• Journal of applied mathematics & informatics
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• v.41 no.4
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• pp.811-819
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• 2023

In this research we have used the definition of standard fractional vector cross product to obtain fractional curl and fractional field of a standing wave, a travelling wave, a transverse wave, a vector field in xy plane, a complex vector field and an electric field. Fractional curl and fractional field for a complex order are also discussed. We have supported the study with calculation of impedance at γ = 0, 0 < γ < 1, γ = 1. The formula discussed in this paper are useful for study of polarization, reflection, impedance, boundary conditions where fractional solutions have applications.

• Geomechanics and Engineering
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• v.37 no.6
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• pp.591-597
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• 2024

The purpose of this paper is to depict the effect of gravity on a nonlocal thermoelastic medium with initial stress. The Lord-Shulman and Green-Lindsay theories with fractional derivative order serve as the foundation for the formulation of the fundamental equations for the problem. To address the problem and acquire the exact expressions of physical fields, appropriate non-dimensional variables and normal mode analysis are used. MATLAB software is used for numerical calculations. The projected outcomes in the presence and absence of the gravitational field, along with a nonlocal parameter, are compared. Additional comparisons are made for various fractional derivative order values. It is evident that the variation of physical variables is significantly influenced by the fractional derivative order, nonlocal parameter and gravity field.

• Current Optics and Photonics
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• v.7 no.2
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• pp.127-135
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• 2023

Radially polarized beams with the ability to generate a sub-wavelength sized spot in a longitudinal field provides significant applications in microscopic imaging, optical tweezers, lithography and so on. However, this excellent property can also be achieved based on conventional circularly polarized beams. Here, we demonstrate its ability to create a strong longitudinal field by comparing the tight focusing characteristics of fractional-order vortex modulated radial polarized and left-handed circular polarized Bessel-Gauss beams. Additionally, the possibility of generating arbitrary fractional-order vortex modulated Bessel-Gauss beams with a strong longitudinal field is demonstrated. A special modulation method of left-handed circularly polarized Bessel-Gauss beams modulated by a fractional-order vortex is adopted creatively and a series of regulation laws are obtained. Specifically, the fractional-order phase modulation parameter n can accurately control the number of optical lobes. The ratio of the pupil radius to the incident beam waist β₁ can control the radius of the optical lobes. The first-order Bessel function amplitude modulation parameter β₂ can control the number of layers of optical lobes. This work not only adds a new modulation method for optical micromanipulation and optical communication, but also enriches the research on fractional vortex beams which has very important academic significance.

• Geomechanics and Engineering
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• v.23 no.3
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• pp.217-225
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• 2020

In this paper, the thermoelastic interactions in a two-dimension porous body are studied. This problem is solved by using the Green and Lindsay (GL) generalized thermoelasticity model under fractional time derivative. The derived approaches are estimated. with numeral results which are applied to the porous mediums in simplifying geometrical. The bounding plane surface of the present half-space continuum is subjected to a pulse heat flux. We use the Laplace-Fourier transforms methods with the eigenvalues approach to solve the problem. The numerical solutions for the field functions are obtained numerically using the numerical Laplace inversion technique. The effects of the fractional parameter and the thermal relaxation times on the temperature field, the displacement field, the change in volume fraction field of voids distribution and the stress fields have been calculated and displayed graphically and the obtained results are discussed.

• Wind and Structures
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• v.28 no.6
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• pp.347-354
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• 2019

This study provides an approximate analytical solution to the fractional vibration problem of thin plate governing anomalous motion of plate subjected to in-plane loads. The method of variable separable is employed to transform the fractional partial differential equations under consideration into a fractional ordinary differential equation in temporal variable and a bi-harmonic plate equation in spatial variable. The technique of conformable fractional derivative is utilized to solve the resulting fractional differential equation and the approach of finite sine integral transform method is used to solve the accompanying bi-harmonic plate equation. The deflection field which measures the transverse displacement of the plate is expressed in terms of product of Bessel and trigonometric functions via the temporal and spatial variables respectively. The obtained solution reduces to the solution of the free vibration problem of thin plate in literature. This work shows that conformable fractional derivative is an efficient mathematical tool for tracking analytical solution of fractional partial differential equation governing anomalous vibration of thin plates.

• Progress in Superconductivity and Cryogenics
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• v.25 no.1
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• pp.1-5
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• 2023

The study of two-dimensional electron systems with extraordinarily low levels of disorder was, for a long time, the exclusive privilege of the epitaxial thin film research community. However, the successful isolation of graphene by mechanical exfoliation has truly disrupted this field. Furthermore, the assembly of heterostructures consisting of several layers of different 2D materials in arbitrary order by exploiting van der Waals forces has been a game-changer in the field of low-dimensional physics. This technique can be generalized to the large class of strictly 2D materials and offers unprecedented parameters to play with in order to tune electronic and other properties. It has led to a paradigm shift in the field of 2D condensed matter physics with bright prospects. In this review article, we discuss three device fabrication techniques towards high mobility devices: suspended structures, dry transfer, and pick-up transfer methods. We also address state-of-the-art device structures, which are fabricated by the van der Waals pick-up transfer method. Finally, we briefly introduce correlated ground states in the fractional quantum Hall regime.

• Structural Engineering and Mechanics
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• v.63 no.1
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• pp.89-102
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• 2017

In this work, we present a theoretical framework to study the thermovisco-elastic responses of homogeneous, isotropic and perfectly conducting medium subjected to inclined load. Based on recently developed generalized thermoelasticity theory with fractional order strain, the two-dimensional governing equations are obtained in the context of generalized magnetothermo-viscoelasticity theory without energy dissipation. The Kelvin-Voigt model of linear viscoelasticity is employed to describe the viscoelastic nature of the material. The resulting formulation of the field equations is solved analytically in the Laplace and Fourier transform domain. On the application of inclined load at the surface of half-space, the analytical expressions for the normal displacement, strain, temperature, normal stress and tangential stress are derived in the joint-transformed domain. To restore the fields in physical domain, an appropriate numerical algorithm is used for the inversion of the Laplace and Fourier transforms. Finally, we have demonstrated the effect of magnetic field, viscosity, mechanical relaxation time, fractional order parameter and time on the physical fields in graphical form for copper material. Some special cases have also been deduced from the present investigation.

• Journal of Power Electronics
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• v.18 no.3
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• pp.723-735
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• 2018

Linear proportional-integral (PI) controllers are an attractive choice for controlling the speed of induction machines because of their simplicity and ease of implementation. Fractional-order PI (FO-PI) controllers, however, perform better than PI controllers because of their nonlinear nature and the underlying iso-damping property of fractional-order operators. In this work, an FO-PI controller based on the proposed first-order plus dead-time induction motor model and integer-order (IO) controllers, such as Ziegler-Nichols PI, Cohen-Coon PI, and a PI controller tuned via trial-and-error method, is designed. Simulation and experimental investigation on an indirect field-oriented induction motor drive system proves that the proposed FO-PI controller has better speed tracking, lesser settling time, better disturbance rejection, and lower speed tracking error compared with linear IO-PI controllers. Our experimental study also validates that the FO-PI controller maximizes the torque per ampere output of the induction machine and can effectively control the motor at low speed, in field-weakening regions, and under detuned conditions.

• Computers and Concrete
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• v.23 no.5
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• pp.303-309
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• 2019

This study investigates axisymmetric fractional vibration of an isotropic hyperelastic semi-linear thin disc with a view to examine effects of finite deformation associated with the material of the disc and effects of fractional vibration associated with the motion of the disc. The generalized three-dimensional equation of motion is reduced to an equivalent time fraction one-dimensional vibration equation. Using the method of variable separable, the resulting equation is further decomposed into second-order ordinary differential equation in spatial variable and fractional differential equation in temporal variable. The obtained solution of the fractional vibration problem under consideration is described by product of one-parameter Mittag-Leffler and Bessel functions in temporal and spatial variables respectively. The obtained solution reduces to the solution of the free vibration problem in literature. Finally, and amongst other things, the Cauchy's stress distribution in thin disc under finite deformation exhibits nonlinearity with respect to the displacement fields whereas in infinitesimal deformation hypothesis, these stresses exhibit linear relation with the displacement field.

• Journal of Mechanical Science and Technology
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• v.20 no.5
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• pp.658-667
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• 2006

Fractional derivative models, which are used to describe the viscoelastic behavior of material, have received considerable attention. Thus it is necessary to put forward the analysis solutions of dynamic systems containing a fractional derivative. Although previously reported such kind of fractional calculus-based constitutive models, it only handles the particularity of rational number in part, has great limitation by reason of only handling with particular rational number field. Simultaneously, the former study has great unreliability by reason of using the complementary error function which can't ensure uniform real number. In this paper, a new approach is proposed for an analytical scheme for dynamic system of a spring-mass-damper system of single-degree of freedom under general forcing conditions, whose damping is described by a fractional derivative of the order $0<{\alpha}<1$ which can be both irrational number and rational number. The new approach combines the fractional Green's function and Laplace transform of fractional derivative. Analytical examples of dynamic system under general forcing conditions obtained by means of this approach verify the feasibility very well with much higher reliability and universality.