Exact solution of a boundary value problem describing the uniform cylindrical or spherical piston motion

In this paper, we construct the exact solution for fluid motion caused by the uniform expansion of a cylindrical or spherical piston into still air. Following Lighthill [1], we introduce velocity potential into the analysis and seek a similarity form of the solution. We find both numerical and analytic solutions of the second order nonlinear differential equation, with the boundary conditions at the shock and at the piston. The results obtained from the analytic solutions justify numerical solution and the approximate solution of Lighthill [1]. We find that although the approximate solution of Lighthill [1] gives remarkably good numerical results, the analytic form of that solution is not mathematically satisfactory. We also find that in case of spherical piston motion Lighthill’s [1] solution differs significantly from that of our analytic and numerical solutions. We use Pade′ approximation to extend the radius of convergence of the series solution. We also carry out some local analysis at the boundary to obtain some singular solutions.     

An approximate solution of one-dimensional piston problem

A new theory of shock dynamics has been developed in the form of a finite number of compatibility conditions along shock rays. It has been used to study the growth or decay of shock strength for accelerating or decelerating piston starting with a nonzero piston velocity. The results show good agreement with those obtained by Harten's high resolution TVD scheme.     

An approximate solution of two-dimensional convex piston problem

A new theory of shock dynamics (NTSD) has been used to study the propagation of curved shocks originating from the motion of two-dimensional convex piston of various shapes. The results have been compared with those obtained by Whitham's classical shock dynamics and by TVD version of MacCormack's finite difference scheme.     

An approximate solution algorithm for the one-dimensional online median problem

The online median problem consists in finding a sequence of incremental solutions of the k-median problem with k increasing. A particular case of the problem is considered: the clients and facilities are located on the real line. The best algorithm available for the one-dimensional case has competitive ratio 8. We give an improved 5.83-competitive algorithm.     

An approximate solution for the torsional properties of cylindrical shafts with multi-symmetrical curved boundaries

Zusammenfassung Vorliegende Arbeit befasst sich mit der genäherten Bestimmung der Drehsteifigkeit und Spannungsverteilung an der Oberfläche unrunder Wellenprofile. Mittels konformer Abbildung der betrachteten Querschnitte auf den Einheitskreis und der ermittelten Spannungsfunktion werden maximale Schubspannungen und Werte der Torsionssteifigkeit bestimmt, die mit den aus einer Versuchsreihe erhaltenen überprüft werden. Ein Vergleich der theoretischen und experimentellen Werte zeigt eine für technische Zwecke ausreichende Übereinstimmung.     

An approximate solution of the axisymmetric contact problem for an elastic sphere

An axisymmetric, fractionally non-linear contact problem for an elastic sphere with a priori unknown boundary of the contactarea is considered. An integral equation for determining the density of the contact pressures is constructed taking account of the shear displacements of the boundary points of the elastic body. An approximate solution, which refines the equations of Hertz' theory, is constructed in the case of a small contact area.     

How parabolic free boundaries approximate hyperbolic fronts

A rather complete study of the existence and qualitative behaviour of the boundaries of the support of solutions of the Cauchy problem for nonlinear first-order and second-order scalar conservation laws is presented. Among other properties, it is shown that, under appropriate assumptions, parabolic interfaces converge to hyperbolic ones in the vanishing viscosity limit.

     

Riemann problem with Delta initial data for the relativistic Chaplygin Euler equations

In this paper, we study the Riemann problem with the initial data containing the Dirac delta function for the relativistic Chaplygin Euler equations. Under the generalized Rankine-Hugoniot conditions and entropy condition, we constructively obtain the global existence of generalized solutions including delta shock waves that explicitly exhibit four kinds of different structures. Moreover, we obtain the stability of generalized solutions by making use of the perturbation of the initial data     

An approximate solution for nonlinear backward parabolic equations

We consider the backward parabolic equation

     

广义Chaplygin气体磁流体力学方程组的Riemann问题

利用特征分析和相平面分析的方法,由Rankine-Hugoniot条件和稳定性条件,构造性地得到了一维等熵广义Chaplygin气体磁流体力学方程组的Riemann解的存在唯一性.同时,详细研究了疏散波曲线和激波曲线的性质.     

Intuitive approach to the approximate analytical solution for the Blasius problem

For the Blasius problem, we propose an approximate analytical solution in the form of a logarithm of the hyperbolic cosine function which satisfies the given boundary conditions and some known properties of the exact solution. Furthermore, adding some hyperbolic tangent functions to this solution, we obtain much more accurate approximate solution with the relative error less than 0.16% over the whole region. The superiority of the proposed solutions is shown by comparison with the existing approximate analytical solution.     

二维零压气体动力学系统的黎曼问题-3J情形

研究二维零压气体动力学系统带有三片常数的黎曼问题.对外波为3J的情形,借助特征分析方法,通过研究基本波的相互作用,构造了两种不同的显式解结构,一种出现了δ-激波,另一种则包含一个三角形真空区域.     

The Riemann problem with delta initial data for the nonsymmetric Keyfitz-Kranzer system with Chaplygin pressure

In this paper, we study the Riemann problem with the initial data containing the Dirac delta function for the nonsymmetric Keyfitz-Kranzer system with Chaplygin pressure. Under the generalized Rankine-Hugoniot conditions and entropy condition, we constructively obtain the global existence of generalized solutions including delta shock waves that explicitly exhibit four kinds of different structures. Moreover, we obtain the stability of generalized solutions by making use of the perturbation of the initial data.     

拟凸多目标优化问题近似解的最优性条件

研究了拟凸多目标优化问题近似弱有效解、近似有效解的最优性条件.首先,在已有拟凸函数次微分的基础上引进4种近似次微分的概念,并给出它们之间的关系.然后,将4种近似次微分的概念应用到拟凸多目标优化问题中,给出了拟凸多目标优化问题近似弱有效解和近似有效解的充分条件和必要条件,并给出实例加以说明.     

An algorithm for finding an approximate solution to the Weber problem on a line with forbidden gaps

Under study is the problem of optimal location of interconnected objects on a line with forbidden gaps. The task is to minimize the total cost of links between objects and between objects and zones. The properties of the problem are found that allowed us to reduce the initial continuous problem to a discrete problem. Some algorithm for obtaining an approximate solution is developed, and the results of a computational experiment are given.     

Linearization of the Riemann problem for a triangular system of conservation laws and delta shock wave formation process

Using the weak asymptotic method, we approximate a triangular system of conservation laws arising from the so‐called generalized pressureless gas dynamics by a diagonal linear system. Then, we apply the usual method of characteristics to find approximate solution to the original system. As a consequence, we shall see how the delta shock wave naturally arises along the characteristics. Also, we propose a procedure that could be applied to more general systems of conservation laws. Copyright © 2009 John Wiley & Sons, Ltd.     

The quadratic three-dimensional assignment problem: Exact and approximate solution methods

This paper reports on algorithm development for solving the quadratic three-dimensional assignment problem (Q3AP). The Q3AP arises, for example, in the implementation of a hybrid ARQ (automatic repeat request) scheme for enriching diversity among multiple packet re-transmissions, by optimizing the mapping of data bits to modulation symbols. Typical practical problem sizes would be 8, 16, 32 and 64.     

An approximate algorithm for a high-multiplicity parallel machine scheduling problem

We consider a high-multiplicity parallel machine scheduling problem where the objective is to minimize the weighted sum of completion times. We suggest an approximate algorithm and we prove that it is asymptotically exact. The algorithm exploits a convex quadratic relaxation of the problem to fix a partial schedule, consisting of most jobs, and then assigns the residual jobs following a simple and general rule. The quality of the obtained solution is evidenced by some numerical tests.