Liquan Mei

An efficient algorithm for solving Troesch's problem

Abstract A new algorithm is presented for solving Troesch’s problem. The numerical scheme based on the modified homotopy perturbation technique is deduced. Some numerical experiments are made. Compared with the variational iteration method and the Adomian decomposition method, the scheme is shown to be highly accurate, and only a few terms are required to obtain accurate computable solutions. Finally, the algorithm is applied to other problems.

Newton Iterative Parallel Finite Element Algorithm for the Steady Navier-Stokes Equations

A combination method of the Newton iteration and parallel finite element algorithm is applied for solving the steady Navier-Stokes equations under the strong uniqueness condition. This algorithm is motivated by applying the Newton iterations of m times for a nonlinear problem on a coarse grid in domain Ω and computing a linear problem on a fine grid in some subdomains Ωj⊂Ω with j=1,…,M in a parallel environment. Then, the error estimation of the Newton iterative parallel finite element solution to the solution of the steady Navier-Stokes equations is analyzed for the large m and small H and h≪H. Finally, some numerical tests are made to demonstrate the the effectiveness of this algorithm.

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.

Numerical solutions of RLW equation using Galerkin method with extrapolation techniques

Abstract In this paper, we present a new Galerkin method for the regularized long wave (RLW) equation. Based on the Galerkin method using linear finite elements, the extrapolation technique is proposed to increase the order of the time discretization accuracy to O ( ( Δ t ) 2 ) , giving O ( ( Δ t ) 2 + h 2 ) overall, which is quite efficient to solve the one-dimensional RLW. A stability analysis based on Von Neumann theory is performed. Propagation of solitary waves, interaction of two solitary waves and undular bores are simulated using the proposed method to validate the method which is found to be accurate and efficient.

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.

Explicit multistep method for the numerical solution of RLW equation

Abstract In this paper, we present explicit multistep method for regularized long wave (RLW) equation. The discretization in space is based on the Galerkin method using linear space finite elements, and for the time discretization the explicit multistep method is used. Propagation of solitary waves and interaction of two solitary waves are simulated using the proposed method to confirm the theoretical results.

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.

(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.

A Galerkin Finite Element Method for Numerical Solutions of the Modified Regularized Long Wave Equation

A Galerkin method for a modified regularized long wave equation is studied using finite elements in space, the Crank-Nicolson scheme, and the Runge-Kutta scheme in time. In addition, an extrapolation technique is used to transform a nonlinear system into a linear system in order to improve the time accuracy of this method. A Fourier stability analysis for the method is shown to be marginally stable. Three invariants of motion are investigated. Numerical experiments are presented to check the theoretical study of this method.