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.

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.

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.

Exact solution of the Rijke tube equation

Exact solution of the Rijke tube equation is presented.

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.