Analytic solution of self-similar strong shocks in an exponential medium for zero temperature gradient

The self-similar one-dimensional propagation of a strong shock wave in a medium with an exponentially decreasing density is studied. The flow behind the shock is assumed to be spatially isothermal rather than adiabatic to simulate the conditions of large radiative transfer behind the shock. The solution in closed form is obtained. An analytic expression for the similarity exponent has also been obtained.     

Stability of travelling wave solutions to a semilinear hyperbolic system with relaxation

We study a semilinear hyperbolic system with relaxation and investigate the asymptotic stability of travelling wave solutions with shock profile. It is shown that the travelling wave solution is asymptotically stable, provided the initial disturbance is suitably small. Moreover, we show that the time convergence rate is polynomially (resp. exponentially) fast as t→∞ if the initial disturbance decays polynomially (resp. exponentially) for x→∞. Our proofs are based on the space–time weighted energy method. Copyright © 2008 John Wiley & Sons, Ltd.     

On exponential ill-conditioning and internal layer behavior

In the limit of small difTusivity the internal layer behavior associated with the initial-boundary value problems for a viscous shock equation and a reaction diffusion equation is analyzed.As a result of the occurrence of exponentially small eigenvalues for the linearized problems the steady state internal layer solutions are shown to very sensitive to small perturbations.For the time dependent problems the small eigenvalues give rise to exponentially slow internal layer motion.Accurate numerical methods are used to compute the steady state internal layer solutions and the slow internal layer motion.The relationship between the viscous shock problem and some exponentially ill-conditioned linear singular perturbation problems is discussed.     

Global Perturbation of the Riemann Problem for the System of Compressible Flow Through Porous Media

In this paper we consider the unperturbatcd and perturbated Riemann problem for the damped quasiliuear hyperbolic system {v_t - u_x = 0 u_t + p(v)_x = -αu, α > 0, p'(v} < 0 with initial structure of two rarefaction waves or one rarefaction wave plus one shock wave. Under certain restrictions, it admits a unique global discontinuous solution in a class of piecewise continuous and piecewise smooth functions and keeps the initial structure. Moreover, the shock strength is found decaying exponentially due to damping for the later case.     

一类非线性边值问题解的激波性态

利用激波理论和匹配原理, 在适当的条件下讨论了一类非线性方程的激波问题, 得出了其激波解及其激波位置的表示式.将其结果用于一类可压缩流体流动模型, 较简捷地得到了该模型解的激波性态.     

On the exponentially slow motion of a viscous shock

Using formal asymptotic methods, we study the internal layer behavior associated with the following viscous shock problem in the limit ε → 0: The convex nonlinearity f(u) satisfies f(α) = f(–α). For the steady problem, we show that the method of matched asymptotic expansions fails to uniquely determine the location of the equilibrium shock layer solution. This indeterminacy, resulting from neglecting certain exponentially small effects, is eliminated by using the projection method, which exploits certain properties of the spectrum associated with the linearized operator. For the time dependent problem, we show that the viscous shock, which is formed from initial data, drifts towards the equilibrium solution on an exponentially long time interval of the order O(e^C/ε), for some C > 0. This exponentially slow behavior is analyzed by deriving an equation of motion for the location of the viscous shock. For Burgers equation (f(u) = u²/2), the results give an analytical characterization of the slow shock layer motion observed numerically in Kreiss and Kreiss; see [11]. We also show that the shock layer behavior is very sensitive to small changes in the boundary operator. In addition, using a WKB-type method, the slow viscous shock motion is studied numerically for small ε, the results comparing favorably with corresponding analytical results. Finally, we relate the slow viscous shock motion to similar slow internal layer motion for the Allen-Cahn equation.     

Multiple solutions for the laminar flow in a porous pipe with suction at slowly expanding or contracting wall

In this paper, the asymptotic solution for the similarity equation of the laminar flow in a porous pipe with suction at expanding and contracting wall has been obtained using the singular perturbation method. However, this solution neglects exponentially small terms in the matching process. To take into account these exponentially small terms, a method involving the inclusion of exponentially small terms in a perturbation series was used to find the two solutions analytically. The series involving the exponentially small terms and expansion ratio predicts dual solutions. Furthermore, the result indicates that the expansion ratio has much important influence on the solutions. When the expansion ratio is zero, it is a special case that Terrill has discussed.     

Borg’s criterion for almost periodic differential equations

Borg’s criterion is used to prove the existence of an exponentially asymptotically stable periodic orbit of an autonomous differential equation and to determine its domain of attraction. In this article, this method is generalized to almost periodic differential equations. Both sufficient and necessary conditions are obtained for the existence of an exponentially stable almost periodic solution. The condition uses a Riemannian metric, and an example for the explicit construction of such a metric is presented.     

Study on a Cross Diffusion Parabolic System

This paper considers a kind of strongly coupled cross diffusion parabolic system,which can be usedas the multi-dimensional Lyumkis energy transport model in semiconductor science.The global existence andlarge time behavior are obtained for smooth solution to the initial boundary value problem.When the initialdata are a small perturbation of an isothermal stationary solution,the smooth solution of the problem under theinsulating boundary condition,converges to that stationary solution exponentially fast as time goes to infinity.     

Anti-periodic solution for delayed cellular neural networks with impulsive effects

In this paper, we discuss anti-periodic solution for delayed cellular neural networks with impulsive effects. By means of contraction mapping principle and Krasnoselski’s fixed point theorem, we obtain the existence of anti-periodic solution for neural networks. By establishing a new impulsive differential inequality, using Lyapunov functions and inequality techniques, some new results for exponential stability of anti-periodic solution are obtained. Meanwhile, an example and numerical simulations are given to show that impulses may change the exponentially stable behavior of anti-periodic solution.     

Exponential stability for stochastic reaction-diffusion BAM neural networks with time-varying and distributed delays

In this paper we study the stability for a class of stochastic bidirectional associative memory (BAM) neural networks with reaction-diffusion and mixed delays. The mixed delays considered in this paper are time-varying and distributed delays. Based on a new Lyapunov-Krasovskii functional and the Poincaré inequality as well as stochastic analysis theory, a set of novel sufficient conditions are obtained to guarantee the stochastically exponential stability of the trivial solution or zero solution. The obtained results show that the reaction-diffusion term does contribute to the exponentially stabilization of the considered system. Moreover, two numerical examples are given to show the effectiveness of the theoretical results.     

Almost periodic solution of Cohen–Grossberg neural networks with bounded and unbounded delays

In this paper, a class of Cohen–Grossberg neural networks with bounded and unbounded delays is discussed. Several new sufficient conditions are obtained ensuring the existence and exponential stability of the almost periodic solution for this model based on inequality analysis technique and combing the exponential dichotomy with fixed point theorem. The obtained results are helpful to design globally exponentially stable almost periodic oscillatory neural networks. Two numerical examples and simulations are also given to show the feasibility of our results.     

Steady supersonic flow past an almost straight wedge with large vertex angle

This paper studies the problem on the steady supersonic flow at the constant speed past an almost straight wedge with a piecewise smooth boundary. It is well known that if each vertex angle of the straight wedge is less than an extreme angle determined by the shock polar, the shock wave is attached to the tip of the wedge and constant states on both side of the shock are supersonic. This paper is devoted to generalizing this result. Under the hypotheses that each vertex angle is less than the extreme angle and the total variation of tangent angle along each edge is sufficiently small, a sequence of approximate solutions constructed by a modified Glimm scheme is proved to be convergent to a global weak solution of the steady problem. A sequence of the corresponding approximate leading shock fronts issuing from the tip is shown to be convergent to the leading shock front of the obtained solution. The regularity of the leading shock front is established and the asymptotic behaviour of the obtained solution at infinity is also studied.     

Group theoretic method for analyzing interaction of a discontinuity wave with a strong shock in an ideal gas

A group theoretic method is used to obtain an exact particular solution to the system of partial differential equations, describing one-dimensional unsteady planar, cylindrically and spherically symmetric motions in an ideal gas, involving shock waves. It is interesting to remark that the exact solution obtained here is precisely the blast wave solution obtained earlier using a different method of approach. Further, the evolution of a discontinuity wave and its interaction with the strong shock are studied within the state characterized by the exact particular solution. The properties of reflected and transmitted waves and the jump in the shock acceleration are completely characterized, and certain observations are noted in respect to their contrasting behavior.     

一类非线性奇摄动方程的激波问题

利用奇摄动理论和匹配原理，讨论了一类非线性奇摄动方程的激波问题．首先，构造了原问题的外部解和内层解．其次，研究了当激波在区间的边界附近和内部的激波解．最后，得出了与边界条件相对应的激波位置及解的表达式．     

常规故障和定期维修的冗余系统的指数稳定性

利用C_0-半群理论证明了具有常规故障和定期维修的冗余系统非负解的存在唯一性,并研究了相应算子的谱特征,通过分析本质谱界经过紧扰动后的变化,得到系统动态解以指数形式收敛于稳态解.     

Evolution of solitons of nonlinear Schrödinger equation with variable parameters

The nonlinear Schrödinger equation with variable parameters is solved by means of variational technique. A set of evolution equations for the solitary-wave solution is derived. The propagation properties of the solitons in an adiabatic amplification system and in a dispersion-decreasing fiber are analyzed. An explicit analytical approximate soliton solution in the exponentially dispersion-decreasing fiber is obtained using the derived dynamical equations.     

Hypersonic flow past a sphere

The effects of dissociation or ionization of air on the analytical solution for hypersonic flow past a sphere are considered here, under certain assumptions. It has been assumed that the shock wave is in the shape of a sphere, that the density ratio across the shock is constant, that the flow behind the shock is at constant density and that dissociation or ionization only occurs behind the shock wave. Thus the effects of the compressibility of the air, variation of density ratio along the shock, and the department of the shock shape from being circular are not taken into account. Here the velocity, pressure, temperature, pressure coefficient and vorticity, etc., at any point between the shock and the surface of the sphere in the presence of dissociation or ionization are obtained. In addition, shock detachment distance, drag coefficient, stagnation point velocity gradient and sonic points on the shock and the surface have also been obtained. The results have been compared with the corresponding results obtained in the case when dissociation or ionization does not occur behind the shock.     

On the global dynamic behaviour for a generalized haematopoiesis model with almost periodic coefficients and oscillating circulation loss rate

A generalized non‐linear nonautonomous model for the haematopoiesis (cell production) with several delays and an oscillating circulation loss rate is studied. We prove a fixed point theorem in abstract cones, from which different results on existence and uniqueness of positive almost periodic solutions are deduced. Moreover, some criteria are given to guarantee that the obtained positive almost periodic solution is globally exponentially stable.     

一类奇摄动非线性边值问题激波解的Sinc-Galerkin方法

讨论了一类非线性奇摄动方程的激波问题.利用Sinc-Galerkin方法,构造出边值问题的激波解,并由Newton法得到其近似解.