A novel method for a fractional derivative with non-local and non-singular kernel

Akgül, Ali

doi:10.1016/j.chaos.2018.07.032

Cited by 273 publications

(76 citation statements)

References 18 publications

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“…In recent decades, there have been extensive studies to examine qualitative properties in problems of mathematical physics governed by linear differential equations that include the effects of periodic parametric excitations, time delays and fractional derivatives [2,3,31,33,39]. Specifically, this part of differential equations and its different variants of linear dynamical systems have mostly appeared in the description of numerous physical problems such as in the field of electrical circuits and small oscillation systems [5,8].…”

Section: Introductionmentioning

confidence: 99%

Floquet analysis of linear dynamic RLC circuits

2020

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AbstractIn this article, the linear dynamic analysis of AC generators modeled as RLC circuits with periodically time-varying inductances via Floquet’s theory is considered. Necessary conditions for the dynamic stability are derived. The harmonic balance method is employed to predict the transition curves and stability domains. An approximate expression for the Floquet form of solution is constructed using Whittaker’s method in the neighborhood of transition curves. Numerical verifications for the obtained theoretical results are considered. In accordance with the experimental results, a satisfactory agreement is relatively achieved with the closed experimental literature of the problem.

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Section: Introductionmentioning

confidence: 99%

Floquet analysis of linear dynamic RLC circuits

2020

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“…Besides calculating the time-evolution of the wave function and the way to minimize the cost function we used as described in ref. [30], there are other mathematical ways about fractional derivatives to calculate differential equations [36][37][38][39][40][41][42][43][44][45]. Figure 3(a) shows the atomic wave function as time when the dimple-ring potential is shaken, resulting in the optimized control to excite the ground state of the dimplering trap for 20 ms and hold the excited state in the dimplering trap for 20 ms. Figure 3(b) shows the optimized control in black and the initial guess in red.…”

Section: ∫ ( ( )mentioning

confidence: 99%

Optimal control for generating excited state expansion in ring potential

Kim

2020

Open Physics

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We applied an optimal control algorithm to an ultra-cold atomic system for constructing an atomic Sagnac interferometer in a ring trap. We constructed a ring potential on an atom chip by using an RF-dressed potential. A field gradient along the radial direction in a ring trap known as the dimple-ring trap is generated by using an additional RF field. The position of the dimple is moved by changing the phase of the RF field [1]. For Sagnac interferometers, we suggest transferring Bose–Einstein condensates to a dimple-ring trap and shaking the dimple potential to excite atoms to the vibrational-excited state of the dimple-ring potential. The optimal control theory is used to find a way to shake the dimple-ring trap for an excitation. After excitation, atoms are released from the dimple-ring trap to a ring trap by adiabatically turning off the additional RF field, and this constructs a Sagnac interferometer when opposite momentum components are overlapped. We also describe the simulation to construct the interferometer.

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“…These have many useful applications due to the non-locality of fractional derivatives: many processes with non-local behaviours can be modelled most efficiently using fractional differential equations [3]. Analytical and numerical solution methods for fractional differential equations have been much studied in the literature [4][5][6][7].…”

Section: Introductionmentioning

confidence: 99%

On a Fractional Operator Combining Proportional and Classical Differintegrals

2020

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The Caputo fractional derivative has been one of the most useful operators for modelling non-local behaviours by fractional differential equations. It is defined, for a differentiable function f (t), by a fractional integral operator applied to the derivative f (t). We define a new fractional operator by substituting for this f (t) a more general proportional derivative. This new operator can also be written as a Riemann-Liouville integral of a proportional derivative, or in some important special cases as a linear combination of a Riemann-Liouville integral and a Caputo derivative. We then conduct some analysis of the new definition: constructing its inverse operator and Laplace transform, solving some fractional differential equations using it, and linking it with a recently described bivariate Mittag-Leffler function.

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A novel method for a fractional derivative with non-local and non-singular kernel

Cited by 273 publications

References 18 publications

Floquet analysis of linear dynamic RLC circuits

Floquet analysis of linear dynamic RLC circuits

Optimal control for generating excited state expansion in ring potential

On a Fractional Operator Combining Proportional and Classical Differintegrals

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