Existence of Solutions for a Non-Autonomous Evolution Equations with Nonlocal Conditions

The existence of mild solutions for non-autonomous evolution equations with nonlocal conditions in Banach space is studied in this article. We obtained the existence of at least one mild solution to the evolution equations by using Krasnoselskii’s fixed point theorem as well as the theory of the evolution family. The interest of this paper is that any assumptions are not imposed on the nonlocal terms and Green’s functions and a new alternative method is applied to prove the existence of mild solutions. The results obtained in this paper may improve some related conclusions on this topic. An example is given as an application of the results.

Existence of Mild Solutions for a Class of Fractional Non-autonomous Evolution Equations with Delay

In this paper,we consider the existence and uniqueness of the mild solutions for a class of fractional non-autonomous evolution equations with delay and Caputo's fractional derivatives.By using the measure of noncompactness,β-resolvent family,fixed point theorems and Banach contraction mapping principle,we improve and generalizes some related results on this topic.At last,we give an example to illustrate the application of the main results of this paper.

ON EXPONENTIAL STABILITY OF NON-AUTONOMOUS STOCHASTIC SEMILINEAR EVOLUTION EQUATIONS

Sufficient conditions for the exponential stability of a class of nonlinear, non-autonomous stochastic differential equations in infinite dimensions are studied. The analysis consists of introducing a suitable approximating solution systems and usig a limiting argument to pass on stability of strong solutions to mild ones. Consequently, under these conditions the random attractors of given stochastic systems are reduced to zero with exponential decay. Lastly, two examples are investigated to illustrate the theory.

The Limit Distribution of Stochastic Evolution Equations Driven by-Stable Non-Gaussian Noise

We study the distribution limit of a class of stochastic evolution equation driven by an additive-stable Non-Gaussian process in the case of α∈(1,2).We prove that,under suitable conditions,the law of the solution converges weakly to the law of a stochastic evolution equation with an additive Gaussian process.

Anti-periodic solutions to a class of second-order evolution equations

In this paper we discuss the anti-periodic problem for a class of abstractnonlinear second-order evolution equations associated with maximal monotone operators in Hilbertspaces and give some new assumptions on operators. We establish the existence and uniqueness ofanti-periodic solutions, which improve andgeneralize the results that have been obtained. Finally weillustrate the abstract theory by discussing a simple example of an anti-periodic problem fornonlinear partial differential equations.

A New Generalization of Extended Tanh—Function Method for Solving Nonlinear Evolution Equations

Making use of a new generalized ans?tze and a proper transformation, we generalized the extended tanh-function method. Applying the generalized method with the aid of Maple, we consider some nonlinear evolution equations. As a result, we can successfully recover the previously known solitary wave solutions that had been found by the extended tanh-function method and other more sophisticated methods. More importantly, for some equations, we also obtain other new and more general solutions at the same time. The results include kink-profile solitary-wave solutions, bell-profile solitary-wave solutions, periodic wave solutions, rational solutions, singular solutions and new formal solutions.

A Comparative Analysis of Computational Stability for Linear and Non-Linear Evolution Equations

For several difference schemes of linear and non-linear evolution equations, taking the one-dimensional linear and non-linear advection equations as examples, a comparative analysis for computational stability is carried out and the relationship between non-linear computational stability, the construction of difference schemes, and the form of initial values is discussed. It is proved through comparative analysis and numerical experiment that the computational stability of the difference schemes of the non-linear evolution equation are absolutely different from that of the linear evolution equation.

Trial Equation Method to Nonlinear Evolution Equations with Rank Inhomogeneous： Mathematical Discussions and Its Applications

A trial equation method to nonlinear evolution equation with rank inhomogeneous is given. As appncations, the exact traveling wave solutions to some higher-order nonlinear equations such as generalized Boussinesq equation, generalized Pochhammer-Chree equation, KdV-Burgers equation, and KS equation and so on, are obtained. Among these, some results are new. The proposed method is based on the idea of reduction of the order of ODE. Some mathematical details of the proposed method are discussed.

TRAVELLING WAVE SOLUTIONS OF NONLINEAR EVOLUTION EQUATIONS BY USING SYMBOLIC COMPUTATION

A Riccati equation involving a parameter and symbolic computation are used to uniformly construct the different forms of travelling wave solutions for nonlinear evolution equations.It is shown that the sign of the parameter can be applied in judging the existence of various forms of travelling wave solutions.An efficiency of this method is demonstrated on some equations,which include Burgers Huxley equation,Caudrey Dodd Gibbon Kawada equation,generalized Benjamin Bona Mahony equation and generalized Fisher equation.

Applications of Computer Algebra in Solving Nonlinear Evolution Equations

With the use of computer algebra, the method that straightforwardly leads to travelling wave solutions is presented. The compound KdV-Burgers equation and KP-B equation are chosen to illustrate this approach. As a result, their abundant new soliton-like solutions and period form solutions are found.

The derivative-dependent functional variable separation for the evolution equations

This paper studies variable separation of the evolution equations via the generalized conditional symmetry. To illustrate, we classify the extended nonlinear wave equation utt = A（u, ux）uxx＋B（u, ux, ut） which admits the derivative- dependent functional separable solutions （DDFSSs）. We also extend the concept of the DDFSS to cover other variable separation approaches.

SMALL-STENCIL PAD SCHEMES TO SOLVE NONLINEAR EVOLUTION EQUATIONS

A set of small-stencil new Pade schemes with the same denominator are presented to solve high-order nonlinear evolution equations. Using this scheme, the fourth-order precision can not only be kept, but also the final three-diagonal discrete systems are solved by simple Doolittle methods, or ODE systems by Runge-Kutta technique. Numerical samples show that the schemes are very satisfactory. And the advantage of the schemes is very clear compared to other finite difference schemes.

Constructing infinite sequence exact solutions of nonlinear evolution equations

To construct the infinite sequence new exact solutions of nonlinear evolution equations and study the first kind of elliptic function, new solutions and the corresponding B^cklund transformation of the equation are presented. Based on this, the generalized pentavalent KdV equation and the breaking soliton equation are chosen as applicable examples and infinite sequence smooth soliton solutions, infinite sequence peak solitary wave solutions and infinite sequence compact soliton solutions are obtained with the help of symbolic computation system Mathematica. The method is of significance to search for infinite sequence new exact solutions to other nonlinear evolution equations.

TAYLOR EXPANSION METHOD FOR NONLINEAR EVOLUTION EQUATIONS

A new numerical method of integrating the nonlinear evolution equations, namely the Taylor expansion method, was presented. The standard Galerkin method can be viewed as the 0_th order Taylor expansion method; while the nonlinear Galerkin method can be viewed as the 1_st order modified Taylor expansion method. Moreover, the existence of the numerical solution and its convergence rate were proven. Finally, a concrete example, namely, the two_dimensional Navier_Stokes equations with a non slip boundary condition,was provided. The result is that the higher order Taylor expansion method is of the higher convergence rate under some assumptions about the regularity of the solution.

Extension of Variable Separable Solutions for Nonlinear Evolution Equations

We give the generalized definitions of variable separable solutions to nonlinear evolution equations, and characterize the relation between the functional separable solution and the derivative-dependent functional separable solution. The new definitions can unify various kinds of variable separable solutions appearing in references. As application, we classify the generalized nonlinear diffusion equations that admit special functional separable solutions and obtain some exact solutions to the resulting equations.

Extended Riccati Equation Rational Expansion Method and Its Application to Nonlinear Stochastic Evolution Equations

In this work, by means of a new more general ansatz and the symbolic computation system Maple, we extend the Riccati equation rational expansion method [Chaos, Solitons ＆ Fractals 25 （2005） 1019] to uniformly construct a series of stochastic nontravelling wave solutions for nonlinear stochastic evolution equation. To illustrate the effectiveness of our method, we take the stochastic mKdV equation as an example, and successfully construct some new and more general solutions including a series of rational formal nontraveling wave and coefficient functions＇ soliton-like solution.s and trigonometric-like function solutions. The method can also be applied to solve other nonlinear stochastic evolution equation or equations.

Application of radial basis functions to evolution equations arising in image segmentation

In this paper, radial basis functions are used to obtain the solution of evolution equations which appear in variational level set method based image segmentation. In this method, radial basis functions are used to interpolate the implicit level set function of the evolution equation with a high level of accuracy and smoothness. Then, the original initial value problem is discretized into an interpolation problem. Accordingly, the evolution equation is converted into a set of coupled ordinary differential equations, and a smooth evolution can be retained. Compared with finite difference scheme based level set approaches, the complex and costly re-initialization procedure is unnecessary. Numerical examples are also given to show the efficiency of the method.

The Relationship between Nonconservative Schemes and Initial Values of Nonlinear Evolution Equations

For the nonconservative schemes of the nonlinear evolution equations, taking the one-dimensional shallow water wave equation as an example, the necessary conditions of computational stability are given. Based on numerical tests, the relationship between the nonlinear computational stability and the construction of difference schemes, as well as the form of initial values, is further discussed. It is proved through both theoretical analysis and numerical tests that if the construction of difference schemes is definite, the computational stability of nonconservative schemes is decided by the form of initial values.

CONTROLLABILITY OF NEUTRAL STOCHASTIC EVOLUTION EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTION

In this paper,we investigate the controllability for neutral stochastic evolution equations driven by fractional Brownian motion with Hurst parameter H ∈（1/2,1） in a Hilbert space.We employ the α-norm in order to reflect the relationship between H and the fractional power α.Sufficient conditions are established by using stochastic analysis theory and operator theory.An example is provided to illustrate the effectiveness of the proposed result.

Symmetry Reduction of Initial-Value Problems for a Class of Third-order Evolution Equations

Symmetry reduction of a class of third-order evolution equations that admit certain generalized conditionalsymmetries (GCSs) is implemented.The reducibility of the initial-value problem for an evolution equation to a Cauchyproblem for a system of ordinary differential equations (ODEs) is characterized via the GCS and its Lie symmetry.Complete classification theorems are obtained and some examples are taken to show the main reduction procedure.