of a Lagrangian, which is the case for (2), one has to resort to a number of adhoc methods to determine these. However, we use a recently developed novel method of 'partial Lagrangian' and 'Noether type generators' [5,8] which enjoys the convenience of the method used in the variational case and a formula equivalent to Noether's theorem to determine exact conservation laws. We, consequently, obtain a number of interesting conserved flows for (2) associated with Noether type generators that may appear to be of typical physical value like conservation of energy, spin and those coming from scaling generators.…”

Section: Introductionmentioning

confidence: 99%

Conservation Laws and Associated Noether Type Vector Fields via Partial Lagrangians and Noether’s Theorem for the Liang Equation

2008

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We show how one can construct conservation laws of the Liang equation which is not variational but may be regarded as Euler-Lagrange in part. This first requires the determination of the Noether-type symmetries associated with the partial Lagrangian. The final construction of the conservation laws resort to a formula equivalent to Noether's theorem. A variety of subclasses are given and, for each, a large number of conserved flows are found-the method is usable for any general choice of the variable speed of sound.

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Symmetries, Conservation Laws and Multipliers via Partial Lagrangians and Noether’s Theorem for Classically Non-Variational Problems

Cited by 13 publications

References 8 publications

On the invariance properties of Vaidya-Bonner geodesics via symmetry operators

On the invariance properties of Vaidya-Bonner geodesics via symmetry operators

(3 + 1)‐Dimensional time‐fractional modified Burgers equation for dust ion‐acoustic waves as well as its exact and numerical solutions

Conservation Laws and Associated Noether Type Vector Fields via Partial Lagrangians and Noether’s Theorem for the Liang Equation

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