Variational problems with fractional derivatives: Invariance conditions and Nöther’s theorem

Atanacković, Teodor M.; Konjik, Sanja; Pilipović, Stevan; Simić, Srboljub

doi:10.1016/j.na.2008.12.043

Cited by 121 publications

(92 citation statements)

References 37 publications

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“…In spite of the significance of Cls in analyzing the integrability and internal properties and in proving the existence and uniqueness of solutions of differential equations [28][29][30], the Cls for PDEs having fractional-order are not investigated in detail. The generalization for investigating Cls for FDEs was presented in [31,32]. In recent time, the fractional generalized Noether operators have been introduced [30] for FPDEs that do not have a Lagrangian in order to find Cls using a new conservation theorem [33].…”

Section: Introductionmentioning

confidence: 99%

Space-time fractional Rosenou-Haynam equation: Lie symmetry analysis, explicit solutions and conservation laws

2018

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This article uses the extension of the Lie symmetry analysis (LSA) and conservation laws (Cls) (Singla et al. in Nonlinear Dyn. 89(1):321-331, 2017; Singla et al. in J. Math. Phys. 58:051503, 2017) for the space-time fractional partial differential equations (STFPDEs) to analyze the space-time fractional Rosenou-Haynam equation (STFRHE) with Riemann-Liouville (RL) derivative. We transform the space-time fractional RHE to a nonlinear ordinary differential equation (ODE) of fractional order using its Lie point symmetries. The reduced equation's derivative is in Erdelyi-Kober (EK) sense. We use the power series (PS) technique to derive explicit solutions for the reduced ODE for the first time. The Cls for the governing equation are constructed using a new conservation theorem.

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Section: Introductionmentioning

confidence: 99%

Space-time fractional Rosenou-Haynam equation: Lie symmetry analysis, explicit solutions and conservation laws

2018

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“…Hence, it is of great significance to study. Up to now many results about Noether theory have been obtained, such as Noether theory based on fractional models, see [7,35,80,81]; Noether theory with time delay, see [46,77]; Noether theory for nonlinear dynamical systems, see [36,82]; as well as Noether theory for fractional systems of variable order, see [62,76]. Recently, Noether theory was extended to time scales, see for instance, [18,29,50,51,55,63,79,83].…”

Section: Introductionmentioning

confidence: 99%

Noether theory for Birkhoffian systems with nabla derivatives

Song¹,

Yi²

2017

J. Nonlinear Sci. Appl.

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There are discrete phenomena which happen only on discrete time or hold discrete space structures such as economy series, population dynamics et al.. Then there is a tool needed for these discrete issues or applications. Time scale is one of the useful tools to solve some discrete problems. In this paper, time scale is used to establish discrete Pfaff-Birkhoff principle and achieve discrete Birkhoff equations, discrete Noether identity and discrete conserved quantity for the discrete Birkhoffian system. Firstly, Birkhoff equations, Noether identity and Noether theorem with nabla derivatives on time scales are investigated by using the isochronous variational principle. Secondly, some special cases, especially the discrete Birkhoffian system are discussed. Thirdly, another method, i.e., the duality principle is introduced for the Birkhoffian system on time scales. And finally, an example is given to illustrate the results and methods.

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“…They reduced the given optimization problem to a system of algebraic equations using polynomial basis functions. For more details about the historical comments for the variational problems, see [7] [8].…”

Section: Introductionmentioning

confidence: 99%

Approximate Technique for Solving Class of Fractional Variational Problems

Solouma¹,

Khader²

2015

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This paper is devoted to implementing the Legendre spectral collocation method to introduce numerical solutions of a certain class of fractional variational problems (FVPs). The properties of the Legendre polynomials and Rayleigh-Ritz method are used to reduce the FVPs to the solution of system of algebraic equations. Also, we study the convergence analysis. The obtained numerical results show the simplicity and the efficiency of the proposed method.

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Variational problems with fractional derivatives: Invariance conditions and Nöther’s theorem

Cited by 121 publications

References 37 publications

Space-time fractional Rosenou-Haynam equation: Lie symmetry analysis, explicit solutions and conservation laws

Space-time fractional Rosenou-Haynam equation: Lie symmetry analysis, explicit solutions and conservation laws

Noether theory for Birkhoffian systems with nabla derivatives

Approximate Technique for Solving Class of Fractional Variational Problems

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