Abeer A. Mahmoud

Time-fractional KdV equation for plasma of two different temperature electrons and stationary ion

Using the time-fractional KdV equation, the nonlinear properties of small but finite amplitude electron-acoustic solitary waves are studied in a homogeneous system of unmagnetized collisionless plasma. This plasma consists of cold electrons fluid, non-thermal hot electrons, and stationary ions. Employing the reductive perturbation technique and the Euler-Lagrange equation, the time-fractional KdV equation is derived and it is solved using variational method. It is found that the time-fractional parameter significantly changes the soliton amplitude of the electron-acoustic solitary waves. The results are compared with the structures of the broadband electrostatic noise observed in the dayside auroral zone.

Time-fractional study of electron acoustic solitary waves in plasma of cold electron and two isothermal ions

In this paper, a homogeneous system of unmagnetized collisionless plasma consisting of a cold electron fluid, low-temperature ion obeying Boltzmann-type distribution and high-temperature ion obeying non-thermal distribution is considered. The perturbation method with two different forms of stretching will be considered to drive the KdV and modified KdV (mKdV) equations. The Agrawals method is applied to formulate the time-fractional KdV and mKdV equations. A variational iteration method is used to solve these equations. The results show that the fractional order of the differential equations can be used to modify the shape of the solitary pulse instead of adding higher order dissipation terms to the equations. This study may be useful to construct the compressive and rarefactive electrostatic potential pulses associated with the broadband electrostatic noise type emissions.

Ion-acoustic waves in plasma of warm ions and isothermal electrons using time-fractional KdV equation

The ion-acoustic solitary wave in collisionless unmagnetized plasma consisting of warm ions-fluid and isothermal electrons is studied using the time fractional KdV equation. The reductive perturbation method has been employed to derive the Korteweg-de Vries equation for small but finite amplitude ion-acoustic wave in warm plasma. The Lagrangian of the time fractional KdV equation is used in a similar form to the Lagrangian of the regular KdV equation with fractional derivative for the time differentiation. The variation of the functional of this Lagrangian leads to the Euler—Lagrange equation that gives the time fractional KdV equation. The variational-iteration method is used to solve the derived time fractional KdV equation. The calculations of the solution are carried out for different values of the time fractional order. These calculations show that the time fractional can be used to modulate the electrostatic potential wave instead of adding a higher order dissipation term to the KdV equation. The results of the present investigation may be applicable to some plasma environments, such as the ionosphere plasma.

Arbitrary amplitude dust-acoustic waves in four-component dusty plasma using non-extensive electrons and ions distributions-soliton solution

The four-component dusty plasma consisting of positive and negative dust species, electrons, and ions is considered for study. The fluid dynamics equations are applied to describe the motion of the two dust species. Both the electrons and ions are described by employing non-extensive distributions. The one-dimensional arbitrary amplitude of an electrostatic solitary structure has been studied using the Sagdeev pseudo-potential and phase-portrait approaches. In addition to the existence of either the compressive or rarefactive solitary wave, the analysis shows that these two types of waves coexist and propagate in the studied plasma model. Due to the complexity involved in the structure of Sagdeev pseudo-potential, a small amplitude approximation is considered. The corresponding potential diagrams and phase portrait are investigated and the analysis supports the existence of both compressive and rarefactive solitary waves in the considered plasma.

Time-fractional effect on pressure waves propagating through a fluid filled circular long elastic tube

Abstract The pressure waves propagating through an incompressible inviscid fluid that moves in a circular cylindrical long elastic tube are considered. The reductive perturbation method is used to derive the KdV equation from the hydrodynamic equations of the system. The Euler–Lagrange variational technique described by Agrawal has been applied to formulate the time-fractional KdV equation. The derived time-fractional KdV equation is solved by employing the variational-iteration method represented by He. The effects of the tube and fluid parameters and the time fractional order on the propagation of pressure waves are investigated.

Cylindrical electron acoustic solitons for modified time-fractional nonlinear equation

The modulation of cylindrical electron acoustic characteristics using a time fractal modified nonlinear equation has been investigated in nonisothermal plasmas. The time fractional cylindrical modified-Korteweg-de Vries equation has been obtained by Agrawals analysis. A cylindrical localized soliton has been obtained via the Adomian decomposition method. The pressure term and cylindrical time fractional effects on the modulated wave properties have been investigated with comparative auroral observations. It is established that the presence of the fractional order factor not only significantly modifies the solitary characteristics but also varies the profile polarity.

Time fractional effect on ion acoustic shock waves in ion-pair plasma

The nonlinear properties of ion acoustic shock waves are studied. The Burgers equation is derived and converted into the time fractional Burgers equation by Agrawal’s method. Using the Adomian decomposition method, shock wave solutions of the time fractional Burgers equation are constructed. The effect of the time fractional parameter on the shock wave properties in ion-pair plasma is investigated. The results obtained may be important in investigating the broadband electrostatic shock noise in D- and F-regions of Earth’s ionosphere.

Space—time fractional KdV—Burgers equation for dust acoustic shock waves in dusty plasma with non-thermal ions

The KdV—Burgers equation for dust acoustic waves in unmagnetized plasma having electrons, singly charged nonthermal ions, and hot and cold dust species is derived using the reductive perturbation method. The Boltzmann distribution is used for electrons in the presence of the cold (hot) dust viscosity coefficients. The semi-inverse method and Agrawal variational technique are applied to formulate the space—time fractional KdV—Burgers equation which is solved using the fractional sub-equation method. The effect of the fractional parameter on the behavior of the dust acoustic shock waves in the dusty plasma is investigated.

Arbitrary amplitude double-layers in four-component dusty plasma with q-non-extensive electrons and ions

The occurrence and propagation of dust acoustic double-layers with arbitrary amplitude have been investigated in four-component dusty plasma with q-non-extensive distributed electrons and ions. Via the Sagdeev pseudo-potential technique, the fluid equations of the plasma under consideration map to a single equation, namely, energy equation. Also, employing Sagdeev pseudo-potential and phase-portrait techniques shows that the profile of the double-layer solution is highly sensitive to the strength of Mach number, non-extensive parameters, and dust temperatures ratio. The results appear that only compressive double-layers are found for certain conditions. In addition, the case of small amplitude double-layer approximation is introduced and the explicit form of double-layer solution is written down where its amplitude and width depend crucially on the plasma parameters. The obtained results are very useful to understand the basic features of Jupiters magnetosphere, Earths mesosphere, and cometary tails whe...

On the Time Fractional Modulation for Electron Acoustic Shock Waves

Recently, applications of fractional nonlinear partial differential equations have received major attention in fluid mechanics and physics plasmas.[1−4] Electron acoustic wave (EAW) propagation in plasmas has been studied experimentally and theoretically.[5−8] Clearly, evolution of small amplitude solitary nonlinear structures in fluids and plasmas has been investigated by using many nonlinear equations (NLEs).[9−15] These equations are obtained by using perturbation methods such as the well-known reductive perturbation theory (RPT). In comparing the solitary solutions of these equations with observation data,[5,8−11] it was found that the amplitude of solitons and shocks would underestimate in a range of 20%. For that, to reduce this variation, different methods have been proposed such as algebraic techniques, higher order approximations and time fractal solutions. Abdelwahed used the renormalization method to modulate the EA soliton properties in magnetized plasma. It is found that the electron acoustic higher order solution is dependent mainly on the external magnetic field magnitude. El-Wakil et al. inspected electrostatic Viking satellite EA solitons observed in the dayside auroral zone by using the nonlinear time-fractional KdV equation. Borhanian et al. derived that the EA solitary structures were affected by hot electron suprathermality and transverse perturbations in non-planar plasmas. Effects of trapped hot electrons on acoustic soliton have been discussed by using the mKdV equation with time fractional term. Guo et al. studied the progress of ionic waves in ion pair plasmas model via the Gardner equation with the time fraction term. They used the method of variational iteration to investigate the nonthermal electrons’ effect on the produced wave. The studies on plasma physics using fractional nonlinear evolution equations have been discussed by many groups.[19−23] In the past decade, the study of acoustic shocks in astrophysical plasma have been investigated.[24−28] Ema et al. discussed the propagation of acoustic shocks in multi ion plasmas. It is noted that the acoustic shocks are modified by the parameter of heavy-to-light ion number density. Also, both polarity potentials exist in the plasma system. The dust-acoustic shock properties have been studied in two-temperature dust plasmas using the Burgers equation with time fractional order. Researchers discussed the time fractional parameter effect on shock wave features using the variational iteration technique. El-Shewy et al. studied shock waves using the space time-fractional KdV Burgers equation. It is noted that the parameter of space time fractional affects the coexistence of shocks. Recently, shock wave modulation in two viscous ion plasmas have been examined via higher order dissipation in Earth’s ionosphere. In this study, an electron acoustic model with superthermal hot electrons is considered. The KdV equation is derived, Agrawal’s process is used to formulate the time fractional TFKdV equation, and the Adomian decomposition method is used to solve them. In this model, a three-component collisionless unmagnetized plasma having ions in the stationary state, viscous fluid of cold electron and κ velocity distributed hot electrons is studied. The normalized equations in a small, finite amplitude are given by