A numerical scheme for solving differential equations with space and time-fractional coordinate derivatives

Khan, Yasir; Beik, Samaneh Panjeh Ali; Sayevand, Khosro; Shayganmanesh, Ahmad

doi:10.2989/16073606.2014.981699

Cited by 19 publications

(6 citation statements)

References 25 publications

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“…Hence, it was shown in [4] that there is no periodic waves for the autonomous Korteweg-de Vries-Burgers equation of dimension two. We follow, in this paper, the same trend of numerical approach by making use of the recently developed fractional derivative with nonsingular kernel [8][9][10][11][12][13], to express a seventh order Korteweg-de Vries (KdV) equation with one perturbation level. This is the first instance where such a model is extended to the scope of fractional differentiation and fully investigated.…”

Section: Introductionmentioning

confidence: 99%

Around Chaotic Disturbance and Irregularity for Higher Order Traveling Waves

Goufo

Toudjeu

2018

Journal of Mathematics

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Many unknown features in the theory of wave motion are still captivating the global scientific community. In this paper, we consider a model of seventh order Korteweg–de Vries (KdV) equation with one perturbation level, expressed with the recently introduced derivative with nonsingular kernel, Caputo-Fabrizio derivative (CFFD). Existence and uniqueness of the solution to the model are established and proven to be continuous. The model is solved numerically, to exhibit the shape of related solitary waves and perform some graphical simulations. As expected, the solitary wave solution to the model without higher order perturbation term is shown via its related homoclinic orbit to lie on a curved surface. Unlike models with conventional derivative (γ=1) where regular behaviors are noticed, the wave motions of models with the nonsingular kernel derivative are characterized by irregular behaviors in the pure factional cases (γ<1). Hence, the regularity of a soliton can be perturbed by this nonsingular kernel derivative, which, combined with the perturbation parameter ζ of the seventh order KdV equation, simply causes more accentuated irregularities (close to chaos) due to small irregular deviations.

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Section: Introductionmentioning

confidence: 99%

Around Chaotic Disturbance and Irregularity for Higher Order Traveling Waves

Goufo

Toudjeu

2018

Journal of Mathematics

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“…Owing to the viscoelastic and dynamic behaviour of the material, the fractional-order physical models (Miller and Ross, 1993; Podlubny, 1999; Samko et al , 2002; Saha Ray and Bera, 2005b; Saha Ray, 2007; Khan et al , 2015; Saha Ray, 2015) have received much attention by researchers during the past decades. Fractional differential equations have often been used to model the behaviour of dynamic systems (Wharmby and Bagley, 2013; Shen and Soong, 1995).…”

Section: Introductionmentioning

confidence: 99%

An efficient and novel technique for solving continuously variable fractional order mass-spring-damping system

Sahoo

Ray

Das

2017

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Purpose In this paper, the formulation and analytic solutions for fractional continuously variable order dynamic models, namely, fractional continuously mass-spring damper (continuously variable fractional order) systems, have been presented. The authors will demonstrate via two cases where the frictional damping given by fractional derivative, the order of which varies continuously – while the mass moves in a guide. Here, the continuously changing nature of the fractional-order derivative for dynamic systems has been studied for the first time. The solutions of the fractional continuously variable order mass-spring damper systems have been presented here by using a successive recursive method, and the closed form of the solutions has been obtained. By using graphical plots, the nature of the solutions has been discussed for the different cases of continuously variable fractional order of damping force for oscillator. The purpose of the paper is to formulate the continuously variable order mass-spring damper systems and find their analytical solutions by successive recursion method. Design/methodology/approach The authors have used the viscoelastic and viscous – viscoelastic dampers for describing the damping nature of the oscillating systems, where the order of the fractional derivative varies continuously. Findings By using the successive recursive method, here, the authors find the solution of the fractional continuously variable order mass-spring damper systems, and then obtain close-form solutions. The authors then present and discuss the solutions obtained in the cases with the continuously variable order of damping for an oscillator through graphical plots. Originality/value Formulation of fractional continuously variable order dynamic models has been described. Fractional continuous variable order mass-spring damper systems have been analysed. A new approach to find solutions of the aforementioned dynamic models has been established. Viscoelastic and viscous – viscoelastic dampers are described. The discussed damping nature of the oscillating systems has not been studied yet.

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“…There are several definitions of fractional derivative [1][2][3][4] that have been proposed in the past. Here, we review the most frequently used definitions of fractional integral, namely, Riemann-Liouville integral, which is defined as follows…”

Section: Definition (Riemann-liouville)mentioning

confidence: 99%

“…In the past few years, the fractional order physical models [1][2][3][4] have seen much attention by researchers due to dynamic behaviour and the viscoelastic behaviour of material. 5 Thus, the fractional order model is remarkably used for describing the frequency distribution of the structural damping systems.…”

Section: Introductionmentioning

confidence: 99%

Formulation and solutions of fractional continuously variable order mass–spring–damper systems controlled by viscoelastic and viscous–viscoelastic dampers

Ray

Sahoo

Das

2016

Advances in Mechanical Engineering

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This article presents the formulation and a new approach to find analytic solutions for fractional continuously variable order dynamic models, namely, fractional continuously variable order mass-spring-damper systems. Here, we use the viscoelastic and viscous-viscoelastic dampers for describing the damping nature of the oscillating systems, where the order of fractional derivative varies continuously. Here, we handle the continuous changing nature of fractional order derivative for dynamic systems, which has not been studied yet. By successive recursive method, here we find the solution of fractional continuously variable order mass-spring-damper systems and then obtain closed-form solutions. We then present and discuss the solutions obtained in the cases with continuously variable order of damping for oscillator through graphical plots.

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A numerical scheme for solving differential equations with space and time-fractional coordinate derivatives

Cited by 19 publications

References 25 publications

Around Chaotic Disturbance and Irregularity for Higher Order Traveling Waves

Around Chaotic Disturbance and Irregularity for Higher Order Traveling Waves

An efficient and novel technique for solving continuously variable fractional order mass-spring-damping system

Formulation and solutions of fractional continuously variable order mass–spring–damper systems controlled by viscoelastic and viscous–viscoelastic dampers

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