The extended Riccati equation method for travelling wave solutions of ZK equation

In this article, the extended Riccati equation method is applied to seeking more general exact travelling wave solutions of the ZK equation. The traveling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. When the parameters are taken as special values, the solitary wave solutions are obtained from the hyperbolic function solutions. Similarly, the periodic wave solutions are also obtained from the trigonometric function solutions. The approach developed in this paper is effective and it may also be used for solving many other nonlinear evolution equations in mathematical physics.     

Generalization of the simplest equation method for nonlinear non-autonomous differential equations

It is known that the simplest equation method is applied for finding exact solutions of autonomous nonlinear differential equations. In this paper we extend this method for finding exact solutions of non-autonomous nonlinear differential equations (DEs). We applied the generalized approach to look for exact special solutions of three Painlevé equations. As ODE of lower order than Painlevé equations the Riccati equation is taken. The obtained exact special solutions are expressed in terms of the special functions defined by linear ODEs of the second order.     

Condition numbers of algebraic Riccati equations in the Frobenius norm

Consider the continuous-time algebraic Riccati equation (CARE) and the discrete-time algebraic Riccati equation (DARE) which arise in linear control and system theory. It is known that appropriate assumptions on the coefficient matrices guarantee the existence and uniqueness of Hermitian positive semidefinite stabilizing solutions. In this note, we apply the theory of condition developed by Rice to define condition numbers of the CARE and DARE in the Frobenius norm, and derive explicit expressions of the condition numbers in a uniform manner. Both the complex case and real case are considered, and connections to certain existing condition numbers of the CARE and DARE are discussed.     

复微分方程组的代数解

本文研究了一类复微分方程组的代数解的存在问题.利用最大模原理和Nevanlinna值分布理论,得到了一个结论,推广和改进了一些文献的结果,例子表明结论精确.     

Stochastic independent modal-space control of distributed-parameter systems

A method is presented for solving time-varying independent modal-space Kalman filter equations in terms of 2×2 transition matrices, rather than in terms of the more commonly used 4×4 transition matrix solution technique. The basic method consists of replacing the well-known product form solution for the differential matrix Riccati equation with an alternate solution form which consists of a steady-state plus transient term.     

Some predictor–corrector‐type iterative schemes for solving nonsymmetric algebraic Riccati equations arising in transport theory

It is as well known that nonsymmetric algebraic Riccati equations arising in transport theory can be translated to vector equations. In this paper, we propose six predictor–corrector‐type iterative schemes to solve the vector equations. And we give the convergence of these schemes. Unlike the previous work, we prove that all of them converge to the minimal positive solution of the vector equations by the initial vector (e,e), where e = (1,1, ? ,1)^T. Moreover, we prove that all the sequences generated by the iterative schemes are strictly and monotonically increasing and bounded above. In addition, some numerical results are also reported in the paper, which confirm the good theoretical properties of our approach. Copyright © 2014 John Wiley & Sons, Ltd.     

Exponential energy decay of solutions for a system of viscoelastic wave equations of Kirchhoff type with strong damping

The initial boundary value problem for a system of viscoelastic wave equations of Kirchhoff type with strong damping is considered. We prove that, under suitable assumptions on relaxation functions and certain initial data, the decay rate of the solutions energy is exponential.     

Initial boundary value problems for nonlinear dispersive wave equations

In this paper we study initial value boundary problems of two types of nonlinear dispersive wave equations on the half-line and on a finite interval subject to homogeneous Dirichlet boundary conditions. We first prove local well-posedness of the rod equation and of the b-equation for general initial data. Furthermore, we are able to specify conditions on the initial data which on the one hand guarantee global existence and on the other hand produce solutions with a finite life span. In the case of finite time singularities we are able to describe the precise blow-up scenario of breaking waves. Our approach is based on sharp extension results for functions on the half-line or on a finite interval and several symmetry preserving properties of the equations under discussion.     

Convergence of the solution of a nonsymmetric matrix Riccati differential equation to its stable equilibrium solution

We consider the initial value problem for a nonsymmetric matrix Riccati differential equation, where the four coefficient matrices form an M-matrix. We show that for a wide range of initial values the Riccati differential equation has a global solution X(t) on [0,∞) and X(t) converges to the stable equilibrium solution as t goes to infinity.     

Exp-function method for Riccati equation and new exact solutions with two arbitrary functions of (2 + 1)-dimensional Konopelchenko-Dubrovsky equations

In this paper, new exact solutions with two arbitrary functions of the (2 + 1)-dimensional Konopelchenko-Dubrovsky equations are obtained by means of the Riccati equation and its generalized solitary wave solutions constructed by the Exp-function method. It is shown that the Exp-function method provides us with a straightforward and important mathematical tool for solving nonlinear evolution equations in mathematical physics.     

声波方程吸收边界条件的稳定性分析

引言对于无界区域中波动现象的数值模拟，必需引进人工边界将计算限制在一个有界区域上．为了确定解，需要在人工边界上加适当的边界条件．对于声波和弹性波方程，这样的一组人工边界条件，也叫吸收边界条件，在［1，2］中被系统地构造出来．对于声波方程，这些吸收边界条件恰好是单程波方程的近似．如山中所指出，减少边界反射，便于在计算中应用和稳定性是构造吸收边界条件的三点关键．Ellgqllist和Maid。用模态分析方法15］证明，带有［IJ中构造的吸收边界条件的波动方程初边值问题是适定的，并且估计了人工边界所产生的误差．对于更广…     

Blow-up of solutions to semilinear wave equations with variable coefficients and boundary

This paper is devoted to studying initial-boundary value problems for semilinear wave equations and derivative semilinear wave equations with variable coefficients on exterior domain with subcritical exponents in n space dimensions. We will establish blow-up results for the initial-boundary value problems. It is proved that there can be no global solutions no matter how small the initial data are, and also we give the life span estimate of solutions for the problems.     

Some results on difference Riccati equations

Some properties of solutions for the difference Riccati equations are obtained. The existence and forms of rational solutions, and the Borel exceptional value, zeros, poles and fixed points of transcendental solutions are researched.     

线性等式约束系统广义Riccati代数方程的求解*

本文基于定常离散LQ控制问题的动力学方程、价值泛函及系统的约束方程,根据极大值原理,给出了线性等式约束系统下的广义Riccati方程,进而对上述方程进行了深入的探讨,并给出了相应的数值例题。     

三维区域中的Navier-Stokes方程的初边值问题

本文在初边值适当小的假设下，建立了任意三维区域中Navier－Stokes方程初边值问题整体强解的存在性定理．     

Exact slow-fast decomposition of the singularly perturbed matrix differential Riccati equation

In this paper the Hamiltonian matrix formulation of the Riccati equation is used to derive the reduced-order pure-slow and pure-fast matrix differential Riccati equations of singularly perturbed systems. These pure-slow and pure-fast matrix differential Riccati equations are obtained by decoupling the singularly perturbed matrix differential Riccati equation of dimension n₁+n₂ into the pure-slow regular matrix differential Riccati equation of dimension n₁ and the pure-fast stiff matrix differential Riccati equation of dimension n₂. A formula is derived that produces the solution of the original singularly perturbed matrix differential Riccati equation in terms of solutions of the pure-slow and pure-fast reduced-order matrix differential Riccati equations and solutions of two reduced-order initial value problems. In addition to its theoretical importance, the main result of this paper can also be used to implement optimal filtering and control schemes for singularly perturbed linear time-invariant systems independently in pure-slow and pure-fast time scales.     

Iterative Methods for a Linearly Perturbed Algebraic Matrix Riccati Equation Arising in Stochastic Control

We start with a discussion of coupled algebraic Riccati equations arising in the study of linear-quadratic optimal control problems for Markov jump linear systems. Under suitable assumptions, this system of equations has a unique positive semidefinite solution, which is the solution of practical interest. The coupled equations can be rewritten as a single linearly perturbed matrix Riccati equation with special structures. We study the linearly perturbed Riccati equation in a more general setting and obtain a class of iterative methods from different splittings of a positive operator involved in the Riccati equation. We prove some special properties of the sequences generated by these methods and determine and compare the convergence rates of these methods. Our results are then applied to the coupled Riccati equations of jump linear systems. We obtain linear convergence of the Lyapunov iteration and the modified Lyapunov iteration, and confirm that the modified Lyapunov iteration indeed has faster convergence than the original Lyapunov iteration.     

Numerical Solution of Boundary Value Problems for Selfadjoint Differential Equations of 2nth Order

The paper is devoted to solving boundary value problems for self-adjoint linear differential equations of 2nth order in the case that the corresponding differential operator is self-adjoint and positive semidefinite. The method proposed consists in transforming the original problem to solving several initial value problems for certain systems of first order ODEs. Even if this approach may be used for quite general linear boundary value problems, the new algorithms described here exploit the special properties of the boundary value problems treated in the paper. As a consequence, we obtain algorithms that are much more effective than similar ones used in the general case. Moreover, it is shown that the algorithms studied here are numerically stable.     

Existence and stability of standing waves for nonlinear fractional Schrödinger equations with Hartree type nonlinearity

In this paper, we consider the nonlinear fractional Schrödinger equations with Hartree type nonlinearity. We obtain the existence of standing waves by studying the related constrained minimization problems via applying the concentration-compactness principle. By symmetric decreasing rearrangements, we also show that the standing waves, up to translations and phases, are positive symmetric nonincreasing functions. Moreover, we prove that the set of minimizers is a stable set for the initial value problem of the equations, that is, a solution whose initial data is near the set will remain near it for all time.