Vibrations of double-string complex system under moving forces. Closed solutions

In this paper the dynamic response of a double-string system traversed by a constant or a harmonically oscillating moving force is considered. The force is moving with a constant velocity on the top string. The strings are identical, parallel, one upon the other and continuously coupled by a linear Winkler elastic element. The classical solution of the response of a double-string system subjected to a force moving with a constant velocity has a form of an infinite series. The main goal of this paper is to show that in the considered case a part of the solution can be presented in a closed, analytical form instead of an infinite series. The presented method of finding the solution in a closed, analytical form is based on the observation that the solution of the system of partial differential equations in the form of an infinite series is also a solution of an appropriate system of ordinary differential equations.     

Toward Analytic Solution of Nonlinear Differential Difference Equations via Extended Sensitivity Approach

In this paper an efficient computational method based on extending the sensitivity approach(SA) is proposed to find an analytic exact solution of nonlinear differential difference equations.In this manner we avoid solving the nonlinear problem directly.By extension of sensitivity approach for differential difference equations(DDEs),the nonlinear original problem is transformed into infinite linear differential difference equations,which should be solved in a recursive manner.Then the exact solution is determined in the form of infinite terms series and by intercepting series an approximate solution is obtained.Numerical examples are employed to show the effectiveness of the proposed approach.     

Bäcklund transformations and exact solutions of the stationary axially symmetric Einstein equations

The Bäcklund transformations of the self-dual Yang-Mills fields are used to derive exact solutions of the stationary axially symmetric Einstein equations. Three solutions which belong to an infinite series of such solutions are explicitly calculated, the first of which is the Lewis solution. Also the first of another infinite series of solutions is calculated, which for some value of a parameter becomes the Papapetrou solution.     

Transverse magnetic scattering of two incident plane waves by a dielectric coated cylindrical reflector

Scattering characteristics of two plane waves are investigated for a circular cylinder covered by a dielectric substance. Fields are assumed to be transverse magnetic (TM) and represented in an exponential series form. The diffracted radiations are found by applying the boundary conditions to the wave functions. The wave transformation method and the orthogonality of the exponential functions are respectively employed to obtain an infinite series in the solution. Numerical results are evaluated by reducing the infinite series to a finite number of terms and comparing estimates with the single plane wave scattering situation.     

一种求解锂离子电池单粒子模型液相扩散方程的新方法

电解液中的锂离子浓度表达是锂离子电池电化学模型求解的基本任务之一.为了平衡单粒子模型的液相动态性能和计算效率,假设反应仅发生在集电极和电解质界面上,为此,提出一种基于液相扩散方程无穷级数解析解的界面浓度求解新方法.在恒流工况下,利用数列单调收敛准则将解析解转化为一个收敛和函数.在动态工况下,将该解析解简化为输入与和函数的无限离散卷积.利用和函数随时间单调衰减并收敛至零的特性对其进行截断,从而得到有限离散卷积求解算法.对比专业有限元分析软件,该方法在恒流工况和动态工况下均能以较少的计算时间获得相当好的精度.而且,该方法仅有一个配置参数.因此,所提方法将有效减小应用于实时电池管理系统上的电化学模型计算负担.     

一种求解锂离子电池单粒子模型液相扩散方程的新方法

电解液中的锂离子浓度表达是锂离子电池电化学模型求解的基本任务之一.为了平衡单粒子模型的液相动态性能和计算效率,假设反应仅发生在集电极和电解质界面上,为此,提出一种基于液相扩散方程无穷级数解析解的界面浓度求解新方法.在恒流工况下,利用数列单调收敛准则将解析解转化为一个收敛和函数.在动态工况下,将该解析解简化为输入与和函数的无限离散卷积.利用和函数随时间单调衰减并收敛至零的特性对其进行截断,从而得到有限离散卷积求解算法.对比专业有限元分析软件,该方法在恒流工况和动态工况下均能以较少的计算时间获得相当好的精度.而且,该方法仅有一个配置参数.因此,所提方法将有效减小应用于实时电池管理系统上的电化学模型计算负担.     

Solution to the MHD flow over a non-linear stretching sheet by homotopy perturbation method

In this study,by means of homotopy perturbation method(HPM) an approximate solution of the magnetohydrodynamic(MHD) boundary layer flow is obtained.The main feature of the HPM is that it deforms a difficult problem into a set of problems which are easier to solve.HPM produces analytical expressions for the solution to nonlinear differential equations.The obtained analytic solution is in the form of an infinite power series.In this work,the analytical solution obtained by using only two terms from HPM soluti...     

Scattering of a Gaussian beam by an anisotropically coated circular cylinder

A solution to the problem of Gaussian beam scattering by an anisotropically coated circular cylinder is presented. The incident Gaussian beam source is expanded as an approximate expression in the simple form with a Tayor’s series. The transmitted field in the anisotropically coated region is expressed as a infinite summation of eigen plane waves with different polar angles. The unknown coefficients of the scattered fields are obtained with the aid of the boundary conditions. The infinite series can be truncated under the prerequisite of achieving the solution convergence. Only the case of transverse-electric polarization is discussed. The similar formulation of transverse-magnetic polarization can be obtained by adopting the similar method. Some numerical results are presented and discussed. The result is in agreement with that available as expected when the Gaussian beam degenerates to a plane wave incidence case.     

Exact solution for guided sound transmission through a double partition

The exact solution is obtained for the problem of the transmission of a symmetric sound wave through a double partition contained in a parallel-plane waveguide with rigid walls. A membrane model is used for the two leaves of the partition. The solution involves a number of infinite series which converge rapidly. It is not necessary to make any special assumptions regarding the parameters describing the model. In particular no modifications are necessary for the region of the critical frequencies. Full details are given for the particular case of an incident plane wave. The result, which is suitable for computation, involves four converging infinite series. In general there are no frequencies of complete transmission or reflection. Similar analysis applies to the case of a circular waveguide.     

Series Solution for Unsteady Boundary-Layer Flows Due to Impulsively Stretching Plate

The third-order nonlinear partial differential equation modelling the unsteady boundary-layer flows caused by an impulsively stretching fiat plate is solved by using the Adomian decomposition method （ADM）. The ADM yields analytic solution in the form of a rapidly convergent infinite series with easily computable terms. The series solution using the ADM for the unsteady flows is presented for the first time.     

Periodic contact between piezoelectric materials and a rigid body with a wavy surface

An exact analysis is conducted for periodic, two-dimensional (2D) contact of piezoelectric materials in contact with a rigid body with a wavy surface pressed by uniform stresses at infinity. For three cases of eigenvalue distribution, three harmonic functions automatically satisfying the periodicity conditions are carefully constructed to facilitate the derivation of the solution of the considered problem. The stresses and electric displacements are obtained as infinite series. It is found that for the full contact case, the disturbance stress and electric displacement fields remain only the first harmonic which has the slowest decay in the y-direction. The convergence behaviours of the infinite series are checked, which shows that the external loading p and different positions have a great effect on the convergence behaviours of the infinite series and 400 terms are enough to get accurate solution at each position. Numerical results are presented to justify the validity of the present derivation and show the effect of the external loading on the contact behaviours.     

An Efficient Semi-Analytical Method to Compute Displacements and Stresses in an Elastic Half-Space with a Hemispherical Pit

This paper presents an efficient method to calculate the displacement and stress fields in an isotropic elastic half-space having a hemispherical pit and being subject to gravity. The method is semi-analytical and takes advantage of the axisymmetry of the problem. The Boussinesq potentials are used to obtain an analytical solution in series form, which satisfies the equilibrium equations of elastostatics, traction-free boundary conditions on the infinite plane surface and decaying conditions at infinity. The boundary conditions on the free surface of the pit are then imposed numerically, by minimising a quadratic functional of surface elastic energy. The minimisation yields a symmetric and positive definite linear system of equations for the coefficients of the series, whose particular block structure allows its solution in an efficient and robust way. The convergence of the series is verified and the obtained semi-analytical solution is then evaluated, providing numerical results. The method is validated by comparing the semi-analytical solution with the numerical results obtained using a commercial finite element software.     

共振辐射的理论研究——Holstein方程的解

解决介质中共振辐射的关键问题是求解Holstein方程。本文直接利用算符运算将方程积分,给出了激发态粒子集居数的含无限项的级数表达式。基于对这个级数高阶系数的极限行为的研究,成功地克服了无穷项求和的困难,整理出一个在任意给定精度下都只含有限项的解式,由它可以计算出任何时刻的结果。 关键词：     

(2+1)维改进的Zakharov-Kuznetsov方程的无穷序列复合型类孤子新解

对(G′/G)展开法做了进一步的研究,利用两次函数变换将二阶非线性辅助方程的求解问题转化为一元二次代数方程与Riccati方程的求解问题.借助Riccati方程的B?cklund变换及解的非线性叠加公式获得了辅助方程的无穷序列解.这样,利用(G′/G)展开法可以获得非线性发展方程的无穷序列解,这一方法是对已有方法的扩展,与已有方法相比可获得更丰富的无穷序列解.以(2+1)维改进的Zakharov-Kuznetsov方程为例得到了它的无穷序列新精确解.这一方法可以用来构造其他非线性发展方程的无穷序列解.     

Series solution of unsteady free convection flow with mass transfer along an accelerated vertical porous plate with suction

The problem of unsteady free convection flow is considered for the series solution (analytic solution). The flow is induced by an infinite vertical porous plate which is accelerated in its own plane. The series solution expressions for velocity field, temperature field and concentration distribution are presented. The influence of important parameters is seen on the velocity, temperature, concentration, skin friction coefficient and temperature gradient with the help of graphs and tables. Convergence is also properly checked for different values of the important parametes for velocity field, temperature and concentration with the help of ħ-curves.     

无穷级数求和的一种量子力学解法

在一维无限深方势阱的解析解的基础上,利用波函数的归一化常数及能量平均值的两种不同算法的等价性,导出了∑n=1^∞ 1/n^2、∑n=1^∞ 1/n^4等24个无穷级数的求和公式。     

Scattering by cavities of arbitrary shape in an infinite plate and associated vibration problems

This paper presents a solution for the displacement of a uniform elastic thin plate with an arbitrary cavity, modelled using the biharmonic plate equation. The problem is formulated as a system of boundary integral equations by factorizing the biharmonic equation, with the unknown boundary values expanded in terms of a Fourier series. At the edge of the cavity we consider free-edge, simply-supported and clamped boundary conditions. Methods to suppress ill-conditioning which occurs at certain frequencies are discussed, and the combined boundary integral equation method is implemented to control this problem. A connection is made between the problem of an infinite plate with an arbitrary cavity and the vibration problem of an arbitrarily shaped finite plate, using the jump discontinuity present in single-layer distributions at the boundary. The first few frequencies and modes of displacement are computed for circular and elliptic cavities, which provide a check on our numerics, and results for the displacement of an infinite plate are given for four specific cavity geometries and various boundary conditions.     

On Chemisorption of Polymers to Solid Surfaces

The irreversible adsorption of polymers to a two-dimensional solid surface is studied. An operator formalism is introduced for chemisorption from a polydisperse solution of polymers which transforms the analysis of the adsorption process to a set of combinatorial problems on a two-dimensional lattice. The time evolution of the number of polymers attached and the surface area covered are calculated via a series expansion. The dependence of the final coverage on the parameters of the model (i.e. the parameters of the distribution of polymer lengths in the solution) is studied. Various methods for accelerating the convergence of the resulting infinite series are considered. To demonstrate the accuracy of the truncated series approach, the series expansion results are compared with the results of stochastic simulation.     

Classification of an infinite series of static solutions to the vacuum Einstein equations

We propose a classification of an infinite series of the static solutions to the vacuum Einstein equation by means of their soliton number and the real and/or complex pole trajectories. We also show that in the 4-soliton solution there appears an intriguing ring solution of which the curvature invariant has a finite limit at the rings.     

Exact L series solution of the Dirac-Coulomb problem for all energies

We obtain exact solution of the Dirac equation with the Coulomb potential as an infinite series of square integrable functions. This solution is for all energies, the discrete as well as the continuous. The spinor basis elements are written in terms of the confluent hypergeometric functions and chosen such that the matrix representation of the Dirac-Coulomb operator is tridiagonal. The wave equation results in a three-term recursion relation for the expansion coefficients of the wavefunction which is solved in terms of the Meixner-Pollaczek polynomials.