The Second Euler-Lagrange Equation of Variational Calculus on Time Scales

Bartosiewicz, Zbigniew; Martins, Natália; Torres, Delfim F. M.

doi:10.3166/ejc.17.9-18

Cited by 47 publications

(37 citation statements)

References 41 publications

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“…However, with time, the new theory has found important applications in several fields that require modeling of discrete and continuous data simultaneously. We can mention here the calculus of variations, control theory, economics, and biology (e.g., [13][14][15][16][17][18][19][20][21][22][23][24]). For a general introduction to calculus on time scales and its applications, we refer the reader to the books [25,26].…”

Section: Preliminary Results On Time Scale Calculusmentioning

confidence: 99%

Krause's model of opinion dynamics on isolated time scales

Girejko

Machado

Malinowska

et al. 2016

Math Methods in App Sciences

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Communicated by M. A. LachowiczWe analyze a bounded confidence model, introduced by Krause, on isolated time scales. In this model, each agent takes into account only the assessments of the agents whose opinions are not too far away from its own opinion. We show that the behavior of the model depends strongly on the graininess function : If takes values in the interval 0, 1, then our discrete time scale model behaves similarly to the classical one, but if takes values in 1, C1OE, then the model has different properties. Simulations are performed to validate the theoretical results.

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Section: Preliminary Results On Time Scale Calculusmentioning

confidence: 99%

Krause's model of opinion dynamics on isolated time scales

Girejko

Machado

Malinowska

et al. 2016

Math Methods in App Sciences

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“…The calculus of variations on time scales is an important subject under strong current research (see [10,13,19,35,36,41] and references therein). Here we review a general necessary optimality condition for problems of the calculus of variations on time scales [37,38].…”

Section: Resultsmentioning

confidence: 99%

The variational calculus on time scales

Torrest

2010

Int. J. Simul. Multidisci. Des. Optim.

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Abstract. The discrete, the quantum, and the continuous calculus of variations, have been recently unified and extended by using the theory of time scales. Such unification and extension is, however, not unique, and two approaches are followed in the literature: one dealing with minimization of delta integrals; the other dealing with minimization of nabla integrals. Here we review a more general approach to the calculus of variations on time scales that allows to obtain both delta and nabla results as particular cases.

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“…Since then, the variational calculus on time scales advanced fairly quickly, as can be verified with the large number of published papers on the subject [3], [12], [21], [22], [25], [33], [35], [37], [39], [41], [42]. Noether's first theorem has been extended to the variational calculus on time scales using several approaches [7], [8], [40], while the second Noether theorem on times scales is still not available in the literature. So there is evidently a need for a time scale analogue of Noether's second theorem.…”

Section: Introductionmentioning

confidence: 94%