The fractional nonlinear $${\mathcal{PT}}$$ dimer

AbstractWe examine a fractional discrete nonlinear Schrodinger dimer, where the usual first-order derivative in the time evolution is replaced by a non integer-order derivative. The dimer is nonlinear (Kerr) and $${\mathcal{{PT}}}$$ PT -symmetric, and for localized initial conditions we examine the exchange dynamics between both sites. By means of the Laplace transformation technique, the linear $${{\mathcal{{PT}}}}$$ PT dimer is solved in closed form in terms of Mittag–Leffler functions, while for the nonlinear regime, we resort to numerical computations using the direct explicit Grunwald algorithm. In general, we see that the main effect of the fractional derivative is to produce a monotonically decreasing time envelope for the amplitude of the oscillatory exchange. In the presence of $${\mathcal{{PT}}}$$ PT symmetry, the oscillations experience some amplification for gain/loss values below some threshold, while beyond threshold, the amplitudes of both sites grow unbounded. The presence of nonlinearity can arrest the unbounded growth and lead to a selftrapped state. The trapped fraction decreases as the nonlinearity is increased past a critical value, in marked contrast with the standard (non-fractional) case.

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The Fractional Nonlinear PT Dimer

10.21203/rs.3.rs-199590/v1 ◽

2021 ◽

Author(s):

Mario I. Molina

Keyword(s):

Order Derivative ◽

Marked Contrast ◽

Standard Case ◽

First Order ◽

Damped Oscillations ◽

Main Effect ◽

Loss Parameter ◽

Small Gain ◽

Gain Loss ◽

Parameter Values

Abstract We examine a fractional Discrete Nonlinear Schrodinger dimer, where the usual first-order derivative of the time evolution is replaced by a non integer-order derivative. The dimer is nonlinear (Kerr) and PT-symmetric, and we examine the exchange dynamics between both sites. By means of the Laplace transformation technique, the linear PT dimer is solved in closed form in terms of Mittag-Leffler functions, while for the nonlinear regime, we resort to numerical computations using the direct explicit Grunwald algorithm. In general, the main effect of the fractional derivative is the onset of a monotonically decreasing time envelope for the amplitude of the oscillatory exchange. In the presence of PT symmetry, the dynamics shows damped oscillations for small gain/loss in both sites, while at higher gain/loss parameter values, the amplitudes of both sites grows unbounded. In the presence of nonlinearity, selftrapping is still possible although the trapped fraction decreases as the nonlinearity is increased past threshold, in marked contrast with the standard case.

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Functions of bounded fractional differential variation – A new concept

Georgian Mathematical Journal ◽

10.1515/gmj-2016-0030 ◽

2016 ◽

Vol 23 (3) ◽

pp. 417-427 ◽

Cited By ~ 1

Author(s):

Jyotindra C. Prajapati ◽

Krunal B. Kachhia

Keyword(s):

Fractional Derivative ◽

Order Derivative ◽

Normed Algebra ◽

Classical Analysis ◽

First Order ◽

Fractional Differential

AbstractThe idea of functions of bounded differential variation was introduced by Bhatt, Dabhi and Kachhia in [2]. In the present paper, we introduce functions of bounded fractional differential variation using the Caputo-type fractional derivative instead of the commonly used first-order derivative. Various properties and relation with some known results of classical analysis are also studied. We prove that the space ${\mathrm{BFDV}[a,b]}$ of all functions of bounded fractional differential variation on ${[a,b]}$ is a normed algebra under certain type of norms.

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A free fractional viscous oscillator as a forced standard damped vibration

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2016-0018 ◽

2016 ◽

Vol 19 (2) ◽

Cited By ~ 6

Author(s):

Giuseppe Devillanova ◽

Giuseppe Carlo Marano

Keyword(s):

Fractional Derivative ◽

Initial Conditions ◽

Free Response ◽

Resolution Method ◽

Single Degree Of Freedom ◽

Caputo’S Fractional Derivative ◽

First Order ◽

Fractional Differential ◽

Free Dynamic ◽

Fractional Displacement

AbstractThis paper contains a survey on one of the mathematical approaches used to solve a fractional differential equation whose solution gives the free dynamic response of viscoelastic single degree of freedom systems (viscosity is actually modelled by a fractional displacement derivative instead of first order one). The paper shall deal with Caputo’s fractional derivative since its Laplace Transform (on which the resolution method is based) only depends by lower integer order (and therefore measurable and physically meaningful) derivatives given as initial conditions. The paper provides a deep mathematical analysis of the properties of the solution expressed in terms of the mechanical parameters meaning. Additionally, important physical implications are reported exhibiting a richer dynamic behavior if compared to the standard damping case (velocity linear dependence). Some important consequences in the use of Caputo’s fractional derivative are reported, and some limitations to possible viscous parameters values are obtained. Finally, it is shown that free response of fractional derivative equation solution is mathematically equivalent to a suitably forced solution of the integer model.

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Dynamics of the helmholtz oscillator with fractional order damping

Acta Universitaria ◽

10.15174/au.2013.584 ◽

2013 ◽

Vol 23 ◽

pp. 12-15

Author(s):

Adolfo Ortiz ◽

Jesús Seoane ◽

J. Yang ◽

Miguel Sanjuán

Keyword(s):

Fractional Derivative ◽

Fractional Order ◽

Numerical Scheme ◽

Initial Conditions ◽

Critical Value ◽

Periodic Motions ◽

Damping Term ◽

Fractional Damping ◽

Runge Kutta Method ◽

History Of

The dynamics of the nonlinear Helmholtz Oscillator with fractional order damping are studied in detail. The discretization of differential equations according to the Grünwald-Letnikov fractional derivative definition in order to get numerical simulations is reported. Comparison between solutions obtained through a fourth-order Runge-Kutta method and the fractional damping system are comparable when the fractional derivative of the damping term a is fixed at 1. That proves the good performance of the numerical scheme. The effect of taking the fractional derivative on the system dynamics is investigated using phase diagrams varying a from 0.5 to 1.75 with zero initial conditions. Periodic motions of the system are obtained at certain ranges of the damping term. On the other hand, escape of the trajectories from a potential well result at a certain critical value of the fractional derivative. The history of the displacement as a function of time is shown also for every a selected.

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Burgers equation with a fractional derivative; hereditary effects on nonlinear acoustic waves

Journal of Fluid Mechanics ◽

10.1017/s0022112091002203 ◽

1991 ◽

Vol 225 ◽

pp. 631-653 ◽

Cited By ~ 108

Author(s):

N. Sugimoto

Keyword(s):

Fractional Derivative ◽

Acoustic Waves ◽

Burgers Equation ◽

Initial Conditions ◽

Near Field ◽

Expansion Method ◽

Second Order ◽

Order Derivative ◽

Matched Asymptotic Expansion ◽

Hereditary Effects

This paper deals with initial-value problems for the Burgers equation with the inclusion of a hereditary integral known as the fractional derivative of order ½. Emphasis is placed on the difference between the local and global dissipation due to the second-order and the half-order derivatives, respectively. Exploiting the smallness of the coefficient of the second-order derivative, an asymptotic analysis is first developed. When a discontinuity appears, the matched-asymptotic expansion method is employed to derive a uniformly valid solution. If the coefficient of the half-order derivative is also small, as is usually the case, the evolution comprises three stages, namely a lossless near field, an intermediate Burgers region, and a hereditary far field. In view of these results, the equation is then solved numerically, under various initial conditions, by finite-difference and spectral methods. It is revealed that the effect of the fractional derivative accumulates slowly to give rise to a significant dissipation and distortion of the waveform globally, which is to be contrasted with the effect of the second-order derivative, significant only locally, in a thin 'shock layer’.

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Numerical exploration of thin film flow of MHD pseudo-plastic fluid in fractional space: Utilization of fractional calculus approach

Open Physics ◽

10.1515/phys-2021-0081 ◽

2021 ◽

Vol 19 (1) ◽

pp. 710-721

Author(s):

Mubashir Qayyum ◽

Farnaz Ismail ◽

Muhammad Sohail ◽

Naveed Imran ◽

Sameh Askar ◽

...

Keyword(s):

Thin Film ◽

Fractional Calculus ◽

Fractional Derivative ◽

Film Flow ◽

Second Order ◽

Order Derivative ◽

Thin Film Flow ◽

First Order ◽

Plastic Fluid ◽

Fractional Space

Abstract In this article, thin film flow of non-Newtonian pseudo-plastic fluid is investigated on a vertical wall through homotopy-based scheme along with fractional calculus. Three cases were examined after considering (i) partial fractional differential equation (PFDE) by altering first-order derivative to fractional derivative in the interval (0, 1), (ii) PFDE by altering second-order derivative to fractional derivative in the interval (1, 2), and (iii) fully FDE by altering first-order derivative to fractional derivative in (0, 1) and second-order derivative to fractional derivative in (1, 2). Different physical quantities such as the velocity profile and volume flux were computed and analyzed. Validity of obtained results was checked by finding residuals. Moreover, consequence of different parameters on the velocity were also explored in fractional space.

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Conservative solute transport with periodic velocity and sinusoidal source concentration in semi-infinite porous media: An analytical solution

JOURNAL OF ADVANCES IN PHYSICS ◽

10.24297/jap.v6i1.1818 ◽

2014 ◽

Vol 6 (1) ◽

pp. 1024-1031

Author(s):

R R Yadav ◽

Gulrana Gulrana ◽

Dilip Kumar Jaiswal

Keyword(s):

Porous Medium ◽

Porous Media ◽

Laplace Transformation ◽

Seepage Velocity ◽

Transformation Technique ◽

Numerical Examples ◽

One Dimensional ◽

Porous Domain ◽

Conservative Solute Transport ◽

Source Concentration

The present paper has been focused mainly towards understanding of the various parameters affecting the transport of conservative solutes in horizontally semi-infinite porous media. A model is presented for simulating one-dimensional transport of solute considering the porous medium to be homogeneous, isotropic and adsorbing nature under the influence of periodic seepage velocity. Initially the porous domain is not solute free. The solute is initially introduced from a sinusoidal point source. The transport equation is solved analytically by using Laplace Transformation Technique. Alternate as an illustration; solutions for the present problem are illustrated by numerical examples and graphs.

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