The analyticity of solutions of the Stefan problem

The Stefan problem of a semi-infinite body with arbitrarily prescribed initial and boundary conditions is studied. One of the objectives of the paper is to investigate the analyticity of the solutions. For this purpose, the prescribed initial and boundary conditions are considered to be series of fractional powers of their arguments. It is found that the exact solutions of the problem for various forms of the initial and boundary conditions can be established in series of parabolic cylinder functions and time t. Existence and convergence of the series solutions are studied and proved. The present solutions include the known exact solutions as special cases. On the basis of the present solutions, the question of the analyticity of solutions of the Stefan problem, raised by Rubinstein in his book, can be answered. Conditions for analyticity of the solutions with various initial and boundary conditions are fully discussed.     

ANALYTICAL SOLUTIONS FOR THE LAYERED GEO-MATERIALS SUBJECTED TO AN ARBITRARY POINT LOAD IN THE CARTESIAN COORDINATE

This paper presents analytical solutions for the stress and displacement field in elastic layered geo-materials induced by an arbitrary point load in the Cartesian coordinate system. The point load solutions can be obtained by referring to the integral transform and the transfer matrix technique. However, former solutions usually exist in the cylindrical coordinate system subjected to axisymmetric loading. Based on the proposed solutions in the Cartesian coordinate, it is very easy to solve asymmetric problems and consider the condition with internal loads in multi-layered geo-materials. Moreover, point load solutions can be used to construct solutions for analytical examination of elastic problems and incorporated into numerical schemes such as boundary element methods. The results discussed in this paper indicate that there is no problem in the evaluation of the point load solutions with high accuracy and efficiency, and that the material non-homogeneity has a significant effect on the elastic field due to adjacent loading.     

A numerical treatment of the periodic solutions of non-linear vibration systems

Direct numerical integration can be used to find the periodicsolutions for the equations of motion of nonlinear vibrationsystems.The initial conditions are iterated so that theycoincide With the terminal conditions.The time interval ofthe integration(i.e.,the period)and certain parameters ofthe equations of motion can be included in the iterations.Theintegration method has a variable stoplength.This Sbooting method can produce periodic solutions witha shorter computex time.The only error occurs in the numeri-cal integration and it can therefore be estimated and madesmall enough.Using this method one can treat a variety ofvibration problems.such as free conservative.forced.para-meter-excited and self-sustained vibrations with one or se-veral degrees-of-freedom.Unstable solutions and those Whichare sensitive to parameter Changes can also be calculated.Thestability of the solutions is investigated based on the thecryof differential equations with periodic coefficients.The ex-trapolation method and the proc     

The approximate solutions to the non-linear heat conduction problems in a semi-infinite medium

The approximate solutions to the non-linear heat conduction problems in a semi-infinite medium are investigated. The entire temperature range is divided into a number of small sub-regions where the thermal properties can be approximated to be constant. The resulting problems can be considered as the Stefan’s problem of a multi-phase with no latent heat and the exact solutions called Neumann’s solution are available. In order to obtain the solutions, however, a set of highly non-linear equations in determining the phase boundaries should be solved simultaneously. This work presents a semi-analytic algorithm to determine the phase boundaries without solving the highly non-linear equations. Results show that the solutions for a set of highly non-linear equations depend strongly on the initial guess, bad initial guess leads to the wrong solutions. However, the present algorithm does not require the initial guess and always converges to the correct solutions.     

Superposition about the 3D vortex solutions of the fluid dynamic equation

ＩｎｔｒｏｄｕｃｔｉｏｎＩｔｉｓｗｅｌｌ＿ｋｎｏｗｎｔｈａｔｓｏｍｅｎｏｎｌｉｎｅａｒｄｙｎａｍｉｃｅｑｕａｔｉｏｎｓ,ｓｕｃｈａｓＫｄＶｅｑｕａｔｉｏｎ ,ｓｉｎｅ＿Ｇｏｒｄｅｎｅｑｕａｔｉｏｎ ,ｈａｖｅｅｘａｃｔｓｏｌｉｔｏｎｓｏｌｕｔｉｏｎｓ.Ａｖｅｒｙｉｍｐｏｒｔａｎｔｓｐｅｃｉａｌｉｔｙｏｆｔｈｅｓｏｌｉｔｏｎｓｏｌｕｔｉｏｎｓｏｆｎｏｎｌｉｎｅａｒｅｑｕａｔｉｏｎｓｉｓｔｈａｔｔｈｅｙｍａｙｈａｖｅａｓｕｐｅｒｐｏｓｉｔｉｏｎ ,ｉ.ｅ .,ｔｈｅｓｅｎｏｎｌｉｎｅａｒｅｑｕａｔｉｏｎｓｍａｙ…     

Periodic solutions in one-dimensional coupled map lattices

IntroductionDynamicsofcoupledmaplattices (CML )haverecentlyattractedconsiderableattention[1- 4].IthasbeenfoundthatCMLsexhibitavarietyofspace_timepatterns:space_timeintermittency ,spatiotemporalchaos,frontsandinterfaces,nonlinearperiodicsolutions.Inthispaper,anone_dimensionalinfinitechainofnonlinearcoupledmapsisconsideredandtheexistenceofperiodicsolutionsisprovedwhenthecouplingstrengthαissmallenough .Ourapproachistheanti_integrablelimitintroducedinRefs.[5-7] .Intheanti_integrablelimit(coup…     

变边界粘弹性轴对称问题的复变函数法

对边界几何形状、位置随时间变化的变边界结构,给出了用复变函数求解粘弹问题的解析方法。文中用拉普拉斯变换结合平面弹性复变方法,对内外边界变化时粘弹性轴对称问题进行求解。引入两个与时间、空间相关的解析函数,给出了变边界情况下应力、位移以及边界条件与解析函数的关系。当解析函数形式部分确定,则可用边界条件求解其中与时间相关的待定函数。求解待定函数的方程一般情况下为一系列积分方程,特殊情况可求得解析解。对轴对称问题中应力边值问题、位移边值问题以及混合边值问题,分别利用边界条件求得相关系数,从而得到了应力与位移的解析表达。当取Boltzmann粘弹模型时,进行不同边值问题的分析。分析显示,应力、位移的形态与大小均与边界变化过程相关,与固定边界粘弹性问题有较大不同。本文解答可用于粘弹性轴对称问题内外边界任意变化及各种边值问题的力学分析。此外,该法可进一步进行荷载非对称、复杂孔型变边界问题的求解。     

On the local stability analysis of the approximate harmonic balance solutions

Periodic response of nonlinear oscillators is usually determined by approximate methods. In the "steady state" type methods, first an approximate solution for the steady state periodic response is determined, and then the local stability of this solution is determined by analyzing the equation of motion linearized about this predicted "solution". An exact stability analysis of this linear variational equation can provide erroneous stability type information about the approximate solutions. It is shown that a consistent stability type information about these solutions can be obtained only when the linearized variational equation is analyzed by approximate methods, and the level of accuracy of this analysis is consistent with that of the approximate solutions. It is demonstrated that these consistent stability results do not imply that the approximate solution is qualitatively correct. It is also shown that the difference between an approximate and the next higher order stability analysis can be used to "guess" the role of higher harmonics in the periodic response. This trial and error procedure can be used to ensure the qualitatively correct and numerically accurate nature of the approximate solutions and the corresponding stability analysis.     

3-periodic traveling wave solutions for a dynamical coupled map lattice

In this paper, the existence of periodic traveling wave solutions with a priori unknown velocity is considered for a coupled map lattice dynamical system. By trasforming our problem into one that involves polynomials, explicit 2- and 3-periodic traveling wave solutions are found, while the other solutions can be computed numerically. Since there does not seem to be any reports on explicit traveling wave solutions, we hope that our results will lead to the discovery of many others.     

Incompressible Euler Equations and the Effect of Changes at a Distance

Because pressure is determined globally for the incompressible Euler equations, a localized change to the initial velocity will have an immediate effect throughout space. For solutions to be physically meaningful, one would expect such effects to decrease with distance from the localized change, giving the solutions a type of stability. Indeed, this is the case for solutions having spatial decay, as can be easily shown. We consider the more difficult case of solutions lacking spatial decay, and show that such stability still holds, albeit in a somewhat weaker form.     

旋转圆盘受均布压力作用时的摄动分析

本文对中心带轴匀速旋转圆盘在均布压力作用下的受力及变形问题进行了摄动分析，得到了受载条件为Ｊ＾２／ＪＨ＜＜１时，圆盘应力及形变的渐近解。其形变渐近解为Ｗｈｉｔａｋｅｒ函数形式比较表明，渐近解与Ｃｈｅｂｙｓｈｅｖ数值解有较好的一致性。     

Dynamic propagation problems concerning surfaces of asymmetrical mode Ⅲ crack subjected to moving loads

With the theory of complex functions, dynamic propagation problems concerning surfaces of asymmetrical mode Ⅲ crack subjected to moving loads are investigated. General representations of analytical solutions are obtained with self-similar functions. The problems can be easily converted into Riemann-Hilbert problems using this technique. Analytical solutions to stress, displacement and dynamic stress intensity factor under constant and unit-step moving loads on the surfaces of asymmetrical extension crack, respectively, are obtained. By applying these solutions, together with the superposition principle, solutions of discretionarily intricate problems can be found.     

CLOSED-FORM SOLUTIONS FOR ELASTOPLASTIC PURE BENDING OF A CURVED BEAM WITH MATERIAL INHOMOGENEITY

The elastoplastic pure bending problem of a curved beam with material inhomo- geneity is investigated based on Tresca＇s yield criterion and its associated flow rule. Suppose that the material is elastically isotropic, ideally elastic-plastic and its elastic modulus and yield limit vary radially according to exponential functions. Closed-form solutions to the stresses and radial displacement in both purely elastic stress state and partially plastic stress state are presented. Numerical examples reveal the distinct characteristics of elastoplastic bending of a curved beam composed of inhomogeneous materials. Due to the inhomogeneity of materials, the bearing capac- ity of the curved beam can be improved greatly and the initial yield mode can also be dominated. Closed-form solutions presented here can serve as benchmark results for evaluating numerical solutions.     

A discrete algorithm for complex frequency-domain convolutions

ＩｎｔｒｏｄｕｃｔｉｏｎＩｔｉｓｗｅｌｌ＿ｋｎｏｗｎｔｈａｔｔｈｅＬａｐｌａｃｅｔｒａｎｓｆｏｒｍｏｆｔｈｅｐｒｏｄｕｃｔｏｆｔｗｏｔｉｍｅ＿ｄｏｍａｉｎｓｉｇｎａｌｓｅｑｕａｌｓｔｈｅｃｏｍｐｌｅｘｆｒｅｑｕｅｎｃｙ＿ｄｏｍａｉｎｃｏｎｖｏｌｕｔｉｏｎｏｆｉｎｄｉｖｉｄｕａｌＬａｐｌａｃｅｔｒａｎｓｆｏｒｍｓｏｆｔｗｏｓｉｇｎａｌｓ.Ｇｅｎｅｒａｌｌｙ,ｔｈｅｃｏｎｖｏｌｕｔｉｏｎｏｆｔｗｏｃｏｍｐｌｅｘｆｒｅｑｕｅｎｃｙ＿ｄｏｍａｉｎａｎａｌｙｔｉｃａｌｆｕｎｃｔｉｏｎｓｉｓｃａｌｃｕｌａｔｅ…     

配置环形线电极的单极机中的磁流体力学流动

给出配置了环形线电极的单极机中的磁流体力学解析解. 由于问题是线性的, 求解可简化为基本解的叠加, 可利用单极机在两个区域中的基本解的衔接来求得解析解. 讨论了基本解的性质. 利用非完全电极可改善Hartmann 边界层、增加装置中的质量流量. 应用基本解讨论了连续电极的单极机.     

层状弹性半空间轴对称动力问题的奇异解

本文利用Laplace-Hankel联合变换及传播矩阵技术导出了任意层数的层状弹性半空间轴对称动力问题时域奇异解的一般解析表达式,并给出了奇异解数值化实施的计算方法。文末的实例计算表明了本文给出解答的正确性以及数值化实施的可靠性,从而为进一步用边界元法直接解决由于层状介质而引起的非匀质动力问题开拓了一条潜在的途径。     

Self-Similar Solutions of Fracture Dynamics Problems on Axially Symmetry

ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｍｐｏｓｉｔｅｍａｔｅｒｉａｌｓａｒｅａｌｍｏｓｔｍａｄｅｕｐｏｆｆｉｂｅｒａｎｄｍａｔｒｉｘｗｉｔｈａｎｉｓｏｔｒｏｐｉｃｍａｃｒｏｓｔｒｕｃｔｕｒｅ ,ｗｈｏｓｅｖａｒｉｏｕｓｓｔｒｕｃｔｕｒｅｓａｒｅｅａｓｙｔｏａｐｐｅａｒａｐｅｎｎｙ＿ｓｈａｐｅｄｍｉｃｒｏｃｒａｃｋｉｎｔｈｅｐｒｏｃｅｓｓｏｆｔｈｅｉｒｗｏｒｋ.Ｗｈｅｎｉｔｅｘｔｅｎｄｓｇｒａｄｕａｌｌｙ ,ｔｈｅｓｔｒｕｃｔｕｒｅｗｉｌｌｌｅａｄｔｏｄｅｓｔｒｕｃｔｉｏｎ .Ｉｔｉｓｗｅｌｌ＿ｋｎｏｗｎｔｈａｔ…     

求解非线性动力系统周期解的改进打靶法

针对有周期解的动力系统边值问题可以转化为初值问题这一特点,改进了周期解的打靶 法数值求解. 在计算边界条件代数方程关于待定初值参数导数的过程中利用前一次 Runge-Kutta方法计算得到的节点函数值并通过再次利用Runge-Kutta方法获得了该导数值. 用此方法求解了Duffing方程及非线性转子---轴承系统的周期解,用Floquet理论判断了 周期解的稳定性,与普通打靶法作了比较,验证了方法的有效性.     

受外激励二阶参变系统的周期解及其稳定性

给出受外激励二阶参变系统周期解的新的计算方法，它对于相当广泛一类方程均可适用，并可对解的稳定性作出判断。     

Solving initial–boundary-value creep problems

The Galerkin–Bubnov method with global approximations is used to find approximate solutions to initial–boundary-value creep problems. It is shown that this approach allows obtaining solutions available in the literature. The features of how the solutions of initial–boundary-value problems for oneand three-dimensional models are found are analyzed. The approximate solutions found by the Galerkin–Bubnov method with global approximations is shown to be invariant to the form of the equations of the initial–boundary-value problem. It is established that solutions of initial–boundary-value creep problems can be classified according to the form of operators in the mathematical problem formulation