Fundamental flows with nonlinear slip conditions: exact solutions

In this article, we use nonlinear slip conditions to investigate three fundamental flows. Constitutive equations of the third grade fluid give rise to nonlinear equations. Exact analytic solutions of the nonlinear equations with nonlinear boundary conditions are developed. Numerical values between the dimensionless third grade and slip parameters are tabulated. Graphs are plotted and discussed.     

Fundamental flows with nonlinear slip conditions: exact solutions

In this article, we use nonlinear slip conditions to investigate three fundamental flows. Constitutive equations of the third grade fluid give rise to nonlinear equations. Exact analytic solutions of the nonlinear equations with nonlinear boundary conditions are developed. Numerical values between the dimensionless third grade and slip parameters are tabulated. Graphs are plotted and discussed.     

The exact solutions and the relevant constraint conditions for two nonlinear Schrödinger equations with variable coefficients

Two nonlinear Schrödinger equations with variable coefficients are researched, and the various exact solutions (including the bright and dark solitary waves) of the nonlinear Schrödinger equations are obtained with the aid of a subsidiary elliptic-like equation (sub-ODEs for short), at the same time, the constraint conditions which the coefficients of the nonlinear Schrödinger equations with variable coefficients satisfy are presented. The exact solutions and the constraint conditions are helpful in the application of the nonlinear Schrödinger equations with variable coefficients studied in this paper.     

Schwarz problem for higher‐order complex partial differential equations in the upper half plane

Linear and nonlinear elliptic complex partial differential equations of higher‐order are considered under Schwarz conditions in the upper‐half plane. Firstly, using the integral representations for the solutions of the inhomogeneous polyanalytic equation with Schwarz conditions, a class of integral operators is introduced together with some of their properties. Then, these operators are used to transform the problem for linear equations into singular integral equations. In the case of nonlinear equations such a transformation yields a system of integro‐differential equations. Existence of the solutions of the relevant boundary value problems for linear and nonlinear equations are discussed via Fredholm theory and fixed point theorems, respectively.     

圆形三向网架非线性动力稳定性分析

用拟板法将网架简化为平板,给出表层应变与中面位移的非线性关系．根据薄板的非线性动力学理论,建立了在直角坐标系中三向网架的非线性动力学方程,又将此方程转化为极坐标系轴对称非线性动力学方程．在周边固定条件下,引入异于等厚度板的无量纲量,对基本方程无量纲化．利用Galerkin法得到一个三次非线性振动方程,在无外激励情况下,讨论了稳定性与分岔问题．在外激励情况下,用Melnikov方法研究了圆形三向网架可能发生的混沌运动．通过数字仿真绘出了发生混沌的相平面图．     

Observability of nonlinear systems

New necessary and sufficient conditions are presented for the observability of systems described by nonlinear ordinary differential equations with nonlinear observations. The conditions are based on extension of the necessary and sufficient conditions for observability of time-varying linear systems to the linearized trajectory of the nonlinear system. The result is that the local observability of any initial condition can be readily determined, and the observability of the entire initial domain can be computed. The observability of constant parameters appearing in the differential equations is also considered. Examples are presented to illustrate the theory.This research was supported by the National Science Foundation, Grant No. NSF GK-10136.     

A method of solving a nonlinear boundary value problem with a parameter for a loaded differential equation

A nonlinear loaded differential equation with a parameter on a finite interval is studied. The interval is partitioned by the load points, at which the values of the solution to the equation are set as additional parameters. A nonlinear boundary value problem for the considered equation is reduced to a nonlinear multipoint boundary value problem for the system of nonlinear ordinary differential equations with parameters. For fixed parameters, we obtain the Cauchy problems for ordinary differential equations on the subintervals. Substituting the values of the solutions to these problems into the boundary condition and continuity conditions at the partition points, we compose a system of nonlinear algebraic equations in parameters. A method of solving the boundary value problem with a parameter is proposed. The method is based on finding the solution to the system of nonlinear algebraic equations composed.     

On the best parametrization

The numerical solution to a system of nonlinear algebraic or transcendental equations with several parameters is examined in the framework of the parametric continuation method. Necessary and sufficient conditions are proved for choosing the best parameters, which provide the best condition number for the system of linear continuation equations. Such parameters have to be sought in the subspace tangent to the solution space of the system of nonlinear equations. This subspace is obtained if the original system of nonlinear equations is solved at the various parameter values from a given set. The parametric approximation of curves and surfaces is considered.     

On nonlinear neutron transport equations

The purpose of this paper is to investigate a class of time-dependent neutron transport equations in which the total and differential scattering cross sections are nonlinear functions of neutron density function. Sufficient conditions on the nonlinear cross sections are given to insure the existence, uniqueness and asymptotic stability of a solution in one, two, or three-dimensional space domains under various boundary and initial conditions. The approach to the problem is based on abstract analysis on nonlinear evolution equations which are closely related to nonlinear semigroup theory.     

Solvability of nonlinear integral equations and surjectivity of nonlinear Markov operators

In the present paper, we consider integral equations, which are associated with nonlinear Markov operators acting on an infinite-dimensional space. The solvability of these equations is examined by investigating nonlinear Markov operators. Notions of orthogonal preserving and surjective nonlinear Markov operators defined on infinite dimension are introduced, and their relations are studied, which will be used to prove the main results. We show that orthogonal preserving nonlinear Markov operators are not necessarily satisfied surjective property (unlike finite case). Thus, sufficient conditions for the operators to be surjective are described. Using these notions and results, we prove the solvability of Hammerstein equations in terms of surjective nonlinear Markov operators.     

Homotopy perturbation method for fractional KdV-Burgers equation

In the paper, we extend the homotopy perturbation method to solve nonlinear fractional partial differential equations. The time- and space-fractional KdV-Burgers equations with initial conditions are chosen to illustrate our method. As a result, we successfully obtain some available approximate solutions of them. The results reveal that the proposed method is very effective and simple for obtaining approximate solutions of nonlinear fractional partial differential equations.     

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.     

抽象空间中非线性算子方程变号解的存在性研究

利用Banach空间中的锥理论和不动点定理讨论了非线性算子方程变号解的存在性,给出了E_u_0空间下非线性算子方程变号解至少有一个变号解、一个正解和一个负解的条件,并讨论了仅通过一个上解条件得出非线性算子方程变号解的存在性定理.     

An effective treatment of nonlinear differential equations with linear boundary conditions using the homotopy analysis method

In this paper, we provide an effective technique to treat nonlinear differential equations with linear boundary conditions that are reduced from a nonlinear problem describing the steady-state boundary-layer flow of a micropolar fluid near the forward stagnation point of a two-dimensional plane surface. The analytical approximations with high accuracy are obtained using the homotopy analysis method, which agree well with the numerical results. This indicates the validity and great potential of the proposed method for solving nonlinear differential equations with linear boundary conditions.     

Robin-type boundary control problems for the nonlinear Boussinesq type equations

In this paper, we study a linear and a nonlinear boundary control problems arising from viscous flows. The equations are of nonlinear Navier-Stokes type for the velocity and pressure, of transport-diffusion type for the temperature and the salinity. The essential difficulties are due to the nonlinear nature of a part of the boundary conditions and to the nature of the equations: time-dependent, coupled and nonlinear. The existence and the conditions of the uniqueness of the solution, for the variational problem, are studied. The control is of linear or nonlinear Robin-type and acts on a part of the boundary during a time T. The cost function measures the distance between the observed and the computed vorticity. The existence of an optimal control in the admissible set of states and controls is proved. A first order necessary conditions of optimality are obtained.     

Fixed point theorems for a class of nonlinear operators in Banach spaces and applications

In this paper, we study a class of nonlinear operator equations with more extensive conditions in ordered Banach spaces. By using the cone theory and Banach contraction mapping principle, the existence and uniqueness of solutions for such equations are investigated without demanding the existence of upper and lower solutions and compactness and continuity conditions. The results in this paper are applied to a class of abstract semilinear evolution equations with noncompact semigroup in Banach spaces and the initial value problems for nonlinear second-order integro-differential equations of mixed type in Banach spaces. The results obtained here improve and generalize many known results.     

Nonautonomous first integrals and general solutions of the KdV‐Burgers and mKdV‐Burgers equations with the source

The method for constructing first integrals and general solutions of nonlinear ordinary differential equations is presented. The method is based on index accounting of the Fuchs indices, which appeared during the Painlevé test of a nonlinear differential equation. The Fuchs indices indicate us the leading members of the first integrals for the origin differential equation. Taking into account the values of the Fuchs indices, we can construct the auxiliary equation, which allows to look for the first integrals of nonlinear differential equations. The method is used to obtain the first integrals and general solutions of the KdV‐Burgers and the mKdV‐Burgers equations with a source. The nonautonomous first integrals in the polynomials form are found. The general solutions of these nonlinear differential equations under at some additional conditions on the parameters of differential equations are also obtained. Illustrations of some solutions of the KdV‐Burgers and the mKdV‐Burgers are given.     

The Existence and Behavior of Global Solutions to a Mixed Problem with Acoustic Transmission Conditions for Nonlinear Hyperbolic Equations with Nonlinear Dissipation

A mixed problem with acoustic transmission conditions for nonlinear hyperbolic equations with nonlinear dissipation is considered. The existence, uniqueness, and exponential decay of global solutions to this problem with focusing nonlinear sources are proved Additionally, the existence of global solutions and the solution blow-up in a finite time are proved for the case of defocusing nonlinear sources.     

非线性中立抛物型时滞偏微分方程解的振动性质

讨论一类多滞量非线性中立抛物型时滞偏微分方程解的振动性质 ,获得了其一切解振动的充分条件 ;指出了与普通抛物型偏微分方程质的差异 .     

Asymptotic behavior of solutions of second-order nonlinear impulsive differential equations

In this work, the asymptotic behavior of all solutions of second-order nonlinear ordinary differential equations with impulses is investigated. By impulsive differential inequality and Riccati transformation, sufficient conditions of asymptotic behavior of all solutions of second-order nonlinear ordinary differential equations with impulses are obtained. An example is also inserted to illustrate the impulsive effect.