An Exact Solution to the Combined KS and EdV Equation

The exact solution for the combined KS and KdV equation is obtained via introducing a simple and effective nonlinear transformations.This method is very concise and primary and can be applied to other unintegrable nonlinear evolution equations.It is common knowledge that the Korteweg de Vries(KdV) equation [1] (1)has been proposed as model equation for the weakly nonlinear long waves which occur in many different physical systems; the Kuramoto-Sivashinsky (KS) equationis one of the simplest nonliaear partial differential equations that exhibit Chaotic behavior frequently encounted in the study of continous media [2-4] . Many interesting mathematical and physical properties of eqs. (1) and (2) have been studied widely. But, in several problems where a lonq wavelength oscilatory instability is found, the noulineai evolution of the perturbations near rriticality is governed by the dispersion modified Kuramoto-Sivashi nsky equation(3)ft is clear that this equation is a combination of the KdV and

Solving (2+1)-dimensional sine-Poisson equation by a modified variable separated ordinary differential equation method

By introducing a more general auxiliary ordinary differential equation （ODE）, a modified variable separated ordinary differential equation method is presented for solving the （2 ＋ 1）-dimensional sine-Poisson equation. As a result, many explicit and exact solutions of the （2 ＋ 1）-dimensional sine-Poisson equation are derived in a simple manner by this technique.

New Generalized Transformation Method and Its Application in Higher-Dimensional Soliton Equation

A new generalized transformation method is differential equation. As an application of the method, we presented to find more exact solutions of nonlinear partial choose the （3＋1）-dimensional breaking soliton equation to illustrate the method. As a result many types of explicit and exact traveling wave solutions, which contain solitary wave solutions, trigonometric function solutions, Jacobian elliptic function solutions, and rational solutions, are obtained. The new method can be extended to other nonlinear partial differential equations in mathematical physics.

Solving the Burgers-Huxley Equation by G'/G Expansion Method

By introducing and extending the G'/G expansion method with the aid of computer algebraic system “Mathematics”, the exact general solutions were obtained for the Burgers-Huxley equation and special form. Final results were represented in hyperbolic function, trigonometric function and rational function with arbitrary parameters.

The quaternion beam model for hard-magnetic flexible cantilevers

The recently developed hard-magnetic soft(HMS)materials manufactured by embedding high-coercivity micro-particles into soft matrices have received considerable attention from researchers in diverse fields,e.g.,soft robotics,flexible electronics,and biomedicine.Theoretical investigations on large deformations of HMS structures are significant foundations of their applications.This work is devoted to developing a powerful theoretical tool for modeling and computing the complicated nonplanar deformations of flexible beams.A so-called quaternion beam model is proposed to break the singularity limitation of the existing geometrically exact(GE)beam model.The singularity-free governing equations for the three-dimensional(3D)large deformations of an HMS beam are first derived,and then solved with the Galerkin discretization method and the trustregion-dogleg iterative algorithm.The correctness of this new model and the utilized algorithms is verified by comparing the present results with the previous ones.The superiority of a quaternion beam model in calculating the complicated large deformations of a flexible beam is shown through several benchmark examples.It is found that the purpose of the HMS beam deformation is to eliminate the direction deviation between the residual magnetization and the applied magnetic field.The proposed new model and the revealed mechanism are supposed to be useful for guiding the engineering applications of flexible structures.

Exact Solution to the Equation of Motion of a Charged Particle Driven by an Electrostatic Wave

An exact solution is derived for the equation of motion of a charged particle driven by an electrostatic wave.The explicit expression of particle velocity is obtained,and the trapping condition of the charged particle in the electrostatic wave is also derived exactly.The interaction between the charged particle and the electrostatic wave is discussed,which is a supplement to the existing textbook of plasma physics.The results are of interest to particle accelerators,microwave tubes,and basic plasma processes.

Data-driven Power Flow Method Based on Exact Linear Regression Equations

Power flow(PF)is one of the most important calculations in power systems.The widely-used PF methods are the Newton-Raphson PF(NRPF)method and the fast-decoupled PF(FDPF)method.In smart grids,power generations and loads become intermittent and much more uncertain,and the topology also changes more frequently,which may result in significant state shifts and further make NRPF or FDPF difficult to converge.To address this problem,we propose a data-driven PF(DDPF)method based on historical/simulated data that includes an offline learning stage and an online computing stage.In the offline learning stage,a learning model is constructed based on the proposed exact linear regression equations,and then the proposed learning model is solved by the ridge regression(RR)method to suppress the effect of data collinearity.In online computing stage,the nonlinear iterative calculation is not needed.Simulation results demonstrate that the proposed DDPF method has no convergence problem and has much higher calculation efficiency than NRPF or FDPF while ensuring similar calculation accuracy.

Implicit Function Theorem and Its Application to a Ulam's Problem for Exact Differential Equations

We prove a global version of the implicit function theorem under a special condition and apply this result to the proof of a modified Hyers-Ulam-Rassias stability of exact differential equations of the form, g（x, y） ＋ h（x, y）y＇ =0.

Symmetries,Integrability and Exact Solutions to the (2+1)-Dimensional Benney Types of Equations

This paper is concerned with the (2+1)-dimensional Benney types of equations. By the complete Lie group classification method, all of the point symmetries of the Benney types of equations are obtained, and the integrable condition of the equation is given. Then, the symmetry reductions and exact solutions to the (2+1)-dimensional nonlinear wave equations are presented. Especially, the shock wave solutions of the Benney equations are investigated by the symmetry reduction and trial function method.

Exact solution of the Rijke tube equation

Exact solution of the Rijke tube equation is presented.

Ricci Collineations of Static Space Times with Maximal Symmetric Transverse Spaces

A complete classification of static space times with maximal symmetric transverse spaces is provided, according to their Ricci collineations. The classification is made when one component of Ricci collineation vector field V is non-zero （cases 1 - 4）, two components of V are non-zero （cases 5 - 10）, and three components of V are non-zero （cases 11 - 14）, respectlvily. Both non-degenerate （detRab ≠ 0） as well as the degenerate （det Rab = 0） cases are discussed and some new metrics are found.

Ricci Symmetries of Static Space-Times with Maximal Symmetric Transverse Spaces

In the paper [M. Akbar and R.G. Cai, Commun. Theor. Phys. 45 （2006） 95], a complete classification is provided with at least one component of the vector field V is zero. In this paper, I consider the vector field V with all non-zero components and the static space times with maximal symmetric transverse spaces are classified according to their Ricci collineations. These are investigated for non-degenerate Ricci tensor det R ≠0. It turns out that the only collineations admitted by these spaces can be ten, seven, six or four. It also covers our previous results as a spacial case. Some new metrics admitting proper Ricci collineations are also investigated.

Exact Solutions of Atmospheric(2+1)-Dimensional Nonlinear Incompressible Non-hydrostatic Boussinesq Equations

Exact solutions of the atmospheric(2+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq(INHB) equations are researched by Combining function expansion and symmetry method. By function expansion, several expansion coefficient equations are derived. Symmetries and similarity solutions are researched in order to obtain exact solutions of the INHB equations. Three types of symmetry reduction equations and similarity solutions for the expansion coefficient equations are proposed. Non-traveling wave solutions for the INHB equations are obtained by symmetries of the expansion coefficient equations. Making traveling wave transformations on expansion coefficient equations, we demonstrate some traveling wave solutions of the INHB equations. The evolutions on the wind velocities, temperature perturbation and pressure perturbation are demonstrated by figures, which demonstrate the periodic evolutions with time and space.

Lump Solution of (2+1)-Dimensional Boussinesq Equation

A class of lump solutions of(2+1)-dimensional Boussinesq equation are obtained with the help of Maple by using Hirota bilinear method.Some contour plots with different determinant values are sequentially made to show that the corresponding lump solution tends to zero when the determinant approaches zero.The particular lump solutions with specific values of the involved parameters are plotted,as illustrative examples.

Solving Space-Time Fractional Differential Equations by Using Modified Simple Equation Method

In this article,we establish new and more general traveling wave solutions of space-time fractional Klein–Gordon equation with quadratic nonlinearity and the space-time fractional breaking soliton equations using the modified simple equation method.The proposed method is so powerful and effective to solve nonlinear space-time fractional differential equations by with modified Riemann–Liouville derivative.

Form invariance of schema and exact schema theorem

One of the most important research questions in GAs is the explanation of the evolutionary process of CAs as a mathematical object. In this paper, we use matrix linear transformations to do it, first. This new method makes the study on mechanism of CAs simpler. We obtain the conditions under which the operators of crossover and mutation are commutative operators of CAs. We also give an exact schema equation on the basis of the concept of schema space. The result is similar to Stephens and Waelbroeck's work, but they have novel meanings and a larger degree of coarse graining.

Study on a General Hopf Hierarchy

By using a general symmetry theory related to invariant functions,strong symmetry operators and hereditary operators,we find a general integrable hopf heirarchy with infinitely many general symmetries and Lax pairs.For the first order Hopf equation,there exist infinitely many symmetries which can be expressed by means of an arbitrary function in arbitrary dimensions.The general solution of the first order Hopf equation is obtained via hodograph transformation.For the second order Hopf equation,the Hopf-diffusion equation,there are five sets of infinitely many symmetries.Especially,there exist a set of primary branch symmetry with which contains an arbitrary solution of the usual linear diffusion equation.Some special implicit exact group invariant solutions of the Hopf-diffusion equation are also given.