Analysis of a Time Implicit Scheme for the Oseen Model Driven by Nonlinear Slip Boundary Conditions

This work is concerned with the time discrete analysis of the Oseen system of equations driven by nonlinear slip boundary conditions of friction type. We study the existence of solutions of the time discrete model and derive several a priori estimates needed to recover the solution of the continuous problem by means of weak compactness. Moreover, for the difference between the exact and approximate solutions, we obtain the rate of convergence of order one with respect to the time step without imposing extra regularity on the weak solution.     

Stability for Monostable Wave Fronts of Delayed Lattice Differential Equations

This paper is concerned with the stability of traveling wave fronts for delayed monostable lattice differential equations. We first investigate the existence non-existence and uniqueness of traveling wave fronts by using the technique of monotone iteration method and Ikehara theorem. Then we apply the contraction principle to obtain the existence, uniqueness, and positivity of solutions for the Cauchy problem. Next, we study the stability of a traveling wave front by using comparison theorems for the Cauchy problem and initial-boundary value problem of the lattice differential equations, respectively. We show that any solution of the Cauchy problem converges exponentially to a traveling wave front provided that the initial function is a perturbation of the traveling wave front, whose asymptotic behaviour at \(-\infty \) satisfying some restrictions. Our results can apply to many lattice differential equations, for examples, the delayed cellular neural networks model and discrete diffusive Nicholson’s blowflies equation.     

Steady Flows of Viscoelastic Fluids in Domains with Outlets to Infinity

The equations governing the motion of incompressible viscoelastic fluids of Rivlin—Ericksen and Oldroyd type are investigated in domains with cylindrical and paraboloidal outlets to infinity. For sufficiently small fluxes, prescribed in each outlet, existence and uniqueness of solutions are proven in weighted Hölder spaces. In domains with paraboloidal outlets the solution is obtained as a perturbation of the corresponding Navier—Stokes solution and in domains with cylindrical outlets as a perturbation of a flux carrier, constructed by joining together the exact solutions found in each outlet. These exact solutions are shown to be either rectilinear flows of Poiseuille type or flows composed of a rectilinear and of a transverse secondary component.     

Free convection from a vertical permeable circular cone with non-uniform surface heat flux

 In this paper, the problem of laminar free convection from a vertical permeable circular cone maintained with non-uniform surface heat flux is considered. The governing boundary layer equations are reduced non-similar boundary layer equations with surface heat flux proportional to x ⁿ (where x is the distance measured from the leading edge). The solutions of the reduced equations are obtained by using three distinct solution methodologies; namely, (i) perturbation solution for small transpiration parameter, ξ, (ii) asymptotic solution for large ξ, and (iii) the finite difference solutions for all ξ. The solutions are presented in terms of local skin-friction and local Nusselt number for smaller values of Prandtl number and heat flux gradient and are displayed in tabular form as well as graphically. Effects of pertinent parameters on velocity and temperature profiles are also shown graphically. Solutions obtained by finite difference method are also compared with the perturbation solutions for small and large ξ and found to be in excellent agreement. Received on 1 October 1999     

Invertibility of the Poiseuille Linearization for Stationary Two-Dimensional Channel Flows: Symmetric Case

We study the invertibility of the linearization of the stationary Navier-Stokes system near the Poiseuille solution with flux F \Phi in infinite or semi-infinite two dimensional channels and in the setting of Sobolev spaces accounting for a natural geometric symmetry. For the case of an infinite straight channel, we show that such a "Poiseuille linearization" is invertible for all the values of the flux F \Phi , whereas invertibility holds only off a (possibly empty) discrete set of singular values in the case of a semi-infinite straight channel. The same result is true for more general symmetric domains with a partly distorted geometry. This yields the existence of solutions of the Navier-Stokes equations approaching the Poiseuille solution at infinity for flux of arbitrarily large magnitude if the perturbation data are small enough in a suitable norm.     

Perturbation methods for nonlinear autonomous discrete-time dynamical systems

Two perturbation methods for nonlinear autonomous discrete-time dynamical systems are presented. They generalize the classical Lindstedt-Poincaré and multiple scale perturbation methods that are valid for continuous-time systems. The Lindstedt-Poincaré method allows determination of the periodic or almost-periodic orbits of the nonlinear system (limit cycles), while the multiple scale method also permits analysis of the transient state and the stability of the limit cycles. An application to the discrete Van der Pol equation is also presented, for which the asymptotic solution is shown to be in excellent agreement with the exact (numerical) solution. It is demonstrated that, when the sampling step tends to zero the asymptotic transient and steady-state discrete-time solutions correctly tend to the asymptotic continuous-time solutions.     

On the large amplitude oscillations of a thin elastic beam

This paper is concerned with the finite amplitude, free, planar oscillations of a thin elastic beam. By assuming the motion to be inextensional but at the same time recognizing the existence of a resultant normal force acting on each cross-section of the beam a system of governing equations is derived which is manageable but still meaningful. For the case of the simply-supported beam a finite difference, Galerkin, and (regular) perturbation solutions are explicitly obtained. The results are compared and discussed. In the course of obtaining the various solutions it is found that an additional simplification in the form of the governing equations is possible. This simplification turns out to be quite important from a general point of view of obtaining approximate analytical solutions.     

梁中非线性弯曲波传播特性的研究

研究了梁中的非线性弯曲波的传播特性，同时考虑了梁的大挠度引起的几何非线性效应和 梁的转动惯性导致的弥散效应，利用Hamilton变分法建立了梁中非线性弯曲波的波动方程. 对该方程进行了定性分析，在不同的条件下，该方程在相平面上存在同宿轨道或异宿轨道， 分别对应于方程的孤波解或冲击波解. 利用Jacobi椭圆函数展开法，对该非线性方程进行 求解，得到了非线性波动方程的准确周期解及相对应的孤波解和冲击波解，讨论了这些解存 在的必要条件，这与定性分析的结果完全相同. 利用约化摄动法从非线性弯曲波动方程中导 出了非线性Schr\"{o}dinger方程，从理论上证明了考虑梁的大挠度和转动惯性时梁中存在 包络孤立波.     

Periodic Traveling Waves for Diffusion Equations with Time Delayed and Non-local Responding Reaction

We develop a singular perturbation technique to study the existence of periodic traveling wave solutions with large wave speed for a class of reaction-diffusion equations with time delay and non-local response. Unlike the classical singular perturbation method, our approach is based on a transformation of the differential equations to integral equations in a Banach space that reduces the singular perturbation problem to a regular perturbation problem. The periodic traveling wave solutions then are obtained by the use of Liapunov-Schmidt method and a generalized implicit function theorem. The general result obtained has been applied to a non-local reaction-diffusion equation derived from an age-structured population model with a logistic type of birth function.     

HOMOTOPY PERTURBATION SOLUTION AND PERIODICITY ANALYSIS OF NONLINEAR VIBRATION OF THIN RECTANGULAR FUNCTIONALLY GRADED PLATES

In this paper nonlinear analysis of a thin rectangular functionally graded piate is formulated in terms of von-Karman＇s dynamic equations. Functionaily Graded Material （FGM） properties vary through the constant thickness of the plate at ambient temperature. By expansion of the solution as a series of mode functions, we reduce the governing equations of motion to a Duffing＇s equation. The homotopy perturbation solution of generated Duffing＇s equation is also obtained and compared with numerical solutions. The sufficient conditions for the existence of periodic oscillatory behavior of the plate are established by using Green＇s function and Schauder＇s fixed point theorem.     

Periodic and quasiperiodic motion for complex non-linear systems

A damped complex non-linear system corresponding to two coupled non-linear oscillators with a periodic damping force is investigated by an asymptotic perturbation method based on Fourier expansion and time rescaling. Four coupled equations for the amplitude and the phase of solutions are derived. Phase-locked solutions with period equal to the damping force period are possible only if the oscillators amplitudes are equal. On the contrary, if the oscillators amplitudes are different, periodic solutions exist only with a period different from the damping force period. These solutions are stable only for perturbations that conserve the phase difference and the square amplitude sum of the oscillators. Energy considerations are used in order to study existence and characteristics of quasiperiodic motion. We demonstrate that modulated motion can be also obtained for appropriate values of the detuning parameter and in this case an approximate analytic solution is easily constructed. If the detuning parameter decreases the modulation period increases and then diverges, an infinite-period bifurcation occurs and the resulting motion becomes unbounded. Analytic approximate solutions are checked by numerical integration.     

A brief review of the homotopy analysis method

本文简述同伦分析方法基本思想、最新理论进展及其在流体力学、固体力学、一般力学、量子力学、应用数学、金融等科学和工程领域的应用.同伦分析方法不依赖物理小参数, 适用范围更广,而且提供了一种简单的途径确保级数解收敛, 适用于强非线性问题.同伦分析方法已被成功应用于求解一些具有挑战性的力学问题,并获得一些全新的、 从未见报道的解. 这些成功的应用,证明了同伦分析方法的普遍有效性和原创性.     

Singular Perturbations of Nonlinear Degenerate Parabolic PDEs: a General Convergence Result

The main result of the paper is a general convergence theorem for the viscosity solutions of singular perturbation problems for fully nonlinear degenerate parabolic PDEs (partial differential equations) with highly oscillating initial data. It substantially generalizes some results obtained previously in [2]. Under the only assumptions that the Hamiltonian is ergodic and stabilizing in a suitable sense, the solutions are proved to converge in a relaxed sense to the solution of a limit Cauchy problem with appropriate effective Hamiltonian and initial data. In its formulation, our convergence result is analogous to the stability property of Barles and Perthame. It should thus reveal a useful tool for studying general singular perturbation problems by viscosity solutions techniques. A detailed exposition of ergodicity and stabilization is given, with many examples. Applications to homogenization and averaging are also discussed.     

同伦分析方法进展综述

本文简述同伦分析方法基本思想、最新理论进展及其在流体力学、固体力学、一般力学、量子力学、应用数学、金融等科学和工程领域的应用.同伦分析方法不依赖物理小参数, 适用范围更广,而且提供了一种简单的途径确保级数解收敛, 适用于强非线性问题.同伦分析方法已被成功应用于求解一些具有挑战性的力学问题,并获得一些全新的、 从未见报道的解. 这些成功的应用,证明了同伦分析方法的普遍有效性和原创性.      

Global existence of solutions to the periodic initial value problems for two-dimensional Newton-Boussinesq equations

A class of periodic initial value problems for two-dimensional Newton- Boussinesq equations are investigated in this paper. The Newton-Boussinesq equations are turned into the equivalent integral equations. With iteration methods, the local existence of the solutions is obtained. Using the method of a priori estimates, the global existence of the solution is proved.     

Singular perturbation of linear algebraic equations with application to stiff equations

In this paper the singular perturbation problem of linear algebraic equations with a small parameter is presented by an example in practice. The existence and uniqueness theorem of its solution is proved by the perturbation method and the estimation of error for its approximate solution is given. Finally, the example mentioned above explaining how to apply the theory to solve the stiff equations is shown.     

On Slightly Compressible Ideal Flow¶in the Half-Plane

We consider the Euler equations of barotropic inviscid compressible fluids in the half-plane. It is well known that, as the Mach number goes to zero, the compressible flows approximate the solution of the equations of motion of inviscid, incompressible fluids. In 2D (two dimensions) such limit solution exists on any arbitrary time interval, with no restriction on the size of the initial data. It is then natural to expect the same for the compressible solution, if the Mach number is sufficiently small. We decompose the solution as the sum of the irrotational part, the incompressible part and the remainder, which describes the interaction between the first two components. First we study the life span of smooth irrotational solutions, i.e., the largest time interval T(?) of existence of classical solutions, when the initial data are a small perturbation of size ? from a constant state. Related to this is a decay property for the irrotational part. Then, we study the interaction between the two components and show the existence on any arbitrary time interval, for any Mach number sufficiently small. This yields the existence of smooth compressible flow on any arbitrary time interval. For the proofs we use a combination of energy and decay estimates.     

Periodic boundary value problems for first-order integro-differential equations of mixed type

ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇｐｅｒｉｏｄｉｃｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍ(ＰＢＶＰ,ｆｏｒｓｈｏｒｔ)ｆｏｒｆｉｒｓｔ＿ｏｒｄｅｒｉｎｔｅｇｒｏ＿ｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｏｆｍｉｘｅｄｔｙｐｅｕ′=ｆ(ｔ,ｕ,Ｔ1ｕ,Ｔ2ｕ)　　(ａ.ｅ.ｔ∈Ｉ),(1)ｕ(0)=ｕ(2π),(2)ｗｈｅｒｅＩ=[0,2π],ｆｓａｔｉｓｆｉｅｓＣａｒａｔｈｅｏｄｏｒｙ’ｓｃｏｎｄｉｔｉｏｎｓ,Ｔ1ｉｓａＶｏｌｔｅｒｒａｉｎｔｅｇｒａｌｏｐｅｒａｔｏｒ,Ｔ2ｉｓａＦｒｅｄｈｏｌｍｉｎｔｅ…     

The difference methods for the solution of singular-perturbation for the elliptic-parabolic differential equation

In this paper it is discussed the difference method for the solution of singular perturbation problems for the elliptic equations, involving small parameter in the higher derivatives. As ε= 0 the original equations are degenerated into the parabolic equations.Authors constructed special difference schemes by means of the boundary layer properties of the solutions of these problems as well as investigated the convergence of this scheme and asymptotic behaviour of the solutions. Finally, a numerical example is given.     

MHD flow and heat transfer of a hybrid nanofluid past a permeable stretching/shrinking wedge

The steady flow and heat transfer of a hybrid nanofluid past a permeable stretching/shrinking wedge with magnetic field and radiation effects are studied. The governing equations of the hybrid nanofluid are converted to the similarity equations by techniques of the similarity transformation. The bvp4c function that is available in MATLAB software is utilized for solving the similarity equations numerically. The numerical results are obtained for selected different values of parameters. The results discover that two solutions exist, up to a certain value of the stretching/shrinking and suction strengths. The critical value in which the solution is in existence decreases as nanoparticle volume fractions for copper and wedge angle parameter increase. It is also found that the hybrid nanofluid enhances the heat transfer rate compared with the regular nanofluid. The reduction of the heat transfer rate is observed with the increase in radiation parameter. The temporal stability analysis is performed to analyze the stability of the dual solutions, and it is revealed that only one of them is stable and physically reliable.