应用数学和力学

Existence and Comparison Results of Solutions for Nonlinear Random Integral and Differential Equations Relative to Weak Topology

Ding Xie-ping

1987, 8(12): 1039-1050.

Abstract(1589) PDF(529)

Abstract:
In this paper, we prove several existence theorems of random solutions to nonlinear random Volterra integral equations under the weak topology of Banach spaces. Then, as applications, we obtain the existence theorems of weak random solutions to random differential equations. Existence of extremal random solutions and a random comparison theorem for these random equations are also obtained. Our theorems improve and extend the corresponding results in [4,5,10,11,12].

A Theory of Classical Spacetime(Ⅰ)——Foundations

Yu Xin

1987, 8(12): 1051-1064.

Abstract(2021) PDF(558)

Abstract:
Despite its beauty and grandeur the theory of GR still appears to be incomplete in the following ways:(1) It cannot accommodate the asymmetric total energy momentum tensor whose asymmetry has been shown to exist in the presence of electromagnetism.(2) The law of angular momentum balance as an exact equation is not an automatic consequence of the field equations as is the. case, with the law of linear momentum balance.(3) The four degrees of arbitrariness left by the contracted second Bianchi identity makes a unique solution of the field equations unattainable without extra (unphysical) postulates.To answer the challenge posed by the above assertions we propose in this paper to complete Einstein's theory by postulating the principle fibre bundle P[M,SU(2)] for the underlying geometry of tile 4-dtmensional spacetime, where the structure group SU (2) is the real representation of the special complex unitary gioup of dimension 2; SU (2) leaves concurrently invariant the metric form dS²=g_αβdx^αdx^β and the fundamental 2-form φ=(1/2_l)a_αβdx^α∧dx^β defined globally on M. The Einstein equation defined in terms of the SU(2)-connection is imposed on the spacetime manifold together with the Maxwell inhomogeneous equation as the supplementary condition where the electromagnetic tensor is identified with a contracted form of the curvature tensor. The result is a set of 16 functionally independent equations to the 16 unknown field variables (g_αβ,a_αβ). Moreover, the law of angular momentum balance is Just the skew-symmetric part of the generalized Einstein equation where the spin angular momentum tensor is shown directly proportional to the torsion tensor.

A Class of Incompletely Exponentially Fitted Difference Schemes for a Singular Perturbation Problem

Lin Peng-cheng, Sun Guang-fu

1987, 8(12): 1065-1074.

Abstract(1628) PDF(505)

Abstract:
Some sufficient conditions are considered, under which the solutions of a class of incompletely exponentially fitted difference schemes converge uniformly in e, with orders one and two, to the solution of the singular perturbation problem: eu"+a(x)u'-b(x)u=f(x), for 0a>0, b(x)≥0. From these conditions.an incompletely exponentially fitted second-order scheme is derived. Finally, the results of some numerical experiments are given.

Reflection and Radiation of a Wave System at the Open End of a Submerged Elastic Pipe

Song Jia-su

1987, 8(12): 1075-1086.

Abstract(1680) PDF(494)

Abstract:
The reflection and radiation of a wave system at the open end of a submerged semi-infinite elastic pipe are studied. This wave system consists of a flexural wave in the pipe, an acoustic surface wave in the fluid exterior to the pipe and an acoustic wave in the pipe's interior. Fourier transform techniques are used to formulate this semi-infinite geometry problem rigorously as a Wiener-Hopf type equation. An approximate solution is obtained by using a perturbation method in which the ratio of the massdensities of the fluid and the pipe material is regarded as a small parameter. The calculation of the reflection coefficient is emphasized, and the polar plots of the radiation coefficient are also presented.

An Analysis on the Flow Resistance in the Entrance Region of Diffuse Laminar Flow between Two Parallel Spherical Surfaces

Wang Zhi-qing, Liu Zhen-bei

1987, 8(12): 1087-1098.

Abstract(2189) PDF(507)

Abstract:
In this paper, V. V. Golubef method is first extended to the diffuse laminar flow between two parallel spherical surfaces. With the boundary layer motion equation in spherical coordinates, we derive the momentum integral equation together with the energy integral equation for the laminar boundary layer of the entrance region between two parallel spherical surfaces. And then by applying Picards gradually approaching method for the momentum integral equation, we get the approximate expression which the entrance region length varies with the thickness of boundary layer. In the end every coefficient of entrace region effect is analyzed and calculated.

The Nonlinear Numerical Analysis Method for Frames

Zhu Ju-fen, Zhou Cheng-fang, Lü He-xiang

1987, 8(12): 1099-1109.

Abstract(1820) PDF(662)

Abstract:
This paper gives the direct formulas of stiffness matrixes of two-kinds of Kirchhoff nonlinear elements under total-Lagrange coordinate. For the first one, it includes not only the quadric terms of increments of strain and displacement but also-the influence of rotations. For the second one, it is simplified and its nonlinear is considered by taking into account the influence of axial force on the equilibrium equation in the linear beam theory.The nonlinear equation obtained from both of the above-Said elements is solved by mixed Newton-Raphson method, and by comparing the results obtained from two kinds of nonlinear beam some important conclusions that we can know how to use them right are given in our paper.

A Direct Method for Deriving Fundamental Solution of Half-Plane Problem

Huang Yu-ying, Yin Lei-fang

1987, 8(12): 1111-1120.

Abstract(1832) PDF(634)

Abstract:
A fundamental solution for half-plane problems which will play a key role in calculation of the stress concentration around a hole embedded in half-plane is derived by a method combining images with direct integrations. It is wore intuitive than the Fourier transform method used by Gladwell[6]. In addition, the principle and procedure of boundary element method to solve the half-plane problems are also presented by means of Betti's reciprocal theorem in this paper.It is shown that the. computing procedure for half-plane problems is much more convenient using the fundamental solution presented here than the one adopted by C.A. Brebbia^[1].

Determining the Stress Intensity Factor by Using the Principle of Minimum Potential Energy

Chen Guang-zu

1987, 8(12): 1121-1129.

Abstract(2099) PDF(441)

Abstract:
Expressing the total potential energy of the system of a cracked body П by Williams' infinite series solution of stress and displacement components containing coefficients A_n(n = 1,2,...), we obtain a set of simultaneous linear equations of unknown coefficients A_n by using the principle of minimum potential energy. When the set of equations is solved, the stress intensity factor K₁ can be easily determined. It is equal to √2πaA₁ Take a sample plate as an example. A single-edgc-cracked plate under tension, with the ratio of crack length to the width of the plate being 0.5 and the ratio of half plate height to the width of the plate being 2.0 and 2. 5, has been calculated. Only 20-30 coefficients are taken, and the errors in stress intensity factors are within 5%.

The Transformation and General Pythagoras’ Theorem on the Linear Manifold

Gu An-hai

1987, 8(12): 1134-1134.

Abstract(1863) PDF(536)

Abstract:
The primary aim of this paper is to describe the transformation function Φ^[1]. to extend the conclusion of Pythagoras' theorem for "squared length" of any triangle and to study some geometrical significance of its application

1987 Vol. 8, No. 12