Shimin Guo

Fractional variational homotopy perturbation iteration method and its application to a fractional diffusion equation

In this paper, we use the fractional variational homotopy perturbation iteration method (FVHPIM) with modified Riemann-Liouville derivative to solve a time-fractional diffusion equation. Using this method, a rapid convergent sequence tending to the exact solution of the equation can be obtained. To show the efficiency of the considered method, some numerical examples are presented.

The extended Riccati equation mapping method for variable-coefficient diffusion–reaction and mKdV equations

Abstract In this paper, the extended Riccati equation mapping method is proposed to seek exact solutions of variable-coefficient nonlinear evolution equations. Being concise and straightforward, this method is applied to certain type of variable-coefficient diffusion–reaction equation and variable-coefficient mKdV equation. By means of this method, hyperbolic function solutions and trigonometric function solutions are obtained with the aid of symbolic computation. It is shown that the proposed method is effective, direct and can be used for many other variable-coefficient nonlinear evolution equations.

Time-fractional Gardner equation for ion-acoustic waves in negative-ion-beam plasma with negative ions and nonthermal nonextensive electrons

Nonlinear propagation of ion-acoustic waves is investigated in a one-dimensional, unmagnetized plasma consisting of positive ions, negative ions, and nonthermal electrons featuring Tsallis distribution that is penetrated by a negative-ion-beam. The classical Gardner equation is derived to describe nonlinear behavior of ion-acoustic waves in the considered plasma system via reductive perturbation technique. We convert the classical Gardner equation into the time-fractional Gardner equation by Agrawals method, where the time-fractional term is under the sense of Riesz fractional derivative. Employing variational iteration method, we construct solitary wave solutions of the time-fractional Gardner equation with initial condition which depends on the nonlinear and dispersion coefficients. The effect of the plasma parameters on the compressive and rarefactive ion-acoustic solitary waves is also discussed in detail.

Auxiliary equation method for the mKdV equation with variable coefficients

Abstract By using solutions of an ordinary differential equation, an auxiliary equation method is described to seek exact solutions of nonlinear evolution equations with variable coefficients. Being concise and straightforward, this method is applied to the mKdV equation with variable coefficients. As a result, new explicit solutions including solitary wave solutions and trigonometric function solutions are obtained with the aid of symbolic computation.

(3 + 1)-dimensional cylindrical Korteweg-de Vries equation for nonextensive dust acoustic waves: Symbolic computation and exact solutions

By combining the effects of bounded cylindrical geometry, azimuthal and axial perturbations, the nonlinear dust acoustic waves (DAWs) in an unmagnetized plasma consisting of negatively charged dust grains, nonextensive ions, and nonextensive electrons are studied in this paper. Using the reductive perturbation method, a (3 + 1)-dimensional variable-coefficient cylindrical Korteweg-de Vries (KdV) equation describing the nonlinear propagation of DAWs is derived. Via the homogeneous balance principle, improved F-expansion technique and symbolic computation, the exact traveling and solitary wave solutions of the KdV equation are presented in terms of Jacobi elliptic functions. Moreover, the effects of the plasma parameters on the solitary wave structures are discussed in detail. The obtained results could help in providing a good fit between theoretical analysis and real applications in space physics and future laboratory plasma experiments where long-range interactions are present.

Modulation instability and ion-acoustic rogue waves in a strongly coupled collisional plasma with nonthermal nonextensive electrons

The nonlinear propagation of ion-acoustic waves is theoretically reported in a collisional plasma containing strongly coupled ions and nonthermal electrons featuring Tsallis distribution. For this purpose, the nonlinear integro-differential form of the generalized hydrodynamic model is used to investigate the strong-coupling effect. The modified complex Ginzburg–Landau equation with a linear dissipative term is derived for the potential wave amplitude in the hydrodynamic regime, and the modulation instability of ion-acoustic waves is examined. When the dissipative effect is neglected, the modified complex Ginzburg–Landau equation reduces to the nonlinear Schrodinger equation. Within the unstable region, two different types of second-order ion-acoustic rogue waves including single peak type and rogue wave triplets are discussed. The effect of the plasma parameters on the rogue waves is also presented.

Modulation instability and dissipative ion-acoustic structures in collisional nonthermal electron-positron-ion plasma: solitary and shock waves*

The nonlinear behavior of an ion-acoustic wave packet is investigated in a three-component plasma consisting of warm ions, nonthermal electrons and positrons. The nonthermal components are assumed to be inertialess and hot where they are modeled by the kappa distribution. The relevant processes, including the kinematic viscosity amongst the plasma constituents and the collision between ions and neutrals, are taken into consideration. It is shown that the dynamics of the modulated ion-acoustic wave is governed by the generalized complex Ginzburg–Landau equation with a linear dissipative term. The dispersion relation and modulation instability criterion for the generalized complex Ginzburg–Landau equation are investigated numerically. In the general dissipation regime, the effect of the plasma parameters on the dissipative solitary (dissipative soliton) and shock waves is also discussed in detail.

An efficient Galerkin spectral method for two-dimensional fractional nonlinear reaction–diffusion-wave equation

Abstract The aim of this paper is to develop an efficient numerical treatment for the two-dimensional fractional nonlinear reaction–diffusion-wave equation with the time-fractional derivative of order α ( 1 α 2 ). For this purpose, we employ the alternating direction implicit (ADI) method based on the Crank–Nicolson scheme for the time stepping, while we apply the Legendre–Galerkin spectral method for the space discretization. The stability and convergence analysis are rigorously set up. In addition, the proposed method is extended to solve the time-fractional Klein–Gordon and sine-Gordon models. Numerical experiments are included, which verifies the theoretical predictions.

The compound (G′G)-expansion method and double non-traveling wave solutions of (2+1) -dimensional nonlinear partial differential equations

Abstract To seek the exact double non-traveling wave solutions of nonlinear partial differential equations, the compound ( G ′ G ) -expansion method is firstly proposed in this paper. With the aid of symbolic computation, this new method is applied to construct double non-traveling wave solutions of (2+1)-dimensional Painleve integrable Burgers equation and (2+1)-dimensional breaking soliton equation. As a result, abundant double non-traveling wave solutions including double hyperbolic function solutions, double trigonometric function solutions, double rational solutions, and a series of complexiton solutions of these two equations are obtained via the proposed method. These exact solutions contain arbitrary functions, which may be helpful to explain some complex phenomena. When the parameters are taken as special values, the double solitary-like wave solutions can be derived from double hyperbolic function solutions. Furthermore, the time evolutions of double solitary-like wave solutions are discussed in detail.

Finite difference/spectral-Galerkin method for a two-dimensional distributed-order time–space fractional reaction–diffusion equation

Abstract In this letter, we consider the numerical approximation of a two-dimensional distributed-order time–space fractional reaction–diffusion equation. The time- and space-fractional derivatives are considered in the senses of Caputo and Riesz, respectively. By using the composite mid-point quadrature, the original fractional problem is approximated by a multi-term time–space fractional differential equation. Then the multi-term Caputo fractional derivatives are discretized by the L 2 - 1 σ formula. We apply the Legendre–Galerkin spectral method for the spatial approximation. Two numerical experiments with smooth and non-smooth initial conditions, respectively, are performed to illustrate the robustness of the proposed method. The results show that: our scheme can arrive at the spectral accuracy (resp. algebraic accuracy) in space for the problem with smooth (resp. non-smooth) initial condition. For both of these two cases, our scheme can lead to the second-order accuracies in time. Additionally, the convergence rates in both spatial and temporal distributed-order variables are two.