Fractional Variational Problems Depending on Indefinite Integrals and with Delay

Almeida, Ricardo

doi:10.1007/s40840-015-0248-4

Cited by 10 publications

(8 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This subject was introduced by Riewe in 1996, where the author generalizes the classical calculus of variations, by using fractional derivatives, and allows to obtain conservations laws with nonconservative forces such as friction (Riewe, 1996(Riewe, , 1997. Later appeared several works on various aspects of the fractional calculus of variations and involving different fractional operators, like the Riemann-Liouville, the Caputo, the Grunwald-Letnikov, the Weyl, the Marchaud or the Hadamard fractional derivatives (Agrawal, 2002;Almeida, 2016;Askari and Ansari, 2016;Atanacković, Konjik and Pilipović, 2008;Baleanu, 2008;Fraser, 1992;Georgieva and Guenther, 2002;Jarad, Abdeljawad and Baleanu, 2010). For the state of the art of the fractional calculus of variations, we refer the readers to the books Malinowska and Torres, 2012).…”

Section: Prefacementioning

confidence: 99%

“…In the second part, we systematize some new recent results on variableorder fractional calculus of (Tavares, Almeida and Torres, 2015, 2016, 2018a. In Chapter 3, considering three types of fractional Caputo derivatives of variable-order, we present new approximation formulas for those fractional derivatives and prove upper bound formulas for the errors.…”

Section: Prefacementioning

confidence: 99%

“…This fact motivates the evaluation of calculation strategies based on delayed signal samples". This subject has already been studied for constant fractional order (Almeida, 2016;Baleanu, Maaraba and Jarad, 2008;Jarad, Abdeljawad and Baleanu, 2010). However, for a variable fractional order, it is, to the authors' best knowledge, an open question.…”

Section: Variational Problems With Time Delaymentioning

confidence: 99%

See 2 more Smart Citations

The Variable-Order Fractional Calculus of Variations

Almeida

Tavares

Torres

2019

SpringerBriefs in Applied Sciences and Technology

Self Cite

134

View full text Add to dashboard Cite

PrefaceThis book intends to deepen the study of the fractional calculus, giving special emphasis to variable-order operators.Fractional calculus is a recent field of mathematical analysis and it is a generalization of integer differential calculus, involving derivatives and integrals of real or complex order (Kilbas, Srivastava and Trujillo, 2006;Podlubny, 1999). The first note about this ideia of differentiation, for non-integer numbers, dates back to 1695, with a famous correspondence between Leibniz and L'Hôpital. In a letter, L'Hôpital asked Leibniz about the possibility of the order n in the notation d n y/dx n , for the nth derivative of the function y, to be a non-integer, n = 1/2. Since then, several mathematicians investigated this approach, like Lacroix, Fourier, Liouville, Riemann, Letnikov, Grünwald, Caputo, and contributed to the grown development of this field. Currently, this is one of the most intensively developing areas of mathematical analysis as a result of its numerous applications. The first book devoted to the fractional calculus was published by Oldham and Spanier in 1974, where the authors systematized the main ideas, methods and applications about this field (Mainardi, 2010).In the recent years, fractional calculus has attracted the attention of many mathematicians, but also some researchers in other areas like physics, chemistry and engineering. As it is well known, several physical phenomena are often better described by fractional derivatives (Herrmann, 2013;Odzijewicz, Malinowska and Torres, 2012a; Sheng, 2012). This is mainly due to the fact that fractional operators take into consideration the evolution of the system, by taking the global correlation, and not only local characteristics. Moreover, integer-order calculus sometimes contradict the experimental results and therefore derivatives of fractional order may be more suitable (Hilfer, 2000).In 1993, Samko and Ross devoted themselves to investigate operators when the order α is not a constant during the process, but variable on time: α(t) ). An interesting recent generalization of the theory of fractional calculus is developed to allow the fractional order of the derivative to be non-constant, depending on time (Chen, Liu and Burrage, VI 2014; Odzijewicz, Torres, 2012b, 2013a). With this approach of variable-order fractional calculus, the non-local properties are more evident and numerous applications have been found in physics, mechanics, control and signal processing (Coimbra, Soon and Kobayashi, 2005; Ingman and Suzdalnitsky, 2004;Odzijewicz, Malinowska and Torres, 2013b; Ostalczyk et al., 2015;Ramirez and Coimbra, 2011; Rapaić and Pisano, 2014; Valério and Costa, 2013).Although there are many definitions of fractional derivative, the most commonly used are the Riemann-Liouville, the Caputo, and the Grünwald-Letnikov derivatives. For more about the development of fractional calculus, we suggest (Samko, Kilbas and Marichev, One difficult issue that usually arises when dealing with such fractional operators, is the extreme di...

show abstract

Section: Prefacementioning

confidence: 99%

Section: Prefacementioning

confidence: 99%

Section: Variational Problems With Time Delaymentioning

confidence: 99%

See 1 more Smart Citation

The Variable-Order Fractional Calculus of Variations

Almeida

Tavares

Torres

2019

SpringerBriefs in Applied Sciences and Technology

Self Cite

134

View full text Add to dashboard Cite

show abstract

“…This fact motivates the evaluation of calculation strategies based on delayed signal samples". This subject has already been studied for constant fractional order [3,8,13,14]. However, for a variable fractional order, it is, to the authors' best knowledge, an open question.…”

Section: Variational Problems With Time Delaymentioning

confidence: 99%

Combined fractional variational problems of variable order and some computational aspects

Tavares

Almeida

Torres

2018

Journal of Computational and Applied Mathematics

Self Cite

View full text Add to dashboard Cite

We study two generalizations of fractional variational problems by considering higherorder derivatives and a state time delay. We prove a higher-order integration by parts formula involving a Caputo fractional derivative of variable order and we establish several necessary optimality conditions for functionals containing a combined Caputo derivative of variable fractional order. Because the endpoint is considered to be free, we also deduce associated transversality conditions. In the end, we consider functionals with a time delay and deduce corresponding optimality conditions. Some examples are given to illustrate the new results. Computational aspects are discussed using the open source software package Chebfun.

show abstract

“…The fractional variational problem is an important dynamical system. In recent years, under different differentiability, several researchers [2][3][4][5][6][7][8][9] studied the fractional Euler-Lagrange equations for general fractional variational problems.…”

Section: Introductionmentioning

confidence: 99%

Generalized Euler-Lagrange Equations for Fuzzy Fractional Variational Problems under gH-Atangana-Baleanu Differentiability

Zhang

Wang

Xiong

et al. 2018

Journal of Function Spaces

View full text Add to dashboard Cite

We study in this paper the Atangana-Baleanu fractional derivative of fuzzy functions based on the generalized Hukuhara difference. Under the condition of gH-Atangana-Baleanu fractional differentiability, we prove the generalized necessary and sufficient optimality conditions for problems of the fuzzy fractional calculus of variations with a Lagrange function. The new kernel of gH-Atangana-Baleanu fractional derivative has no singularity and no locality, which was not precisely illustrated in the previous definitions.

show abstract

Fractional Variational Problems Depending on Indefinite Integrals and with Delay

Cited by 10 publications

References 23 publications

The Variable-Order Fractional Calculus of Variations

The Variable-Order Fractional Calculus of Variations

Combined fractional variational problems of variable order and some computational aspects

Generalized Euler-Lagrange Equations for Fuzzy Fractional Variational Problems under gH-Atangana-Baleanu Differentiability

Contact Info

Product

Resources

About