On the boundary layer methods

In this paper, the defect of the traditionary boundary layer methods (including the method of matched asymptotic expansions and the method of Visik-Lyusternik) is noted, from those methods we can not construct the asymptotic expansion of boundary layer term substantially. So the method of multiple scales is proposed for constructing the asymptotic expansion of boundary layer term, the reasonable result is obtained. Furthermore, we compare this method with the method used by Levinson, and find that both methods give the same asymptotic expansion of boundary layer term, but our method is simpler.Again, we apply this method to study some known works on singular perturbations. The limitations of those works have been noted, and the asymptotic expansion of solution is constructed in general condition.     

Application of reconstitution multiple scale asymptotics for a two-to-one internal resonance in Magnetic Resonance Force Microscopy

In this paper we formulate an initial-boundary-value-problem describing the three-dimensional motion of a cantilever in a Magnetic Resonance Force Microscopy setup. The equations of motion are then reduced to a modal dynamical system using a Galerkin ansatz and the respective nonlinear forces are expanded to cubic order. The direct application of the asymptotic multiple scales method to the truncated quadratic modal system near a 2:1 internal resonance revealed conditions for periodic and quasiperiodic energy transfer between the transverse in-plane and out-of-plane modes of the MRFM cantilever. However, several discrepancies are found when comparing the asymptotic results to numerical simulations of the full nonlinear system. Therefore, we employ the reconstitution multiple scales method to a modal system incorporating both quadratic and cubic terms and derive an internal resonance bifurcation structure that includes multiple coexisting in-plane and out-of-plane solutions. This structure is verified and reveals a strong dependency on initial conditions in which orbital instabilities and complex out-of-plane non-stationary motions are found. The latter are investigated via numerical integration of the corresponding slowly-varying evolution equations which reveal that breakdown of quasiperiodic tori is associated with symmetry-breaking and emergence of irregular solutions with a dense spectral content.     

Application of the modified method of multiple scales to solving the problem of a thin clamped circular plate of a very large deflection

In this paper,asymptotic behaviour of the solution to the problem of a thin clamped circular plate under uniform normal pressure at very large deflection is restudied by means of the modified method of multiple scales given in[1?].The result presented herein is in good agreement with the one obtained by professor Chien Wei-zang who first proposed the method of composite expansions to solve this problem in[3].However,by contrast,the advantage of the modified method of multiple scales it seems to be relatively simpler than the method used in[3].It is also shown that the restriction of the method of paper[1-2]pointed out in paper[4]is not essential,and several computation errors in[3]are corrected as well.     

Uniform asymptotics for the linearized Boltzmann equation describing sound wave propagation

The multiple-scale expansionmethod is used for constructing a uniformly applicable asymptotic approximation of the solution of the linearized Boltzmann equation for small Knudsen numbers. The asymptotic expansion is constructed for the particular example of a sound wave generated by a plane oscillation source and dissipating in a half-space. The simplicity of the problem makes it possible clearly to demonstrate the appearance of secular terms in the expansion and the introduction of multiple scales opens the way to eliminating them.     

A class of boundary value problems for third-order differential equation with a turning point

A class of boundary value problems for a third-order differential equation with a turning point is considered. Using the method of multiple scales and others, the uniformly valid asymptotic expansion of solution for the boundary value problem is constructed.     

Analysis of one-to-one autoparametric resonances in cables—Discretization vs. direct treatment

We discuss solution methods for nonlinear vibrations of cables having small initial sag-to-span ratios. One-to-one internal resonances between the in-plane and out-of-plane modes as well as primary resonances of the in-plane mode are considered. Approximate solutions are obtained by two different approaches. In the first approach, the method of multiple scales is applied directly to the governing partial-differential equations and boundary conditions. In the second approach, the equations are first discretized, and then the method of multiple scales is applied to the resulting ordinary-differential equations. It is shown that treatment of the discretized system is inaccurate compared to direct treatment of the partial-differential system. Discrepancies between the two solutions appear even at the first level of approximation. Stability analyses of the amplitude and phase modulation equations for both methods are also performed.     

An analysis of the imperfection sensitivity of square elastic-plastic plates under axial compression

For a simply supported elastic-plastic square plate under axial compression the post-bifurcation behaviour and the sensitivity to initial imperfections are investigated. An exact asymptotic expansion is given for the initial post-bifurcation behaviour of a perfect plate compressed into the plastic range. The imperfection sensitivity is studied through an asymptotic analysis of the behaviour of the hypoelastic plate that results from neglecting the effect of elastic unloading. The results of the asymptotic analyses are compared with results of a numerical incremental solution by means of a combined finite element—Rayleigh Ritz method. The paper considers the effect of different in-plane boundary conditions and the effect of various degrees of strain hardening.     

Perturbation method for thin plate bending problems

In this paper, problems of bending of a thin plate under the action of in-plane forces are studied by using the method of multiple scales.     

悬索在考虑1:3内共振情况下的动力学行为

研究了悬索在受到外激励作用下考虑1∶3内共振情况下的两模态非线性响应.对于一定范围内悬索的弹性-几何参数而言,悬索的第三阶面内对称模态的固有频率接近于第一阶面内对称模态固有频率的三倍,从而导致1∶3内共振的存在.首先利用Galerkin方法把悬索的面内运动方程进行离散,然后利用多尺度法对离散的运动方程进行摄动得到主共振情况下的平均方程.接下来对平均方程的稳态解、周期解以及混沌解进行了研究.最后利用Runge-Kutta法研究了悬索两自由度离散模型的非线性响应.     

An application of the method of multiple scales to magnetohydrodynamic flow

In this paper we study the flow of an incompressible conducting fluid along an elliptic duct imposed in an uniform magnetic field. In case Hartmann number of the flow is sufficiently large, the method of multiple scales is adopted for constructing the asymptotic approximation of solution up to any order. Our method can also be applied to study the magnetohydrodynamic flow along any duct whose cross section has smooth boundary.     

Development of a swirling jet in infinite space

We examine the problem of swirling-jet development in an infinite space filled with the same fluid. The fourth term of the asymptotic expansion of the tangential-velocity component is obtained. The constant appearing in the solution is obtained semlempirically. Results are presented of calculations of the velocities and pressure in swirling jets and of experimental studies.Swirling jet flows play an important role in the process of combustion intensification and stabilization and are widely used in engineering.The formulation and first solution of the problem of swirling-jet development in an infinite space filled with the same fluid at rest were accomplished by Loitsyanskii [1], who found the first two terms of the asymptotic expansion of the solution of the boundary-layer equations. The third and fourth terms of the asymptotic expansion of the axial-velocity component were found in [2], which made it possible to study the effect of jet swirl on the axial-velocity-component profile.In the present study we obtain the fourth term of the asymptotic expansion of the tangential-velocity component and present results of experimental studies on swirling jets.The authors wish to thank L. G. Loitsyanskii for valuable comments.     

Singular perturbation solution of boundary-value problem for a secord-order differential-difference equation

In this paper, the method of two-variables expansion is used to construct boundary layer terms of asymptotic solution of the boundary-value problem for a second-order DDE. The n-order formal asymptotic solution is obtained and the error is estimated. Thus the existence of uniformly valid asymptotic solution is proved.     

Equivalent multipolar point-source modeling of small spheres for fast and accurate electromagnetic wave scattering computations

In this paper, we develop reduced models to approximate the solution of the electromagnetic scattering problem in an unbounded domain which contains a small perfectly conducting sphere. Our approach is based on the method of matched asymptotic expansions. This method consists in defining an approximate solution using multi-scale expansions over outer and inner fields related in a matching area. We make explicit the asymptotics up to the second order of approximation for the inner expansion and up to the fifth order for the outer expansion. We validate the results with numerical experiments which illustrate theoretical orders of convergence for the asymptotic models requiring negligible computational cost.     

COMPUTER COMPUTATION OF THE METHOD OF MULTIPLE SCALES-DIRICHLET PROBLEM FOR A CLASS OF SYSTEM OF NONLINEAR DIFFERENTIAL EQUATIONS

The method of boundary layer with multiple scales and computer algebra were applied to study the asymptotic behavior of solution of boundary value problems for a class of system of nonlinear differential equations . The asymptotic expansions of solution were constructed. The remainders were estimated. And an example was analysed. It provides a new foreground for the application of the method of boundary layer with multiple scales .     

A long wave low frequency motion model for a finitely sheared elastic layer

We consider two-dimensional long wave low frequency motion in a pre-stressed layer composed of neo-Hookean material. Specifically, the pre-stress is a simple shear deformation. Derivation of the dispersion relation associated with traction-free boundary conditions is briefly reviewed. Appropriate approximations are established for the two associated long wave modes. From these approximations it is clear that there may be either two, one or no real long wave limiting phase speeds. These approximations are also used to establish the relative asymptotic orders of the displacement components and pressure increment. Using these relative orders to motivate the introduction of appropriate a scales, an asymptotically consistent model long wave low frequency motion is established. It is shown that in the presence of shear there is neither bending nor extension, or analogues of their previously established pre-stressed counterparts. In fact, both the in-plane and normal displacement components have the same asymptotic orders and the derived governing equation is of vector form.     

Computer computation of the method of multiple scales—dirichlet problem for a class of system of nonlinear differential equations

The method of boundary layer with multiple scales and computer algebra were applied to study the asymptotic behavior of solution of boundary value problems for a class of system of nonlinear differential equations. The asymptotic expansions of solution were constructed. The remainders were estimated. And an example was analysed. It provides a new foreground for the application of the method of boundary layer with multiple scales. Contributed by Jiang Fu-ru, Original Member of Editorial Committe, AMM Biography: Xie La-bing (1976∼); Jiang Fu-ru(1927∼)     

On the boundary value problems for a class of ordinary differential equations with turning points

In this paper we study the boundary value problems for a class of ordinary differential equations with turning points by the method of multiple scales. The paradox in [1] and the variational approach in [2] are avoided. The uniformly valid asymptotic approximations of solutions have been constructed. We also study the case which does not exhibit resonance.     

The effect of two distinct fast time scales in the rotating,stratified Boussinesq equations: variations from quasi-geostrophy

Inspired by the use of fast singular limits in time-parallel numerical methods for a single fast frequency, we consider the limiting, nonlinear dynamics for a system of partial differential equations when two fast, distinct time scales are present. First-order slow equations are derived via the method of multiple time scales when the two small parameters are related by a rational power. We find that the resultant system depends only on the relationship of the two fast time scales, i.e. which fast time is fastest? Using the theory of cancellation of fast oscillations, we show that with the appropriate assumptions on the nonlinear operator of the full system, this reduced slow system is exactly that which the solution will converge to if each asymptotic limit is considered sequentially. The same result is also obtained via the method of renormalization. The specific example of the rotating, stratified Boussinesq equations is explored in detail, indicating that the most common distinguished limit of this system—quasi-geostrophy, is not the only limiting asymptotic system.     

Asymptotic expansions and the possibilities to drop the hypotheses in the prandtl problem

The plane problem on the quasistatic compression of a thin perfectly plastic layer between undeformable rough plates (the Prandtl problem) has a well-known analytic solution at all points sufficiently far from the midsection and endpoints of the layer. Both the static and the kinematic component of this solution were obtained on the basis of the Prandtl hypothesis [1] stating that the tangential stress is linear along the layer thickness and is maximal in absolute value on the plate surfaces. (If the plates are perfectly rough, then this maximum value coincides with the shear yield stress.) The Prandtl hypothesis was widely confirmed in experiments carried out after the paper [1] had been published.At the same time, it is natural to ask whether one can construct a classical solution of this problem without imposing any static or kinematic hypotheses on the unknown variables and whether there exist any other mathematical solutions in which these hypotheses do not hold and which themselves are not observed in experiments.In the present paper, we use asymptotic analysis with a natural small geometric parameter and uniquely determine an exact solution (in the sense of finiteness of the number of terms in the asymptotic expansion), which coincides with the Prandtl solution generalized to the case of an arbitrary roughness coefficient of the plates. We rigorously show that such asymptotics cannot hold near the layer midsection, where we construct another, internal asymptotic expansion. In the abovementioned sense, the solution corresponding to the internal expansion is also exact and models the compression of a thin vertical strip in the middle of the layer. We realize two possible versions of matching of the two expansions in the cross-section whose distance from the midsection is equal to the layer thickness.     

Nonlinear resonance of a subsatellite on a short constant tether

The nonlinear resonant behavior of a subsatellite on a short constant tether during station-keeping phase is investigated in this paper. The nonlinear dynamic equations of in-plane motion of the system are derived based on Kane’s method first. Then an approach of multiple scales expressed in matrix form is employed in solving the simplified nonlinear system of cubic nonlinearity near its local equilibrium position. Analysis shows that there exists a three-to-one resonance in such a nonlinear system with two degrees of freedom. Afterward, the approximate solution up to third order determined analytically by the Weierstrass elliptic function is obtained and the comparison between the approximate and numerical solutions presented as well. The results show that the approximate solution is coincide well with the numerical solution of original system. The nonlinear resonance of the subsatellite on short tether exhibits coexistent quasiperiodic motions or a quasiperiodic oscillation near local equilibrium position.