应用数学和力学

Asymptotic Behavior of the 2D Generalized Stochastic Ginzburg-Landau Equation With Additive Noise

LI Dong-long, GUO Bo-ling

2009, 30(8): 883-894. doi: 10.3879/j.issn.1000-0887.2009.08.001

Abstract(1762) PDF(831)

Abstract:
The 2D generalized stochastic Ginzburg-Landau equation with additive noise is considered. The compactness of the random dynamical system was established by a priori estimates method, which shows that the random dynamical system possesses a random attractor in H₀¹

Prediction of Nanoparticle Transport and Deposition in Bends

LIN Pei-feng, LIN Jian-zhong

2009, 30(8): 895-960. doi: 10.3879/j.issn.1000-0887.2009.08.002

Abstract(1766) PDF(975)

Abstract:
Nanoparticle transport and deposition in bends with circular cross-section were solved for different Reynolds number and Schmidt number. The perturbation method was used to solve the equations. The results show that the particle transport patterns are similar and not dependent on the particle size and other parameters when suspended nanoparticles are flowing in the straight tube. Particle deposition at the outside edge is most intensive, and oppositely, the deposition at the inside edge is the weakest. At the upper and lower tube, the depositions are approximately the same for different Schmidt number. Curvatures of tube, Reynolds number and Schmidt number have effects of second order, fourth order and first order on the relative deposition efficiency, respectively.

Free Vibration of Functionally Graded Material Beams With Surface-Bonded Piezoelectric Layers in Thermal Environment

LI Shi-rong, SU Hou-de, CHENG Chang-jun

2009, 30(8): 907-918. doi: 10.3879/j.issn.1000-0887.2009.08.003

Abstract(2108) PDF(945)

Abstract:
Free vibration of statically thermal post-buckled functionally graded material beams with surface-bonded piezoelectric layers subjected to both temperature rise and voltage is studied. By accurately considering the axial extension and based on Euler-Bernoulli beam theory, geometrically nonlinear dynamic governing equations for FGM beams with surfacebonded piezoelectric layers subjected to thermo-electro-mechanical loadings were formulated. It was assumed that the material properties of the middle FGM layer vary continuously as a power law function of the thickness coordinate and that the piezoelectric layers are isotropic and homogenous. By assuming that the amplitude of beam. s vibration is small and its response harmonic, the above mentioned non-linear partial differential equations were reduced to two sets of coupled ordinary differential equations; the one for the postbuckling, and the other for linear vibration of the beam superimposed upon the post buckled configuration. By using a shooting method to solve the two sets of ordinary differential equations with fixed-fixed boundary conditions numerically, response of postbuckling and free vibration in the vicinity of the postbuckled configuration of the beam with fixed-fixed ends and subjected to transversely non-uniform heating and uniform electric field were obtained. Thermo-electric postbuckling equilibrium paths and characteristic curves of the first three natural frequencies versus the temperature, the electricity and the material gradient parameters were plotted. It is found that the three lowest frequencies of the prebuckled beam decrease with an increase in the temperature, but those of a buckled beam increase monotonically with the temperature rise. The results also show that the tensional force produced in the piezoelectric layers by the voltage can efficiently increase the critical buckling temperature and the natural frequency.

Fractal Geometry and Topology Abstracted From Hair Fibers

YIN Ya-jun, YANG Fan, LI Ying, FAN Qin-shan

2009, 30(8): 919-926. doi: 10.3879/j.issn.1000-0887.2009.08.004

Abstract(1886) PDF(1577)

Abstract:
Based on the concepts of fractal super fibers, the (3, 9+2)-circle and (9+2, 3)-circle binary fractal sets were abstracted from such prototypes as wool fibers and human hairs, with the (3)circle and the (9+2)circle fractal sets as subsets. As far as the (9+2) topological patterns are concerned, the following propositions were proved: The (9+2) topological patterns accurately exist, but they are of no uniqueness. Their total number is 9. Among them there are only 2 allotropes. In another word, among the 9 topological patterns only 2 are independent (or fundamental). Besides, it was demonstrated that the (3, 9+2)-circle and (9+2, 3)-circle fractal sets are golden ones with symmetry breaking.

Numerical Method and Analysis of Computational Fluid Mechanics for Photoelectric Semiconducting Detector

YUAN Yi-rang, LI Chang-feng, LIU Yun-xin, MA Li-qin

2009, 30(8): 927-938. doi: 10.3879/j.issn.1000-0887.2009.08.005

Abstract(1637) PDF(763)

Abstract:
For computational fluid mechanics simulations of a threedimensional photoelectric semiconducting detector, a modified upwind finite difference fractional step scheme was proposed. The optimal error estimated by using techniques including calculus of variations, energy method, induction hypothesis, and a priore stimates were obtained. The proposed scheme has been applied to simulate the photoelectric semiconducting detector.

Asymptotic Solution of Nonlocal Nonlinear Reaction Diffusion Robin Problems With Two Parameters

MO Jia-qi

2009, 30(8): 939-944. doi: 10.3879/j.issn.1000-0887.2009.08.006

Abstract(1712) PDF(800)

Abstract:
A class of nonlocal nonlinear reaction diffusion singularly perturbed Robin problem with two parameters was studied. Using singular perturbation method, the structure of solutions for the problem related two small parameters was discussed. The asymptotic solutions of the problem were given.

Uses of Solar Radiation for Formation Flying Control Around the Sun-Earth Libration Point

GONG Sheng-ping, LI Jun-feng, BAOYIN He-xi

2009, 30(8): 945-952. doi: 10.3879/j.issn.1000-0887.2009.08.007

Abstract(1587) PDF(690)

Abstract:
Solar radiation pressure is made use to control the formation flying around L₂ libration point in the Sun-Earth system. Controlling formation flying around a Halo orbit requires very small thrust that cannot be satisfied by latest thrusters. Key contribution of this paper is the low continuous thrust produced by the solar radiation pressure to achieve the tight formation flying around libration point. However, only certain families of formation types can be controlled by solar radiation pressure force since the direction of solar radiation pressure is restricted to certain range. Two types of feasible formation using solar radiation pressure force control were designed. The conditions of the feasible formations were given analytically. Simulations were given for each case and the results show that the formations are well controlled by the solar radiation pressure.

Symmetry Solutions of a Non-Linear Elastic Wave Equation With Third Order Anharmonic Corrections

M. T. Mustafa, Khalid Masood

2009, 30(8): 953-962. doi: 10.3879/j.issn.1000-0887.2009.08.008

Abstract(1887) PDF(967)

Abstract:
Lie symmetry method was applied to analyze a non-linear elastic wave equation for longitudinal deformations with third order anharmonic corrections to the elastic energy. Symmetry algebra was found and reductions to second order ODEs were obtained through invariance under different symmetries. The reduced ODEs were further analyzed to obtain several exact solutions in explicit form. Apostol(Apostol B F. On a non-linear wave equation in elasticity. Phys Lett A, 2003, 318(6):545552) had observed that anharmonic corrections generally lead t o solutions with time-dependent singularities in finite time. Along with solutions with time-dependent singularities are obtained, also solutions which do not exhibit time-dependent singularities were obtained.

Multi-Symplectic Runge-Kutta Methods for Landau-Ginzburg-Higgs Equation

HU Wei-peng, DENG Zi-chen, HAN Song-mei, FAN Wei

2009, 30(8): 963-969. doi: 10.3879/j.issn.1000-0887.2009.08.009

Abstract(1748) PDF(786)

Abstract:
The nonlinear wave equation, describing many important physical phenomena, has been investigated widely in last several decades. Landau-Ginzburg-Higgs equation, a typical nonlinear wave equation, was sdudied based on the multisymplectic theory in Hamilton space. The multi symplectic Runge-Kutta method was reviewed and a semiimplicit scheme with certain discrete conservation laws was constructed to solve the first order partial differential equations that were derived from the LandauGinzburg-Higgs equation. The results of numerical experiment for soliton solution of the Landau-Ginzburg-Higgs equation were reported finally, which show that the multi symplectic Runge-Kutta method is an efficient algorithm with excellent long-time numerical behaviors.

Wavelet Multiscale Method for the Inversion of Maxwell’s Equation

DING Liang, HAN Bo, LIU Jia-qi

2009, 30(8): 970-978. doi: 10.3879/j.issn.1000-0887.2009.08.010

Abstract(1547) PDF(806)

Abstract:
The estimation of the electrical conductivity in Maxwell's equation is concerned with. The primary difficulty is the presence of numerous local minima in the objective functional. A wavelet multiscale method was introduced and applied to the inversion of Maxwell equations. The inverse problem was then decomposed to multiple scales by wavelet transform and hence the original inverse problem was reformulated to a set of subinverse problems corresponding to different scales solved successively according to the size of scale from the shortest to the longest. On each scale, the stable and fast regularized Gauss-Newton method was carried out. The results of numerical simulation showed that this method is an available method, especially on aspects of wide convergence, computational efficiency and precision.

Boundary Value Problems for Nonlinear Second Order Difference Equations With Impulse

Huseyin Bereketoglu, Aydin Huseynov

2009, 30(8): 979-989. doi: 10.3879/j.issn.1000-0887.2009.08.011

Abstract(1661) PDF(738)

Abstract:
A boundary value problem with impulse (BVPI) for nonlinear second order difference equations is considered. Green's function of the BVPI was constructed and then the nonlinear BVPI was reduced to a fixed point problem. Banach fixed point theorem and Lipschitz condition were applied to show the uniqueness of solutions for the nonlinear BVPI. Finally, the theorem existence of solutions for the nonlinear BVPI was proved.

Solvability of a Class of Second-Order Quasilinear Boundary Value Problems

YAO Qing-liu

2009, 30(8): 990-996. doi: 10.3879/j.issn.1000-0887.2009.08.012

Abstract(1565) PDF(770)

Abstract:
A class of second-order quasilinear boundary value problems was considered when the non-linear term is singular and the limit growth function at infinite exists. By introducing the height function of nonlinear term on bounded set and considering integration of the height function, an existence theorem of solution was proved. The existence theorem shows that the problem has a solution if the integration of the limit growth function has appropriate value.

Difference Equation Approach of Statistical Mechanics of Complex Networks

GUO Jin-li

2009, 30(8): 997-1002. doi: 10.3879/j.issn.1000-0887.2009.08.013

Abstract(1406) PDF(922)

Abstract:
The difference equation approach of estimating degree distribution in growing networks was proposed after analyzing the disadvantages of some existing approaches. This approach avoids not only logic conflicts brought by continuum of discrete problem, but also the assumption of existence of the stationary degree distribution in network analysis. The degree distribution formula of Poisson growth and preferential attachment network was obtained by this approach. It was strictly proved that this network is scale free based on Poisson process theory and properties of Gamma distribution.

Finite-Time Stability With Respect to a Closed Invariant Set for a Class of Discontinuous Systems

CHENG Gui-fang, MU Xiao-wu

2009, 30(8): 1003-1008. doi: 10.3879/j.issn.1000-0887.2009.08.014

Abstract(1485) PDF(1059)

Abstract:
The problem of finite-time stability is mainly discussed with respect to a closed (not necessarily compact) invariant set for a class of nonlinear systems with discontinuous righthand sides in the sense of Filippov solutions. When Liapunov function is Lipschitz continuous and regular, Liapunov theorem on finite-time stability with respect to a closed invariant set was presented.

2009 Vol. 30, No. 8