STABILITY OF SYSTEM OF TWO-DIMENSIONAL NON-HYDROSTATIC REVOLVING FLUIDS

Applying the theory of stratification, it is proved that the system of the two-dimensional non-hydrostatic revolving fluids is unstable in the two-order continuous function class. The construction of solution space is given and the solution approach is offered. The sufficient and necessary conditions of the existence of formal solutions are expressed for some typical initial and boundary value problems and the calculating formulae to formal solutions are presented in detail.     

Analytical study of solitons to nonlinear time fractional parabolic equations

The some of the well-known nonlinear time fractional parabolic partial differential equations is studied in this paper. The fractional complex transform and the first integral method are employed to construct one-soliton solutions of these equations. The power of this manageable method is confirmed. The obtained solutions include solitary wave solutions, periodic wave solutions and combined formal solutions.     

ON THE WELL POSEDNESS OF INITIAL VALUE PROBLEM FOR EULER EQUATIONS OF INCOMPRESSIBLE INVISCID FLUID (Ⅰ)

1　IntroductionandSignsForthestudyofwellposednessontheinitialorboundaryproblemsabouttheEulerequations,therearemanyimportantresultsgivenbydifferentmethodsindifferentfunctionclasses[1- 3].Andthereislessdiscussiononitsillposedness[4 ,5 ].Inthedifferentialfunctionclasses,BaouondiandGoulaouicprovedthattheCauchyproblemonthecompactmanifoldofEulerequationshasuniquesolutionwellexpectaconstant[4 ].InthebooktitledSolutionsAnalytiquesdeQuelquesEquationsauxDerivesPartiellesenMecaniquedesFluides,thest…     

Singular solutions and Green's method in micropolar theory of elasticity

The matrices of fundamental solutions are constructed for a concentrated force as well as a concentrated couple varying harmonically in time and acting in an unbounded micropolar elastic continuum. These solutions are then used to obtain solutions for some other loading singularities. Integral representations, for the displacement and the rotation vectors are obtained by making use of the basic singular solutions. The formal solutions to two fundamental boundary value problems are expressed in terms of integrals which include given surface and body data and Green's functions.     

Schauder estimates for parabolic equation of bi-harmonic type

Global Schauder estimates for the initial-value parabolic problem of the bi-harmonic type are proved,and the existence and uniqueness of the solutions in the suitable space are obtained.Similarly to the second-order case,first a formal expression of solutions by the Fourier transform is obtained,and then the regularity,uniqueness and existence of solutions using the potential theory and approximation argument are got. out approach is simple and straightforward.     

Asymptotic theory of initial value problems for nonlinear perturbed Klein-Gordon equations

Introduction InthispaperasymptotictheoryofthefollowinginitialvalueproblemforanonlinearKlein Gordonequationisconsidered.tt-Δ =εF(t,x,,ε),t>0,x∈R2,(0,x,ε)=0(x,ε),t(0,x,ε)=1(x,ε),x∈R2,(1)where(t,x)isarealvaluedunknownfunction,Δ=2i     

An asymptotic analysis of composite beams with kinematically corrected end effects

A finite element-based beam analysis for anisotropic beams with arbitrary-shaped cross-sections is developed with the aid of a formal asymptotic expansion method. From the equilibrium equations of the linear three-dimensional (3D) elasticity, a set of the microscopic 2D and macroscopic 1D equations are systematically derived by introducing the virtual work concept. Displacements at each order are split into two parts, such as fundamental and warping solutions. First we seek the warping solutions via the microscopic 2D cross-sectional analyses that will be smeared into the macroscopic 1D beam equations. The variations of fundamental solutions enable us to formulate the macroscopic 1D beam problems. By introducing the orthogonality of asymptotic displacements to six beam fundamental solutions, the end effects of a clamped boundary are kinematically corrected without applying the sophisticated decay analysis method. The boundary conditions obtained herein are applied to composite beams with solid and thin-walled cross-sections in order to demonstrate the efficiency and accuracy of the formal asymptotic method-based beam analysis (FAMBA) presented in this paper. The numerical results are compared to those reported in literature as well as 3D FEM solutions.     

A COMPLETE EXPRESSION OF THE ASYMPTOTIC SOLUTION OF DIFFERENTIAL EQUATION WITH A N－TH ORDER TURNING POINT

ＡＣＯＭＰＬＥＴＥＥＸＰＲＥＳＳＩＯＮＯＦＴＨＥＡＳＹＭＰＴＯＴＩＣＳＯＬＵＴＩＯＮＯＦＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＷＩＴＨＡＮ－ＴＨＯＲＤＥＲＴＵＲＮＩＮＧＰＯＩＮＴＺｈａｎｇＪｕ－ｌｉｎｇ（张居铃）（ＳｉｃｈｕａｎＩｎｓｔｉｔｕｔｅｏ...     

A COMPLETE EXPRESSION OF THE ASYMPTOTIC SOLUTION OF DIFFERENTIAL EQUATION WITH A∩-TH ORDER TURNING POINT

A second order linear ordinary differential equation has been studied, and the complete expresson, of the formal uniformly valid asymptotic solutions to the equation near turning point is obtained by using extended Airy function.     

On the slow evolution of quasi-stationary shock waves

We consider the Burgers equation with a nonhomogeneous drift term, in the limit of small dissipation. A finite-dimensional manifold of slowly varying shock-like solutions is described, and a formal derivation of the dynamics on this manifold, in terms of a system of ordinary differential equations, is given. We also discuss the interpretation of the stationary solutions to the Burgers equation imbedded on the slow manifold.     

ON THE WELL POSEDNESS OF INITIAL VALUE PROBLEM FOR EULER EQUATIONS OF INCOMPRESSIBLE INVISCID FLUID(Ⅰ)

The ill posed initial value problem of the Euler equations and the formal solvability of ill posed problem based on stratification theory are discussed. For some ill posed initial value problems, the existence conditions of formal solutions and the methods of how to construct a formal solution are given. Finally, an example is given to discuss the ill posedness of the initial value problem on hyper plane {t=0} in R⁴, and explain that the problem has more than one solution. Foundation item: the National Natural Science Foundation of China (19971054) Biography: Shen Zhen (1977−)     

Torsional-mode transients in variable-thickness shells of revolution

For thin shells of revolution the existence of torsional-vibration modes, uncoupled from bending and extensional modes, has been established[1]. Here a linear second-order differential equation for the uncoupled torsional stress mode is obtained and its solution for impact loading of shells is sought. The mode-superposition method which utilizes the natural modes of vibration predicted by elementary theory, is, in general, not satisfactory for sharp impact loading as many modes are often required for convergence. Hence we employ two novel techniques for solving the impact problems. Firstly a formal asymptotic procedure, based on extensions to geometrical optics, is employed to generate asymptotic wavefront expansions. Rigorous justifications for this formal technique are provided in an appendix. Secondly a transform technique whereby solutions are sought in terms of Bessel functions is discussed and applied to particular impact loading problems. The Bessel function solutions found here can be used to determine the natural frequencies of the shells. Shells both finite and infinite in extent are discussed and reflections at a stress-free end are examined.     

SINGULAR PERTURBATION OF INITIAL BOUNDARY VALUE FROBLEMS OF REACTION DIFFUSlON EQUATION WITH DELAY

ＳＩＮＧＵＬＡＲＰＥＲＴＵＲＢＡＴＩＯＮＯＦＩＮＩＴＩＡＬＢＯＵＮＤＡＲＹＶＡＬＵＥＦＲＯＢＬＥＭＳＯＦＲＥＡＣＴＩＯＮＤＩＦＦＵＳｌＯＮＥＱＵＡＴＩＯＮＷＩＴＨＤＥＬＡＹＺｈａｎｇＸｉａｎｇ（张祥）（ＡｎｈｕｉＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ）Ｗｕ...     

ON THE WELL POSEDNESS OF INITIAL VALUE PROBLEM FOR EULER EQUATIONS OF INCOMPRESSIBLE INVISCID FLUID (Ⅱ)

IntroductionForthestudyofwellposednessontheinitialorboundaryproblemsabouttheEulerequations,therearemanyimportantresultsgivenbydifferentmethodsindifferentfunctionclasses[1]- [4 ].Thepaper (Ⅰ )oftheauthordiscussedtheill_posednessoftheCauchyproblemswhichsatisfythesameconditionsoftheinitialvalueproblematthehypersurfacet=t( 0 ) +g(x) ,g(x) ≠constant,g(0 ) =0 R4,andgavetwodifferentformalsolutions[5 ].Italsogavetheformulasforcomputation .Inthepresentpaper,twokindsofillposednessoftheEulerequation…     

On a class of scattering equations

In this paper a class of conservative wave equations treated by Broer and Peletier is considered when a single localised scatterer is present. A criterion for stability in the energy norm is derived. The formal solution of the scattering problem is given. The relation between the location of poles of the scattering matrix, the occurrence of standing wave solutions and the energy criterion is discussed.     

Asymptotic properties of solutions of the Cauchy problem for a degenerate singularly perturbed system of differential equations in the case of the multiple spectrum of the principal operator

We prove the asymptotic character of a solution of the Cauchy problem for a singularly perturbed linear system of differential equations with degenerate matrix of the coefficients of derivatives in the case where the limit matrix pencil is regular and has multiple “finite” and “infinite” elementary divisors. We establish conditions under which the constructed formal solutions are asymptotic expansions of the corresponding exact solutions. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 2, pp. 247–257, April–June, 2007.     

Classification of the Spatial Equilibria of the Clamped Elastica: Symmetries and Zoology of Solutions

We investigate the configurations of twisted elastic rods under applied end loads and clamped boundary conditions. We classify all the possible equilibrium states of inextensible, unshearable, isotropic, uniform and naturally straight and prismatic rods. We show that all solutions of the clamped boundary value problem exhibit a π-flip symmetry. The Kirchhoff equations which describe the equilibria of these rods are integrated in a formal way which enable us to describe the boundary conditions in terms of 2 closed form equations involving 4 free parameters. We show that the flip symmetry property is equivalent to a reversibility property of the solutions of the Kirchhoff differential equations. We sort these solutions according to their period in the phase plane. We show how planar untwisted configurations as well as circularly closed configurations play an important role in the classification. This revised version was published online in June 2006 with corrections to the Cover Date.     

Unsteady Free Convection along an Infinite Vertical Flat Plate Embedded in a Stably Stratified Fluid-Saturated Porous Medium

The problem of unsteady free convection heat transfer from a one-dimensional (parallel) flow along an infinite vertical flat plate embedded in a thermally stratified fluid-saturated porous medium is considered. Flows are induced by a sudden change in the arbitrary temporal plate temperature. By a formal reduction of the corresponding boundary value problems to well-known Fourier heat conduction problems, analytical solutions of the Darcy and energy equations are obtained. Several special cases are discussed in detail.     

Application of the Krylov–Bogoliubov–Mitropolsky method to weakly damped strongly non-linear planar Hamiltonian systems

In this paper, an analytical approximation of damped oscillations of some strongly non-linear, planar Hamiltonian systems is considered. To apply the Krylov–Bogoliubov–Mitropolsky method in this strongly non-linear case, we mainly provide the formal and exact solutions of the homogeneous part of the variational equations with periodic coefficients resulting from the Hamiltonian systems. It is shown that these are simply expressed in terms of the partial derivatives of the solutions, written in action-angle variables, of the Hamiltonian systems. Two examples, including a non-linear harmonic oscillator and the Morse oscillator, are presented to illustrate this extension of the method. The approximate first order solution obtained in each case is observed to be quite satisfactory.     

二维拉氏辐射流体力学人为解构造方法

人为构造解方法是复杂多物理过程耦合程序正确性验证的重要方法之一，适用于二维拉氏大变形网格的流体、辐射耦合人为解模型较为少见。针对拉氏辐射流体力学程序正确性验证的需要，从二维拉氏辐射流体力学方程组出发，基于坐标变换技术，给出了拉氏空间到欧氏空间的物理变量导数关系式，开展了辐射流体耦合的人为解构造方法研究，构造了一类质量方程无源项的二维人为解模型，并应用于非结构拉氏程序LAD2D辐射流体力学计算的正确性考核，为流体运动网格上的辐射扩散计算提供了一种有效手段。数值结果显示观测到的数值模拟收敛阶与理论分析一致。