Element-free Galerkin （EFG） method for a kind of two-dimensional linear hyperbolic equation

The present paper deals with the numerical solution of a two-dimensional linear hyperbolic equation by using the element-free Galerkin (EFG) method which is based on the moving least-square approximation for the test and trial functions. A variational method is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. Compared with numerical methods based on mesh, the EFG method for hyperbolic problems needs only the scattered nodes instead of meshing the domain of the problem. It neither requires any element connectivity nor suffers much degradation in accuracy when nodal arrangements are very irregular. The effectiveness of the EFG method for two-dimensional hyperbolic problems is investigated by two numerical examples in this paper.     

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.     

Element-free Galerkin (EFG) method for analysis of the time-fractional partial differential equations

The present paper deals with the numerical solution of time-fractional partial differential equations using the element-free Galerkin (EFG) method, which is based on the moving least-square approximation. Compared with numerical methods based on meshes, the EFG method for time-fractional partial differential equations needs only scattered nodes instead of meshing the domain of the problem. It neither requires element connectivity nor suffers much degradation in accuracy when nodal arrangements are very irregular. In this method, the first-order time derivative is replaced by the Caputo fractional derivative of order α (0<α ≤1). The Galerkin weak form is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. Several numerical examples are presented and the results we obtained are in good agreement with the exact solutions.     

An element-free Galerkin (EFG) method for generalized Fisher equations (GFE)

A generalized Fisher equation(GFE) relates the time derivative of the average of the intrinsic rate of growth to its variance.The exact mathematical result of the GFE has been widely used in population dynamics and genetics,where it originated.Many researchers have studied the numerical solutions of the GFE,up to now.In this paper,we introduce an element-free Galerkin(EFG) method based on the moving least-square approximation to approximate positive solutions of the GFE from population dynamics.Compared with other numerical methods,the EFG method for the GFE needs only scattered nodes instead of meshing the domain of the problem.The Galerkin weak form is used to obtain the discrete equations,and the essential boundary conditions are enforced by the penalty method.In comparison with the traditional method,numerical solutions show that the new method has higher accuracy and better convergence.Several numerical examples are presented to demonstrate the effectiveness of the method.     

An improved element-free Galerkin method for solving generalized fifth-order Korteweg-de Vries equation

In this paper, an improved element-free Galerkin (IEFG) method is proposed to solve the generalized fifth-order Korteweg-de Vries (gfKdV) equation. When the traditional element-free Galerkin (EFG) method is used to solve such an equation, unstable or even wrong numerical solutions may be obtained due to the violation of the consistency conditions of the moving least-squares (MLS) shape functions. To solve this problem, the EFG method is improved by employing the improved moving least-squares (IMLS) approximation based on the shifted polynomial basis functions. The effectiveness of the IEFG method for the gfKdV equation is investigated by using some numerical examples. Meanwhile, the motion of single solitary wave and the interaction of two solitons are simulated using the IEFG method.     

A meshless method for the compound KdV-Burgers equation

The element-free Galerkin (EFG) method for numerically solving the compound Korteweg-de Vries-Burgers (KdVB) equation is discussed in this paper.The Galerkin weak form is used to obtain the discrete equation and the essential boundary conditions are enforced by the penalty method.The effectiveness of the EFG method of solving the compound Korteweg-de Vries-Burgers (KdVB) equation is illustrated by three numerical examples.     

An improved element-free Galerkin method for solving the generalized fifth-order Korteweg–de Vries equation

In this paper, an improved element-free Galerkin (IEFG) method is proposed to solve the generalized fifth-order Korteweg-de Vries (gfKdV) equation. When the traditional element-free Galerkin (EFG) method is used to solve such an equation, unstable or even wrong numerical solutions may be obtained due to the violation of the consistency conditions of the moving least-squares (MLS) shape functions. To solve this problem, the EFG method is improved by employing the improved moving least-squares (IMLS) approximation based on the shifted polynomial basis functions. The effectiveness of the IEFG method for the gfKdV equation is investigated by using some numerical examples. Meanwhile, the motion of single solitary wave and the interaction of two solitons are simulated using the IEFG method.     

Analysis of the generalized Camassa and Holm equation with the improved element-free Galerkin method

In this paper, we analyze the generalized Camassa and Holm （CH） equation by the improved element-free Galerkin （IEFG） method. By employing the improved moving least-square （IMLS） approximation, we derive the formulas for the generalized CH equation with the IEFG method. A variational method is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. Because there are fewer coefficients in the IMLS approximation than in the MLS approximation, and in the IEFG method, fewer nodes are selected in the entire domain than in the conventional EFG method, the IEFG method should result in a higher computing speed. The effectiveness of the IEFG method for the generalized CH equation is investigated by numerical examples in this paper.     

An element-free Galerkin(EFG) method for numerical solution of the coupled Schrdinger-KdV equations

The present paper deals with the numerical solution of the coupled Schrdinger-KdV equations using the elementfree Galerkin(EFG) method which is based on the moving least-square approximation.Instead of traditional mesh oriented methods such as the finite difference method(FDM) and the finite element method(FEM),this method needs only scattered nodes in the domain.For this scheme,a variational method is used to obtain discrete equations and the essential boundary conditions are enforced by the penalty method.In numerical experiments,the results are presented and compared with the findings of the finite element method,the radial basis functions method,and an analytical solution to confirm the good accuracy of the presented scheme.     

NEW EXPLICIT AND EXACT TRAVELLING WAVE SOLUTIONS FOR A COMPOUND KdV－BURGERS EQUATION

In this paper, new explicit and exact travelling wave solutions for a compound KdV－Burgers equation are obtained by using the hyperbola function method and the Wu elimination method, which include new solitary wave solutions and periodic solutions. Particularly important cases of the equation, such as the compound KdV, mKdV－Burgers and mKdV equations can be solved by this method. The method can also solve other nonlinear partial differential equations.     

A deep learning method for solving third-order nonlinear evolution equations

It has still been difficult to solve nonlinear evolution equations analytically. In this paper, we present a deep learning method for recovering the intrinsic nonlinear dynamics from spatiotemporal data directly. Specifically, the model uses a deep neural network constrained with given governing equations to try to learn all optimal parameters. In particular, numerical experiments on several third-order nonlinear evolution equations, including the Korteweg–de Vries (KdV) equation, modified KdV equation, KdV–Burgers equation and Sharma–Tasso–Olver equation, demonstrate that the presented method is able to uncover the solitons and their interaction behaviors fairly well.     

截断展开方法和广义变系数KdV方程新的精确类孤子解

利用特殊的截断展开方法求出了广义变系数KdV方程新的类孤子解.这种方法的基本思想是假定形式解具有截断展开形式,以致可把广义变系数KdV方程转化为一组待定函数的代数方程组,进而给出待定函数容易积分的常微分方程.利用例子证明了这种方法是十分有效的. 关键词： 截断展开法 变系数 KdV方程 孤波解     

New explicit and exact solutions for a compound KdV-Burgers equation

In this paper, new explicit and exact solutions for a compound KdV-Burgers equation are obtained using the hyperbolic function method and the Wu elimination method, which include new solitary wave solutions and periodic solutions. Particularly important cases of the equation, such as the compound KdV, mKdV-Burgers and mKdV equations can be solved by this method. The method can also be applied to solve other nonlinear partial differential equation and equations.     

On the theory of acoustic solitons in a liquid with distributed gas bubbles

A close relation is established between numerical solutions to two systems of equations, viz., the two-level nonlinear wave dynamic model of a liquid with gas bubbles and the Korteweg-de Vries (KdV) equation. This model is used for deriving the KdV equation in the long-wave approximation for any dependent variable of the gas-liquid mixture. The KdV equations derived earlier using radically different approximations are particular cases of our equations.     

A discrete KdV equation hierarchy: continuous limit,diverse exact solutions and their asymptotic state analysis

In this paper, a discrete KdV equation that is related to the famous continuous KdV equation is studied. First, an integrable discrete KdV hierarchy is constructed, from which several new discrete KdV equations are obtained. Second, we correspond the first several discrete equations of this hierarchy to the continuous KdV equation through the continuous limit. Third, the generalized (m, 2N − m)-fold Darboux transformation of the discrete KdV equation is established based on its known Lax pair. Finally, the diverse exact solutions including soliton solutions, rational solutions and mixed solutions on non-zero seed background are obtained by applying the resulting Darboux transformation, and their asymptotic states and physical properties such as amplitude, velocity, phase and energy are analyzed. At the same time, some soliton solutions are numerically simulated to show their dynamic behaviors. The properties and results obtained in this paper may be helpful to understand some physical phenomena described by KdV equations.     

New approximate solution for time-fractional coupled KdV equations by generalised differential transform method

In this paper,the generalised two-dimensional differential transform method (DTM) of solving the time-fractional coupled KdV equations is proposed.The fractional derivative is described in the Caputo sense.The presented method is a numerical method based on the generalised Taylor series expansion which constructs an analytical solution in the form of a polynomial.An illustrative example shows that the generalised two-dimensional DTM is effective for the coupled equations.     

Lie Symmetry and Nonlinear Instability in Computation of KdV Solitons

The soliton calculation method put forward by Zabusky and Kruskal has played an important role in the development of soliton theory, however numerous numerical results show that even though the parameters satisfy the linear stability condition, nonlinear instability will also occur. We notice an exception in the numerical calculation of soliton, gain the linear stability condition of the second order Leap-frog scheme constructed by Zabusky and Kruskal, and then draw the perturbed equation with the finite difference method. Also, we solve the symmetry group of the KdV equation with the knowledge of the invariance of Lie symmetry group and then discuss whether the perturbed equation and the conservation law keep the corresponding symmetry. The conservation law of KdV equation satisfies the scaling transformation, while the perturbed equation does not satisfy the Galilean invariance condition and the scaling invariance condition. It is demonstrated that the numerical simulation destroy some physical characteristics of the original KdV equation. The nonlinear instability in the calculation of solitons is related to the breaking of symmetry.     

KdV Equation with Self-consistent Sources in Non-uniform Media

Two non-isospectral KdV equations with self-consistent sources are derived. Gauge transformation between the first non-isospectral KdV equation with self-consistent sources （corresponding to λt = -2aA） and its isospectral counterpart is given, from which exact solutions for the first non-isospectral KdV equation with self-consistent sources is easily listed. Besides, the soliton solutions for the two equations are obtained by means of Hirota＇s method and Wronskian technique, respectively. Meanwhile, the dynamical properties for these solutions are investigated.     

Multi-symplectic method for generalized fifth-order KdV equation

This paper considers the multi-symplectic formulations of the generalized fifth-order KdV equation in Hamiltonian space. Recurring to the midpoint rule, it presents an implicit multi-symplectic scheme with discrete multi-symplectic conservation law to solve the partial differential equations which are derived from the generalized fifth-order KdV equation numerically. The results of the numerical experiments show that this multi-symplectic algorithm is good in accuracy and its long-time numerical behaviour is also perfect.     

Newton-conjugate-gradient methods for solitary wave computations

In this paper, the Newton-conjugate-gradient methods are developed for solitary wave computations. These methods are based on Newton iterations, coupled with conjugate-gradient iterations to solve the resulting linear Newton-correction equation. When the linearization operator is self-adjoint, the preconditioned conjugate-gradient method is proposed to solve this linear equation. If the linearization operator is non-self-adjoint, the preconditioned biconjugate-gradient method is proposed to solve the linear equation. The resulting methods are applied to compute both the ground states and excited states in a large number of physical systems such as the two-dimensional NLS equations with and without periodic potentials, the fifth-order KdV equation, and the fifth-order KP equation. Numerical results show that these proposed methods are faster than the other leading numerical methods, often by orders of magnitude. In addition, these methods are very robust and always converge in all the examples being tested. Furthermore, they are very easy to implement. It is also shown that the nonlinear conjugate gradient methods are not robust and inferior to the proposed methods.