Galerkin methods applied to some model equations for non-linear dispersive waves

The application of a dissipative Galerkin scheme to the numerical solution of the Korteweg de Vries (KdV) and Regularised Long Wave (RLW) equations, is investigated. The accuracy and stability of the proposed schemes is derived using a localised Fourier analysis. With cubic splines as basis functions, the errors in the numerical solutions of the KdV equation for different mesh-sizes and different amounts of dissipation is determined. It is shown that the Galerkin scheme for the RLW equation gives rise to much smaller errors (for a given mesh-size), and allows larger steps to be taken for the integrations in time (for a specified error tolerance). Also, the interaction of two solitons is compared for the KdV and RLW equations, and several differences in their behaviour are found.     

Application of homotopy perturbation method to the RLW and generalized modified Boussinesq equations

In this Letter, He's homotopy perturbation method (HPM) is implemented for finding the solitary-wave solutions of the regularized long-wave (RLW) and generalized modified Boussinesq (GMB) equations. We obtain numerical solutions of these equations for the initial conditions. We will show that the convergence of the HPM is faster than those obtained by the Adomian decomposition method (ADM). The obtained solutions, in comparison with the exact solutions admit a remarkable accuracy. A clear conclusion can be drawn from the numerical results that the HPM provides highly accurate numerical solutions for nonlinear differential equations.     

The element-free Galerkin method of numerically solving a regularized long-wave equation

The element-free Galerkin (EFG) method is used in this paper to find the numerical solution to a regularized long-wave (RLW) equation. The Galerkin weak form is adopted to obtain the discrete equations, and the essential boundary conditions are imposed by the penalty method. The effectiveness of the EFG method of solving the RLW equation is investigated by two numerical examples in this paper.     

Application of Improved (G'/G)–Expansion Method to Traveling Wave Solutions of Two Nonlinear Evolution Equations

In this work, the improved (G'/G)-expansion method is proposed for constructing more general exact solutions of nonlinear evolution equation with the aid of symbolic computation. In order to illustrate the validity of the method we choose the RLW equation and SRLW equation. As a result, many new and more general exact solutions have been obtained for the equations. We will compare our solutions with those gained by the other authors.     

New periodic wave solutions via Exp-function method

The Exp-function method with the aid of symbolic computational system is used to obtain the generalized solitary solutions and periodic solutions for nonlinear evolution equations arising in mathematical physics, namely, nonlinear partial differential (BBMB) equation, generalized RLW equation and generalized shallow water wave equation. It is shown that the Exp-function method, with the help of symbolic computation, provides a powerful mathematical tool for solving other nonlinear evolution equations arising in mathematical physics.     

Solutions to Class of Linear and Nonlinear Fractional Differential Equations

In this paper, the fractional auxiliary sub-equation expansion method is proposed to solve nonlinear fractional differential equations. To illustrate the effectiveness of the method, we discuss the space-time fractional Kd V equation, the space-time fractional RLW equation, the space-time fractional Boussinesq equation, and the(3+1)-spacetime fractional ZK equation. The solutions are expressed in terms of fractional hyperbolic and fractional trigonometric functions. These solutions are useful to understand the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time. The analytical solution of homogenous linear FDEs with constant coefficients are obtained by using the series and the Mittag–Leffler function methods. The obtained results recover the well-know solutions when α = 1.     

Numerical soliton-like solutions of the potential Kadomtsev–Petviashvili equation by the decomposition method

In this Letter we present an Adomian's decomposition method (shortly ADM) for obtaining the numerical soliton-like solutions of the potential Kadomtsev–Petviashvili (shortly PKP) equation. We will prove the convergence of the ADM. We obtain the exact and numerical solitary-wave solutions of the PKP equation for certain initial conditions. Then ADM yields the analytic approximate solution with fast convergence rate and high accuracy through previous works. The numerical solutions are compared with the known analytical solutions.     

Exact traveling wave solution of nonlinear variants of the RLW and the PHI-four equations

By means of the modified extended tanh-function (METF) method the multiple traveling wave solutions of some different kinds of nonlinear partial differential equations are presented and implemented in a computer algebraic system. The solutions for the nonlinear equations such as variants of the RLW and variant of the PHI-four equations are exactly obtained and so the efficiency of the method can be demonstrated.     

Simulations of solitary waves of RLW equation by exponential B-spline Galerkin method

In this paper,an approximate function for the Galerkin method is composed using the combination of the exponential B-spline functions.Regularized long wave equation(RLW)is integrated fully by using an exponential B-spline Galerkin method in space together with Crank–Nicolson method in time.Three numerical examples related to propagation of single solitary wave,interaction of two solitary waves and wave generation are employed to illustrate the accuracy and the efficiency of the method.Obtained results are compared with some early studies.     

On numerical solution of Burgers' equation by homotopy analysis method

In this Letter, we present the Homotopy Analysis Method (shortly HAM) for obtaining the numerical solution of the one-dimensional nonlinear Burgers' equation. The initial approximation can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions. Convergence of the solution and effects for the method is discussed. The comparison of the HAM results with the Homotopy Perturbation Method (HPM) and the results of [E.N. Aksan, Appl. Math. Comput. 174 (2006) 884; S. Kutluay, A. Esen, Int. J. Comput. Math. 81 (2004) 1433; S. Abbasbandy, M.T. Darvishi, Appl. Math. Comput. 163 (2005) 1265] are made. The results reveal that HAM is very simple and effective. The HAM contains the auxiliary parameter ?, which provides us with a simple way to adjust and control the convergence region of solution series. The numerical solutions are compared with the known analytical and some numerical solutions.     

跨音速定常势流计算中的解不唯一问题

本文采用近似因式分解(AF2)方法,数值求解二元跨音速小扰动速势方程。大量的数值实验表明:1.选取不同的松驰因子或加速收敛参数可能得到不唯一解。2.选取不同的初场,无论是采用Murman-Cole守恒或非守恒格式,还是采用Engquist-Osher格式,都可能得到不唯一解。     

New types of solitary wave solutions for the higher order nonlinear Schrodinger equation

We present new types of solitary wave solutions for the higher order nonlinear Schrodinger (HNLS) equation describing propagation of femtosecond light pulses in an optical fiber under certain parametric conditions. Unlike the reported solitary wave solutions of the HNLS equation, the novel ones can describe bright and dark solitary wave properties in the same expressions and their amplitude may approach nonzero when the time variable approaches infinity. In addition, such solutions cannot exist in the nonlinear Schrodinger equation. Furthermore, we investigate the stability of these solitary waves under some initial pertubations by employing the numerical simulation methods.     

正则长波方程的格子Boltzmann模型

提出了用于正则长波方程的5-Bit格子Boltzmann模型。应用Chapman-Enskog展开和多重尺度技术,通过选择平衡态分布函数的高阶矩,得到了时间尺度t₀上的守恒律,从而给出三阶精度的算法。模型中的参数通过稳定性分析给出。     

Second harmonic generation: the solution for an amplitude-modulated initial pulse

We address the initial value problem for one-dimensional second harmonic generation starting from a purely amplitude-modulated fundamental wave. A general method to solve the problem in terms of a Schrödinger equation is presented, in which the initial pulse-shape is taken as a potential. Several examples with the complete solution given in analytical form are discussed. A much broader class of solutions can be found with the help of a single numerical integration. In particular, solutions with incident pulses approximating a sech -shape have been obtained.     

Nonlinear Langevin Equation for a System of Coulomb Particles

The nonlinear Langevin equation for a system of Coulomb particles with random processes, which are functionals of the velocity distribution function of such particles, has been derived and analyzed. It is shown by direct numerical solutions that this equation correctly describes the collisional relaxation of such a system even in the case of anomalous deviation of the initial velocity distribution of particles from the equilibrium distribution. The equation can be conveniently used in the Monte Carlo methods and in “particle-in-cell” methods.     

The variational iteration method for solitary patterns solutions of gBBM equation

We consider solitary patterns solutions of generalized Benjamin–Bona–Mahony equations (shortly gBBM). The variational iteration method (shortly VIM) is applied for the numerical solution subject to appropriate initial condition. The numerical solutions of our model equation are calculated in the form of convergence power series with easily computable components. The VIM performs extremely well in terms of accuracy, efficiently, simplicity, stability and reliability.     

Numerical Solutions of Fractional Boussinesq Equation

Based upon the Adomian decomposition method, a scheme is developed to obtain numerical solutions of a fractional Boussinesq equation with initial condition, which is introduced by replacing some order time and space derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. So the traditional Adomian decomposition method for differential equations of integer order is directly extended to derive explicit and numerical solutions of the fractional differential equations. The solutions of our model equation are calculated in the form of convergent series with easily computable components.     

Evolution of two-dimensional lump nanosolitons for the Zakharov-Kuznetsov and electromigration equations

The evolution of lump solutions for the Zakharov-Kuznetsov equation and the surface electromigration equation, which describes mass transport along the surface of nanoconductors, is studied. Approximate equations are developed for these equations, these approximate equations including the important effect of the dispersive radiation shed by the lumps as they evolve. The approximate equations show that lump-like initial conditions evolve into lump soliton solutions for both the Zakharov-Kuznetsov equation and the surface electromigration equations. Solutions of the approximate equations, within their range of applicability, are found to be in good agreement with full numerical solutions of the governing equations. The asymptotic and numerical results predict that localized disturbances will always evolve into nanosolitons. Finally, it is found that dispersive radiation plays a more dominant role in the evolution of lumps for the electromigration equations than for the Zakharov-Kuznetsov equation.     

交通拥堵相变问题的同伦分析法

利用同伦分析法研究了一类基于洛伦兹系统的交通拥堵相变问题的非线性方程. 通过选取不同的初始解和不同的线性算子,分别得到了问题的近似解和相应的残留误差. 通过与前人结果的比较得出,在研究该类问题时同伦分析法优于微分变换法; 在应用同伦分析法时,要选取尽可能接近原算子线性部分作为线性算子. 本文还给出了一种新的初始解选取方法(双同伦分析法). 数值模拟的结果证实了理论分析的正确性. 关键词： 同伦分析法 交通拥堵 近似解 残留误差     

气体放电击穿过程的物理和数值研究

本文对低气压（１０＾－２Ｐａ）热阴极气体放电的击穿过程给出了物理描述和相应的双流体数学型，并发展了一种选择和调整未知初始条件的有效算法，可以通过伴随试射法得到对初始条件十分敏感的非线性两点边值常微分方程组的数值解，从而给出这类气体放电中击穿过程的定量描述。