Fractional calculus of variations of several independent variables

Odzijewicz, Tatiana; Malinowska, Agnieszka B.; Torres, Delfim F. M.

doi:10.1140/epjst/e2013-01966-0

Cited by 11 publications

(10 citation statements)

References 37 publications

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“…The Fractional Calculus of Variations (FCV) consider a new class of variational functionals that depend on fractional derivatives and/or fractional integrals [14]. Here we reviewed necessary conditions of optimality for problems of the FCV with generalized operators [16,19,20,21,22,23]. The study of such variational problems is a subject of strong current study because of its numerous applications.…”

Section: Resultsmentioning

confidence: 99%

“…The Generalized Fractional Calculus of Variations (GFCV) concerns operators depending on general kernels, unifying different perspectives to the FVC. As particular cases, such operators reduce to the standard fractional integrals and derivatives (see, e.g., [2,12,19,20,21,22,23]). Before presenting the GFCV, we define the concept of minimizer.…”

Section: The General Fractional Calculus Of Variationsmentioning

confidence: 99%

“…(cf. [21]) Suppose thatȳ ∈ A(ζ) is a minimizer of (22). Then, y satisfies the following generalized Euler-Lagrange equation:…”

Section: The Generalized Fractional Calculus Of Variations 21mentioning

confidence: 99%

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Generalized fractional calculus with applications to the calculus of variations

Odzijewicz

Malinowska

Torres

2012

Computers & Mathematics with Applications

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Section: Resultsmentioning

confidence: 99%

Section: The General Fractional Calculus Of Variationsmentioning

confidence: 99%

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Generalized fractional calculus with applications to the calculus of variations

Odzijewicz

Malinowska

Torres

2012

Computers & Mathematics with Applications

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“…These papers could be grouped into four categories, namely, survey of fractional calculus [22], numerical methods to solve FPDEs [17,23,[26][27][28][32][33][34], qualitative analysis of FPDEs [16,18,21,24,30,35] and application of FPDEs in various fields [19,20,25,29,31].…”

mentioning

confidence: 99%

“…Paper [21] investigates multidimensional integration by parts formulae for generalised fractional derivatives and integrals. The new results allow the authors to obtain optimality conditions for multidimensional fractional variational problems with Lagrangians depending on generalised partial integrals and derivatives.…”

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confidence: 99%

Editorial

Zhou

Trujillo

Garrappa

2013

Eur. Phys. J. Spec. Top.

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In the past forty years, fractional calculus had played a very important role in various fields such as mechanics, electricity, chemistry, biology, economics, control theory, and signal and image processing [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15].In the last decade, fractional calculus has been recognised as one of the best tools to describe long-memory processes. Such models are interesting for engineers and physicists but also for pure mathematicians. The most important among such models are those described by partial differential equations (PDEs) containing fractional derivatives. Their evolutions behave in a much more complex way than in the classical integer-order case and the study of the corresponding dynamics is a hugely demanding task. Although some results of qualitative analysis for fractional partial differential equations (FPDEs) can be similarly obtained, many classical PDEs' methods are hardly applicable directly to FPDEs. New theories and methods are thus required to be specifically developed for FPDEs, whose investigation becomes more challenging. Comparing with PDEs' classical theory, the research on FPDEs is only at an initial stage of development.This special issue on Dynamics of Fractional Partial Differential Equations consists of 20 original articles covering various aspects of FPDEs and their applications. These papers could be grouped into four categories, namely, survey of fractional calculus [22], numerical methods to solve FPDEs [17,23,[26][27][28][32][33][34], qualitative analysis of FPDEs [16,18,21,24,30,35] and application of FPDEs in various fields [19,20,25,29,31].In paper [16] the theory of Hausdorff measure of noncompactness and fixed point theorems are applied to study the abstract nonlocal Cauchy problem of a class a

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Generalized fractional operators for nonstandard Lagrangians

Taverna

Torres

2014

Math Methods in App Sciences

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In this note we study the application of generalized fractional operators to a particular class of nonstandard Lagrangians. These are typical of dissipative systems and the corresponding Euler-Lagrange and Hamilton equations are analyzed. The dependence of the equation of motion on the generalized kernel permits to obtain a wide range of different configurations of motion. Some examples are discussed and analyzed.Comment: This is a preprint of a paper whose final and definite form will appear in Mathematical Methods in the Applied Sciences, ISSN 0170-4214. Paper submitted 31/Jan/2014; revised 23/Apr/2014; accepted for publication 25/Apr/201

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Fractional calculus of variations of several independent variables

Cited by 11 publications

References 37 publications

Generalized fractional calculus with applications to the calculus of variations

Generalized fractional calculus with applications to the calculus of variations

Editorial

Generalized fractional operators for nonstandard Lagrangians

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