A class of singular perturbation solutions to semilinear equations of fourth order

A class of singularly perturbed boundary value problems for semilinear equations of fourth order with two parameters are considered. Under suitable conditions, using the method of lower and upper solutions, the existence and the asymptotic behavior of the solution to the boundary value problem are studied, In the present paper, the solution to the original singularly perturbed problem with two parameters has only one boundary layer.     

Solitary waves in a viscous liquid film flowing down a thin vertical cylinder

Several equations to describe the flow of a viscous liquid film on a thin cylinder are derived. The solitary-wave solutions to these equations are studied. The families of solutions are constructed for the first two eigenvalues that correspond to single-humped and double-humped waves. It is found that these families become similar as the similarity parameter increases. The dependencies of phase velocities and wave amplitudes on the free parameters of the problem are analyzed. The resulting solutions are compared with solitary waves in films on a flat surface.     

优化迭代步长的两种改进增量谐波平衡法

增量谐波平衡法(IHB法)是一个半解析半数值的方法, 其最大优点是适合于强非线性系统振动的高精度求解. 然而, IHB法与其他数值方法一样, 也存在如何选择初值的问题, 如初值选择不当, 会存在不收敛的情况. 针对这一问题, 本文提出了两种基于优化算法的IHB法: 一是结合回溯线搜索优化算法(BLS)的改进IHB法(GIHB1), 用来调节IHB法的迭代步长, 使得步长逐渐减小满足收敛条件; 二是引入狗腿算法的思想并结合BLS算法的改进IHB法(GIHB2), 在牛顿-拉弗森(Newton-Raphson)迭代中引入负梯度方向, 并在狗腿算法中引入2个参数来调节BSL搜索方式用于调节迭代的方式, 使迭代方向沿着较快的下降方向, 从而减少迭代的步数, 提升收敛的速度. 最后, 给出的两个算例表明两种改进IHB法在解决初值问题上的有效性.      

Asymptotic solutions of the two-dimensional equations for a thin viscous shock layer in a rarefied gas

Two-dimensional hypersonic rarefied gas flow around blunt bodies is investigated for the continuum to free-molecular transition regime. In [1], as a result of an asymptotic analysis, three rarefied gas flow regimes, depending on the relationship between the problem parameters, were detected and one of these regimes was investigated. In the present study, asymptotic solutions of the thin viscous shock layer equations at small Reynolds numbers are obtained for the other two flow regimes. Analytical expressions for the heat transfer, friction and pressure coefficients are obtained as functions of the incident flow parameters and the body geometry and temperature. As the Reynolds number tends to zero, the values of these coefficients approach their values in free-molecular flow. The scaling parameters of hypersonic rarefied gas flow around bodies are determined for different regimes. The asymptotic solutions are compared with the results of direct Monte Carlo simulation.     

High‐resolution,monotone solution of the adjoint shallow‐water equations

A monotone, second‐order accurate numerical scheme is presented for solving the differential form of the adjoint shallow‐water equations in generalized two‐dimensional coordinates. Fluctuation‐splitting is utilized to achieve a high‐resolution solution of the equations in primitive form. One‐step and two‐step schemes are presented and shown to achieve solutions of similarly high accuracy in one dimension. However, the two‐step method is shown to yield more accurate solutions to problems in which unsteady wave speeds are present. In two dimensions, the two‐step scheme is tested in the context of two parameter identification problems, and it is shown to accurately transmit the information needed to identify unknown forcing parameters based on measurements of the system response. The first problem involves the identification of an upstream flood hydrograph based on downstream depth measurements. The second problem involves the identification of a long wave state in the far‐field based on near‐field depth measurements. Copyright © 2002 John Wiley & Sons, Ltd.     

Slip effects on the peristaltic flow of a Jeffrey fluid in an asymmetric channel under the effect of induced magnetic field

In the present study, we investigated the effects of slip and induced magnetic field on the peristaltic flow of a Jeffrey fluid in an asymmetric channel. The governing two‐dimensional equations for momentum, magnetic force function and energy are simplified by using the assumptions of long wavelength and low but finite Reynolds number. The reduced problem has been solved by Adomian decomposition method (ADM) and closed form solutions have been presented. Further, the exact solution of the proposed problem has also been computed and the mathematical comparison shows that both solutions are almost similar. The effects of pertinent parameters on the pressure rise per wavelength are investigated using numerical integration. The expressions for pressure rise, friction force, velocity, temperature, magnetic force function and the stream lines against various physical parameters of interest are shown graphically. Moreover, the behavior of different kinds of wave shape are also discussed. Copyright © 2009 John Wiley & Sons, Ltd.     

Dynamic behavior of two unequal parallel permeable interface cracks in a piezoelectric layer bonded to two half piezoelectric materials planes

IntroductionDuetotheintrinsicelectro_mechanicalcouplingbehavior,piezoelectricmaterialsareveryusefulinelectronicdevices.However,mostpiezoelectricmaterialsarebrittlesuchasceramicsandcrystals.Therefore ,piezoelectricmaterialshaveatendencytodevelopcriticalcracksduringthemanufacturingandthepolingprocesses.So ,itisimportanttostudytheelectro_elasticinteractionandfracturebehaviorsofpiezoelectricmaterials.Theincreasingattentiontothestudyofcrackproblemsinpiezoelectricmaterialshasledtoalotofsignificantw…     

Theoretical study of the limiting regimes of hydrate formation during contact of gas and water

This paper presents a mathematical model and analytical solutions of the problem of the growth of a hydrate layer during contact of gas and water for two limiting regimes of gas hydrate formation determined by mass transfer and heat transfer. Critical values are obtained for thermal parameters and parameters that determine the flow properties of the hydrate layer (diffusion coefficient and permeability), on which the hydrate formation regime depends.     

Initial-value problem for coupled Boussinesq equations and a hierarchy of Ostrovsky equations

We consider the initial-value problem for a system of coupled Boussinesq equations on the infinite line for localised or sufficiently rapidly decaying initial data, generating sufficiently rapidly decaying right- and left-propagating waves. We study the dynamics of weakly nonlinear waves, and using asymptotic multiple-scale expansions and averaging with respect to the fast time, we obtain a hierarchy of asymptotically exact coupled and uncoupled Ostrovsky equations for unidirectional waves. We then construct a weakly nonlinear solution of the initial-value problem in terms of solutions of the derived Ostrovsky equations within the accuracy of the governing equations, and show that there are no secular terms. When coupling parameters are equal to zero, our results yield a weakly nonlinear solution of the initial-value problem for the Boussinesq equation in terms of solutions of the initial-value problems for two Korteweg-de Vries equations, integrable by the Inverse Scattering Transform. We also perform relevant numerical simulations of the original unapproximated system of Boussinesq equations to illustrate the difference in the behaviour of its solutions for different asymptotic regimes.     

Real solutions of the direct geometrico-static problem of under-constrained cable-driven parallel robots with 3 cables: a numerical investigation

This paper addresses the direct geometrico-static problem of under-constrained cable-driven parallel robots with 3 cables. The task at hand consists in finding all equilibrium configurations of the end-effector when the cable lengths are assigned. This problem is known to admit 156 solutions in the complex field, but the upper bound on the number of real solutions is as yet an open issue. Finding this bound is the objective of the paper. For this purpose, three numerical approaches are developed, namely a continuation procedure adapted from an algorithm originally proposed by Dietmaier and two evolutionary techniques based on a genetic algorithm and particle swarm optimization. In all cases, a number of sets of robot parameters for which the direct geometrico-static problem provides at the most 54 real configurations is found. The coherence of the obtained results leads to conjecture that the achieved bound is tight. However, formal proof is yet to be discovered.     

Multi-parameter aerodynamic modeling for aeroelastic coupling in turbomachinery

A multi-parameter method based on multivariate spline function approximation and minimum state smoothing for modeling the generalized aerodynamic forces is proposed in order to reduce the cost of the aerodynamic computations in the solution of the coupled fluid–structure problem in turbomachinery. This method allows simultaneous variations of several parameters and provides the solutions of the coupled systems at arbitrary values of the parameters using the generalized aerodynamic forces computed at the few values of the parameters. This multi-parameter aerodynamic modeling method is applied to a large-chord blade, for which the two chosen parameters are the rotation speed and the inter-blade phase angle.     

MHD flow of a viscous fluid on a nonlinear porous shrinking sheet with homotopy analysis method

The present paper investigates the magnetohydrodynamic（MHD） flow of a viscous fluid towards a nonlinear porous shrinking sheet.The governing equations are simplified by similarity transformations.The reduced problem is then solved by the homotopy analysis method.The pertinent parameters appearing in the problem are discussed graphically and presented in tables.It is found that the shrinking solutions exist in the presence of MHD.It is also observed from the tables that the solutions for f（0） with different values of parameters are convergent.     

Similarity solutions of energy and momentum boundary layer equations for a power-law shear driven flow over a semi-infinite flat plate

The problem of a steady forced convection thermal boundary-layer driven by a power-law shear is investigated. The search for similarity solutions reduces the problem to a couple of ordinary differential equations containing three parameters: the exponent of the decaying exterior velocity profile, the exponent of the power-law prescribing the thermal condition on the wall and Prandtl number. The effects of these parameters on the existence and form of similarity solution are investigated and the functional dependence of the local Nusselt number on these parameters is reported and discussed. An analysis of the assumptions usually accepted to derive similarity solutions is also reported in order to show the range of values of the exterior velocity power-law exponent for which such solutions may exist.     

Stability of an equilibrium position of a pendulum with step parameters

A mathematical pendulum affected by parametric disturbance with potential energy being periodic step function is considered. Non-linear equation of the pendulum depends on two parameters characterizing the mean value in time of the parametric disturbance and range of its “ripple”. Values of the parameters can be set arbitrarily. The non-linear problem of stability for two particular solutions of the equation corresponding to a hanging and inverse pendulum is solved.     

Buckling and nonlinear dynamics of elastically coupled double-beam systems

This paper deals with damped transverse vibrations of elastically coupled double-beam system under even compressive axial loading. Each beam is assumed to be elastic, extensible and supported at the ends. The related stationary problem is proved to admit both unimodal (only one eigenfunction is involved) and bimodal (two eigenfunctions are involved) buckled solutions, and their number depends on structural parameters and applied axial loads. The occurrence of a so complex structure of the steady states motivates a global analysis of the longtime dynamics. In this regard, we are able to prove the existence of a global regular attractor of solutions. When a finite set of stationary solutions occurs, it consists of the unstable manifolds connecting them.     

Wave Regimes of Viscous Film Flow down a Vertical Cylinder

The flow of a viscous liquid film down a vertical cylinder in the gravity field is considered. In the case of small Reynolds numbers for long-wave perturbations on a cylinder of radius much greater than the film thickness, the problem can be reduced to a single nonlinear equation for the evolution of the film thickness perturbation. For axially symmetric solutions, this equation coincides with the well-known Sivashinsky-Kuramoto equation. The results of a numerical analysis of this equation for three-dimensional stationary traveling solutions of the problem are reported. The effect of the problem parameters on the solution behavior is demonstrated. Soliton type solutions are presented.     

Nonlinear Spherical Caps and Associated Plate and Membrane Problems

This paper deals with the existence and multiplicity problem of the equilibrium solutions of an elastic spherical cap within nonlinear strain theory. We pose the problem in the form of a three parameter bifurcation problem, one parameter being related to the load, the others to the geometry. When the geometrical parameters are different from zero, the so-called generic case, we revisit the nonuniqueness results, and explore the solutions in the parameter space. Then we analyze the formal limits as the geometrical parameters tend to zero. When the curvature tends to zero, we obtain from the nonlinear shell a von Kármán plate, a new, although natural, result. When the thickness parameter tends to zero, we get a nonlinear membrane problem. A study of the latter gives infinitely many solutions, and we discuss the construction, shapes, and stability in detail. This revised version was published online in August 2006 with corrections to the Cover Date.     

Two-dimensional problem of electrochemical metal machining

The two-dimensional problem of the electrochemical dimensional machining of a metal is investigated within the framework of the model of an ideal stationary process, which makes it possible to use the analogy with the problems of fluid flows with free surfaces. In the problem considered the cathode (machining tool) takes the form of two parallel semi-infinite rectangular electrodes. The blank (anode) is a half-plane whose boundary is perpendicular to the cathodes. Depending on the relationship between the physical and geometrical parameters of the problem, on the machined part (anode) a projection symmetrical about the center line between the cathodes may be formed. Additional mechanical machining of the part is then required. In order to exclude such solutions, a condition is obtained for the mathematical parameters which determine the solution of the problem in the auxiliary complex plane. General and particular limiting cases are considered. For the cases considered the calculation results are presented in the form of plots of the shape of the part machined.     

Shakedown of a cohesive-frictional half-space subjected to rolling and sliding contact

The problem of rolling and sliding contact of a cylinder on the surface of a half-space of cohesive-frictional material is considered. Three shakedown multipliers, of which two are upper bounds and one is exact are computed using a simple numerical procedure. This latter solution differs significantly from previously published analytical solutions which, for realistic material parameters, typically overestimate the shakedown load by a factor of 1.5–2.5.     

流固耦合介质轴对称动力问题解法的改进

用直接求解常微分方程组解文［１］所得的控制方程，减少了传递矩阵计算工作量，避免了子阵求逆，使问题的求解得到了简化