Numerical studies for the variable-order nonlinear fractional wave equation

Sweilam, N. H.; Khader, M. M.; Almarwm, H. M.

doi:10.2478/s13540-012-0045-9

Cited by 43 publications

(27 citation statements)

References 21 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

127

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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

127

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“…In this section, we briefly recall the main ideas of fractional calculus, when the order of integration/differentiation is allowed to vary over time. Fractional operators with variable order can be successfully used to describe several physical phenomena, as discussed in References , and among these we have the aging phenomenon, as discussed in Reference .…”

Section: Fractional Hereditary‐aging Materialsmentioning

confidence: 99%

A numerical integration approach for fractional‐order viscoelastic analysis of hereditary‐aging structures

Beltempo

Bonelli

Bursi

et al. 2019

Numerical Meth Engineering

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Summary In this paper, a novel numerical integration scheme is proposed for fractional‐order viscoelastic analysis of hereditary‐aging structures. More precisely, the idea of aging is first introduced through a new phenomenological viscoelastic model characterized by variable‐order fractional operators. Then, the presented fractional‐order viscoelastic model is included in a variational formulation, conceived for any viscous kernel and discretized in time by employing a discontinuous Galerkin method. The accuracy of the resulting finite element (FE) scheme is analyzed through a model problem, whose exact solution is known; and the most significant variables affecting the solution quality, such as the number of Gaussian quadrature points and time subintervals, are then investigated in terms of error and computational cost. Moreover, the proposed FE integration scheme is applied to study the short‐ and long‐term behavior of concrete structures, which, due to the severe aging exhibited during their service life, represents one of the most challenging time‐dependent behavior to be investigated. Eventually, also the Euler implicit method, commonly used in commercial software, is compared.

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“…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

161

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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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Numerical studies for the variable-order nonlinear fractional wave equation

Cited by 43 publications

References 21 publications

The Variable-Order Fractional Calculus of Variations

The Variable-Order Fractional Calculus of Variations

A numerical integration approach for fractional‐order viscoelastic analysis of hereditary‐aging structures

Caputo derivatives of fractional variable order: Numerical approximations

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