Higher-order Hahn’s quantum variational calculus

Cruz, Artur M. C. Brito da; Martins, Natália; Torres, Delfim F. M.

doi:10.1016/j.na.2011.01.015

Cited by 33 publications

(35 citation statements)

References 16 publications

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“…For example, one interesting direction is to extend our results to the higher-order case. In our opinion, the main difficulty is to prove the higherorder fundamental lemma of the calculus of variations because of the generality of the function β (see Lemma 3.8 of [8] for the particular case of Hahn's variational calculus). Another open question is to consider problems of the general quantum variational calculus with time delay.…”

Section: Discussionmentioning

confidence: 99%

“…One year later, the Hahn variational calculus started with the publication of Malinowska and Torres [20]. Due to its importance, new results in Hahn quantum variational calculus have been published in the literature [8,19]; we also recommend the book [21] for more details on the topic. We refer the interested reader to [2,3,4,7,10,22] for details in calculus of variations based in different quantum operators.…”

Section: Introductionmentioning

confidence: 99%

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General quantum variational calculus

Cruz

Martins

2018

Stat., optim. inf. comput.

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We develop a new variational calculus based in the general quantum difference operator recently introduced by Hamza et al. In particular, we obtain optimality conditions for generalized variational problems where the Lagrangian may depend on the endpoints conditions and a real parameter, for the basic and isoperimetric problems, with and without fixed boundary conditions. Our results provide a generalization to previous results obtained for the q-and Hahn-calculus.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

General quantum variational calculus

Cruz

Martins

2018

Stat., optim. inf. comput.

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“…Quantum calculus has several different dialects (Brito da Cruz et al 2012; Brito da Cruz et al 2013b, c;Ernst 2008;Kac and Cheung 2002). The most common one is based on Jackson's q-operators, where q stands for quantum (Annaby and Mansour 2012;Jackson 1908;Jackson 1910;Kac and Cheung 2002).…”

mentioning

confidence: 99%

“…The Hahn quantum variational calculus was started only in 2010 with the work . Quantum variational problems involving Hahn's derivatives of higher-order were first investigated in (Brito da Cruz et al 2012). Several quantum variational problems have been recently posed and studied (Aldwoah et al 2012;Almeida and Torres 2009;Almeida and Torres 2011;Bangerezako 2004;Bangerezako 2005; Brito da Cruz et al 2013a;Cresson 2005;Cresson et al 2009;Frederico and Torres 2013;Martins and Torres 2012).…”

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confidence: 99%

Quantum Variational Calculus

Malinowska

Torres

2014

SpringerBriefs in Electrical and Computer Engineering

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“…However, only in 2009, Aldwoah [1] defined the inverse of Hahn's difference operator, and short after, Malinowska and Torres [24] introduced and investigated the Hahn quantum variational calculus. For a deep understanding of quantum calculus, we refer the reader to [2,5,6,11,15,16] and references therein.For a fixed q ∈ ]0, 1[ and an ω ≥ 0, we introduce here the Hahn symmetric difference operator of function f at point t = ω 1 − q bỹ …”

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Hahn's symmetric quantum variational calculus

Cruz¹,

Martins²,

Torres³

2013

Numerical Algebra, Control &Amp; Optimization

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Abstract. We introduce and develop the Hahn symmetric quantum calculus with applications to the calculus of variations. Namely, we obtain a necessary optimality condition of Euler-Lagrange type and a sufficient optimality condition for variational problems within the context of Hahn's symmetric calculus. Moreover, we show the effectiveness of Leitmann's direct method when applied to Hahn's symmetric variational calculus. Illustrative examples are provided. 1.Introduction. Due to its many applications, quantum operators are recently subject to an increase number of investigations [24][25][26]. The use of quantum differential operators, instead of classical derivatives, is useful because they allow to deal with sets of nondifferentiable functions [4,10]. Applications include several fields of physics, such as cosmic strings and black holes [27], quantum mechanics [12,29], nuclear and high energy physics [18], just to mention a few. In particular, the q-symmetric quantum calculus has applications in quantum mechanics [17].In 1949, Hahn introduced his quantum difference operator [13], which is a generalization of the quantum q-difference operator defined by Jackson [14]. However, only in 2009, Aldwoah [1] defined the inverse of Hahn's difference operator, and short after, Malinowska and Torres [24] introduced and investigated the Hahn quantum variational calculus. For a deep understanding of quantum calculus, we refer the reader to [2,5,6,11,15,16] and references therein.For a fixed q ∈ ]0, 1[ and an ω ≥ 0, we introduce here the Hahn symmetric difference operator of function f at point t = ω 1 − q bỹ

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Higher-order Hahn’s quantum variational calculus

Cited by 33 publications

References 16 publications

General quantum variational calculus

General quantum variational calculus

Quantum Variational Calculus

Hahn's symmetric quantum variational calculus

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