Caputo-Based Fractional Derivative in Fractional Fourier Transform Domain

Singh, Kulbir; Saxena, Rajiv; Kumar, Sanjay

doi:10.1109/jetcas.2013.2272837

Cited by 40 publications

(12 citation statements)

References 22 publications

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“…In the literature of fractional calculus, several different definitions of derivatives are found . One of those, introduced by (Caputo, 1967) and studied independently by other authors, like (Džrbašjan and Nersesjan, 1968) and (Rabotnov, 1969), has found many applications and seems to be more suitable to model physical phenomena (Dalir and Bashour, 2010;Diethelm, 2004;Machado et al, 2010;Murio and Mejía, 2008;Singh, Saxena and Kumar, 2013;Sweilam and AL-Mrawm, 2011;Yajima and Yamasaki, 2012).…”

Section: Caputo-type Fractional Operators Of Variable-ordermentioning

confidence: 99%

The Variable-Order Fractional Calculus of Variations

Almeida

Tavares

Torres

2019

SpringerBriefs in Applied Sciences and Technology

128

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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...

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Section: Caputo-type Fractional Operators Of Variable-ordermentioning

confidence: 99%

The Variable-Order Fractional Calculus of Variations

Almeida

Tavares

Torres

2019

SpringerBriefs in Applied Sciences and Technology

128

View full text Add to dashboard Cite

show abstract

“…In the literature of fractional calculus, several different definitions of derivatives are found [28]. One of those, introduced by Caputo in 1967 [3] and studied independently by other authors, like Džrbašjan and Nersesjan in 1968 [10] and Rabotnov in 1969 [25], has found many applications and seems to be more suitable to model physic phenomena [6,8,9,15,16,31,33,35]. Before generalizing the Caputo derivative for a variable order of differentiation, we recall two types of special functions: the Gamma and Psi functions.…”

Section: Fractional Calculus Of Variable Ordermentioning

confidence: 99%

Caputo derivatives of fractional variable order: Numerical approximations

Tavares

Almeida

Torres

2016

Communications in Nonlinear Science and Numerical Simulation

162

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We present a new numerical tool to solve partial differential equations involving Caputo derivatives of fractional variable order. Three Caputo-type fractional operators are considered, and for each one of them an approximation formula is obtained in terms of standard (integerorder) derivatives only. Estimations for the error of the approximations are also provided. We then compare the numerical approximation of some test function with its exact fractional derivative. We end with an exemplification of how the presented methods can be used to solve partial fractional differential equations of variable order.

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“…Fact 1: Let and be defined in (1) and (3), then it can be shown the following equality is valid: (6) where coefficients and are given by (7) That is, is the linear combination of and . Proof: From (2), we have (8) Substituting (2), (3), and (8) into (6), we have the results (9) Eliminate the term at both sides, we have (10) This means that the following equalities hold:…”

Section: Designs Of Variable Generalized Fod and Foimentioning

confidence: 99%

Designs of Discrete-Time Generalized Fractional Order Differentiator, Integrator and Hilbert Transformer

Tseng

Lee

2015

IEEE Trans. Circuits Syst. I

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In this paper, the design of a generalized fractional order differentiator (FOD) whose magnitude and phase responses can be controlled independently is investigated. First, a relation between conventional FOD and generalized FOD is studied such that the design tools of conventional FOD in the literature can be used to design variable generalized FOD directly. Then, the similar method is applied to design generalized fractional order integrator (FOI). Next, the proposed generalized FOD and FOI are used to generate a secure single side band (SSB) signal for saving the transmission bandwidth. The parameters of variable generalized FOD and FOI can be used as the secure keys for construction and reconstruction. Finally, the relation between fractional Hilbert transformer and generalized FOD is studied and the edge detection application is demonstrated to show the flexibility and effectiveness of the proposed generalized FOD.Index Terms-Fractional calculus, fractional Hilbert transformer, fractional order differentiator, fractional order integrator, single side band. 1549-8328

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Caputo-Based Fractional Derivative in Fractional Fourier Transform Domain

Cited by 40 publications

References 22 publications

The Variable-Order Fractional Calculus of Variations

The Variable-Order Fractional Calculus of Variations

Caputo derivatives of fractional variable order: Numerical approximations

Designs of Discrete-Time Generalized Fractional Order Differentiator, Integrator and Hilbert Transformer

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