Some alternative results for the nonlinear viscoelasticity equations

In this article, the spatial behaviour of a class of nonlinear viscoelasticity equations is studied. We obtained some alternative results for the solutions under suitable conditions on the nonlinear terms. The main tool used is the weighted energy method. Our results can be viewed as a version of Saint-Venant's principle to the viscoelasticity equations.     

The “Gauss-Seidelization” of iterative methods for solving nonlinear equations in the complex plane

In this paper we introduce a process we have called “Gauss-Seidelization” for solving nonlinear equations. We have used this name because the process is inspired by the well-known Gauss-Seidel method to numerically solve a system of linear equations. Together with some convergence results, we present several numerical experiments in order to emphasize how the Gauss-Seidelization process influences on the dynamical behavior of an iterative method for solving nonlinear equations.     

时间测度链上具正负系数的二阶阻尼动力方程的振动准则

研究了时间测度链上一类具正负系数和阻尼项及非线性中立项的二阶变时滞非线性动力方程的振荡性.利用时间测度链上的有关理论及广义Riccati变换,结合大量不等式技巧,建立了该方程若干新的振动准则,这些准则不仅推广和改进了一些已知的结果,而且在时间测度链上统一了二阶非线性时滞阻尼微分方程和二阶非线性时滞阻尼差分方程的振动性质.     

Existence of analytic solutions to analytic nonlinear q-difference equations

In this paper, some analytic approaches are formulated for the existence of analytic solutions of analytic nonlinear difference equations. From the point of view of dynamical systems, analytic solutions of such kinds of equations can be generally expressed by formal power series of exponential variables, so we are interested in considering a difference equation as a q-difference equation via a suitable coordinate transformation. After stating analytic results for formal series solutions of nonlinear q-difference equations, we also derive some results for the existence of analytic solutions to autonomous rational difference equations.     

Approximations of the nonlinear Volterra's population model by an efficient numerical method

In this paper, we apply the new homotopy perturbation method to solve the Volterra's model for population growth of a species in a closed system. This technique is extended to give solution for nonlinear integro‐differential equation in which the integral term represents the total metabolism accumulated fromtime zero. The approximate analytical procedure only depends on two components. The newhomotopy perturbationmethodwas applied to nonlinear integro‐differential equations directly and by converting the problem into nonlinear ordinary differential equation. We also compare this method with some other numerical results and show that the present approach is less computational and is applicable for solving nonlinear integro‐differential equations and ordinary differential equations as well. Copyright © 2011 John Wiley & Sons, Ltd.     

Some sufficient conditions for the observability of nonlinear systems

Some sufficient conditions are presented for the observability of systems described by nonlinear, ordinary differential equations with linear observations. The point which the authors especially emphasize is the fact that the results obtained are global in some sense. In the beginning, the observability problem is reduced to the uniqueness problem of solutions of some nonlinear integral equations for general nonlinear systems with linear observations. Then, for some restricted systems, a simple sufficient condition is derived. The relation between global and local observability for nonlinear systems is also considered.The authors wish to thank their colleagues in Nagoya University.     

长水波近似方程组的新精确解

依据齐次平衡法的思想 ,首先提出了求非线性发展方程精确解的新思路 ,这种方法通过改变待定函数的次序 ,优势是使求解的复杂计算得到简化 .应用本文的思路 ,可得到某些非线性偏微分方程的新解 .其次我们给出了长水波近似方程组的一些新精确解 ,其中包括椭圆周期解 ,我们推广了有关长波近似方程的已有结果 .     

Oscillation of Third-order Delay Dynamic Equations on Time Scales

This paper is concerned with the oscillatory behavior of a class of third-order noonlinear variable delay neutral functional dynamic equations on time scale. By using the generalized Riccati transformation and inequality technique, we establish some new oscilla- tion criteria for the equations. Our results extend and improve some known results, but also unify the oscillation of third-order nonlinear variable delay functional differential equations and functional difference equations with a nonlinear neutral term. Some examples are given to illustrate the importance of our results.     

Exact solutions of some nonlinear systems of partial differential equations by using the first integral method

In recent years, many approaches have been utilized for finding the exact solutions of nonlinear systems of partial differential equations. In this paper, the first integral method introduced by Feng is adopted for solving some important nonlinear systems of partial differential equations, including, KdV, Kaup–Boussinesq and Wu–Zhang systems, analytically. By means of this method, some exact solutions for these systems of equations are formally obtained. The results obtained confirm that the proposed method is an efficient technique for analytic treatment of a wide variety of nonlinear systems of partial differential equations.     

On the stability and boundedness of solutions of nonlinear vector differential equations of third order

In this paper, by using Liapunov’s second method, we establish some new results for stability and boundedness of solutions of nonlinear vector differential equations of third order. By constructing a Liapunov function, sufficient conditions for stability and boundedness of solutions of equations considered are obtained. Concerning to the subject, some explanatory examples are also given. Our results improve and include a result existing in the literature.     

Modified Camassa–Holm and Degasperis–Procesi Equations Solved by Adomian’s Decomposition Method and Comparison with HPM and Exact Solutions

Many physical and scientific phenomena are modeled by nonlinear partial differential equations (NPDEs); it is difficult to handle nonlinear part of these equations. Recently some analytical methods are applied to solve such equations. In this work, modified Camassa–Holm and Degasperis–Procesi equation is studied. Adomian’s decomposition method (ADM) is applied to obtain solution of this equation. The results are compared to those of homotopy perturbation method (HPM) and exact solution. The study highlights the significant features of the employed method and its ability to handle nonlinear partial differential equations.     

Numerical solutions for some coupled nonlinear evolution equations by using spectral collocation method

In this paper, we introduce a spectral collocation method based on Lagrange polynomials for spatial derivatives to obtain numerical solutions for some coupled nonlinear evolution equations. The problem is reduced to a system of ordinary differential equations that are solved by the fourth order Runge–Kutta method. Numerical results of coupled Korteweg–de Vries (KdV) equations, coupled modified KdV equations, coupled KdV system and Boussinesq system are obtained. The present results are in good agreement with the exact solutions. Moreover, the method can be applied to a wide class of coupled nonlinear evolution equations.     

Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations

We prove some new Strichartz estimates for a class of dispersive equations with radial initial data. In particular, we obtain the full radial Strichartz estimates up to some endpoints for the Schrödinger equation. Using these estimates, we obtain some new results related to nonlinear problems, including small data scattering and large data LWP for the nonlinear Schrödinger and wave equations with radial critical initial data and the well-posedness theory for the fractional order Schrödinger equation in the radial case.     

A Chebyshev spectral collocation method for solving Burgers’-type equations

In this paper, we elaborated a spectral collocation method based on differentiated Chebyshev polynomials to obtain numerical solutions for some different kinds of nonlinear partial differential equations. The problem is reduced to a system of ordinary differential equations that are solved by Runge–Kutta method of order four. Numerical results for the nonlinear evolution equations such as 1D Burgers’, KdV–Burgers’, coupled Burgers’, 2D Burgers’ and system of 2D Burgers’ equations are obtained. The numerical results are found to be in good agreement with the exact solutions. Numerical computations for a wide range of values of Reynolds’ number, show that the present method offers better accuracy in comparison with other previous methods. Moreover the method can be applied to a wide class of nonlinear partial differential equations.     

An extended subequation rational expansion method with symbolic computation and solutions of the nonlinear Schrödinger equation model

To construct exact analytical solutions of nonlinear evolution equations, an extended subequation rational expansion method is presented and used to construct solutions of the nonlinear Schrödinger equation with varing dispersion, nonlinearity, and gain or absorption. As a result, many previous known results of the nonlinear Schrödinger equation can be recovered by means of some suitable selections of the arbitrary functions and arbitrary constants. With computer simulation, the properties of a new non-travelling wave soliton-like solutions with coefficient functions and some elliptic function solutions are shown by some figures.     

On a fixed point theorem in Banach algebras with applications

In this article it is shown that some of the hypotheses of a fixed point theorem of the present author [B.C. Dhage, On some variants of Schauder’s fixed point principle and applications to nonlinear integral equations, J. Math. Phys. Sci. 25 (1988) 603–611] involving two operators in a Banach algebra are redundant. Our claim is also illustrated with the applications to some nonlinear functional integral equations for proving the existence results.     

Travelling wave solutions of nonlinear evolution equations by using the first integral method

In this paper, we established travelling wave solutions for some (2 + 1)-dimensional nonlinear evolution equations. The first integral method was used to construct travelling wave solutions of nonlinear evolution equations. The travelling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. The first integral method presents a wider applicability for handling nonlinear wave equations.     

Exact solutions of the nonlinear Schrödinger equation by the first integral method

The first integral method is an efficient method for obtaining exact solutions of some nonlinear partial differential equations. This method can be applied to nonintegrable equations as well as to integrable ones. In this paper, the first integral method is used to construct exact solutions of the nonlinear Schrödinger equation.     

Solitary waves for the Klein-Gordon-Maxwell system with critical exponent

In this paper we study the existence of solutions for nonlinear Klein-Gordon-Maxwell equations coupled with Maxwell’s equations when the nonlinearity exhibits critical growth. We improve some previous existence results in Azzollini et al. (2009) [5], Carrião et al. (2009) [4] and Cassani (2004) [3].     

Global existence and nonexistence of the non-Newtonian polytropic filtration equations coupled with nonlinear boundary conditions

In this article, we deal with the global existence and nonexistence of solutions to the non-Newtonian polytropic filtration equations coupled with nonlinear boundary conditions. By constructing various kinds of sub- and super-solutions and using the basic properties of M-matrix, we give the necessary and sufficient conditions for global existence of nonnegative solutions. The critical curve of Fujita type is conjectured with the aid of some new results, which extend the recent results of Zheng, Song, and Jiang [Critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux, J. Math. Anal. Appl. 298 (2004), pp. 308–324], Zhou and Mu [Critical curve for a non-Newtonian polytropic filtration system coupled via nonlinear boundary flux, Nonlinear Anal. 68 (2008), pp. 1–11], and Zhou and Mu [Algebraic criteria for global existence or blow-up for a boundary coupled system of nonlinear diffusion equations, Appl. Anal. 86 (2007), pp. 1185–1197] to more general equations.