Accurate numerical resolution of transients in initial-boundary value problems for the heat equation

If the initial and boundary data for a PDE do not obey an infinite set of compatibility conditions, singularities will arise in the solution at the corners of the initial time–space domain. For dissipative equations, such as the 1-D heat equation or 1-D convection–diffusion equations, the impacts of these singularities are short lived. However, they can cause a very severe loss of numerical accuracy if we are interested in transient solutions. The phenomenon has been described earlier from a theoretical standpoint. Here, we illustrate it graphically and present a simple remedy which, with only little extra cost and effort, restores full numerical accuracy.     

A new approach with piecewise-constant arguments to approximate and numerical solutions of oscillatory problems

This paper is devoted to the development of a novel approximate and numerical method for the solutions of linear and non-linear oscillatory systems, which are common in engineering dynamics. The original physical information included in the governing equations of motion is mostly transferred into the approximate and numerical solutions. Therefore, the approximate and numerical solutions generated by the present method reflect more accurately the characteristics of the motion of the systems. Furthermore, the solutions derived are continuous everywhere with good accuracy and convergence in comparing with Runge-Kutta method. An approximate solution is developed for a linear oscillatory problem and compared with its corresponding exact solution. A non-linear oscillatory problem is also solved numerically and compared with the solutions of Runge-Kutta method. Both the graphical and numerical comparisons are provided in the paper. The accuracy of the approximate and numerical solutions can be controlled as desired by the number of terms in the Taylor series and the value of a single parameter used in the present work. Formulae for numerical computation in solving various linear and non-linear oscillatory problems by the new approach are provided in the paper.     

质点自由下落和上抛运动的精确解及偏移量和能量的讨论

基于质点自由下落和上抛在非惯性系下的动力学微分方程,得到质点坐标的精确解,然后获得两类问题的东西方向偏移量.得到了动能和势能的表达式,验证了运动过程中机械能守恒的特点.算例讨论了不同初始高度和不同初始速度下的偏移量值.     

Three-dimensional numerical simulation of crown spike due to coupling effect between bubbles and free surface

The motion of gas bubbles beneath a free surface will lead to a spike of fluid on the free surface. The distance of the bubbles to the free surface is the key factor to different phenomena. When the inception distance varies in some range, crown phenomenon would happen after the impact of weak buoyancy bubbles, so this kind of spike is defined as crown spike in the present paper. Based on potential flow theory, a three-dimensional numerical model is established to simulate the motion of the free-surface spike generated by one bubble or a horizontal line of two in-phase bubbles. After the downward jet formed near the end of the collapse phase, the simulation of the free surface is performed to study the crown spike without regard to the toroidal bubble＇s effect. Calculations about the interaction between one bubble and free surface agree well with the experimental results conducted with a high-speed camera, and relative error is within 15%. Crown spike in both single- and two-bubble cases are simulated numerically. Different features and laws of the motion of crown spike, depending on the bubble-boundary distances and the inter-bubble distances, have been investigated.     

基于微分方程数值解的自由曲面透镜的设计

为满足目标面上均匀照明的需求,设计了一款自由曲面透镜。根据LED光源的发光特点,结合光学成像的特性,非成像光学原理和能量守恒定理,推导出实现光能量在目标面上均匀分布的自由曲面面形的微分方程,采用matlab的ode算法求出面形上的离散坐标点,对离散坐标点拟合后得到透镜模型,通过光学模拟仿真软件对透镜模型进行光线追迹。结果表明,配光角度为80°的透镜,透镜的口径与光源发光面宽度之比大于等于9时,目标面上的照度均匀性大于0.9,光能利用率约为85%。该设计方法可以快速精确地设计出所需的透镜,而且透镜结构紧凑,单颗就能实现均匀照明,有利于LED光源照明系统的小型化。     

Analytic and numerical solutions to the Lane-Emden equation

In this Letter, we present analytical solutions to the Lane-Emden equation describing the thermal behavior of a spherical cloud of gas acting under the mutual attraction of its molecules. Solutions are obtained by using the traditional power series approach and by using the Homotopy Analysis Method (HAM). We show that the series solutions obtained by the Homotopy Analysis Method converge in a larger interval than in the case of the corresponding traditional series solutions. Furthermore, we obtained numerical solutions (using Runge-Kutta-Fehlberg 4-5 technique) which are used to validate the analytical solutions.     

Polyelectrolytes tethered to a free surface

Several attempts have been already carried out in order to tether charged chains by an end at a free fluctuating surface. We review here most of these attempts and focus on how close the physics of charged brushes can be investigated by such an approach. We first describe results about films of charged-neutral diblock copolymers spread at the surface of water. Results can be mostly rationalized in terms of charged brushes although additional structurations and fluctuations of the interface can be observed. The latter deformations are also observed when adsorbed layers of charged-neutral diblock copolymers are considered. At last, we examine how free suspended films of charged-neutral diblock copolymers can be viewed as two opposing charged brushes, both in terms of thickness and pressure. Received 9 May 2000     

Counting symmetry-breaking solutions to symmetric variational problems

By combining Michel's geometric theory of symmetry breaking and classical results from variational analysis, we obtain a lower bound on the number of critical points with given symmetryH G of a potential symmetric underG. The result is obtained by applying the Ljusternik-Schnirelman category in the group orbit space, and can be extended along the same lines to more general situations.     

Direct analytical solutions to non-uniform beam problems

The direct analytical solution to the vibration of non-uniform beams with and without discontinuities and with various boundary conditions is presented. Results are compared to results from the exact solution for certain cases where the exact solution has been obtained. It is shown that the direct solution converges to the exact solution, in fact, with “indefinite accuracy” just as Hamilton stated that it would.     

Characteristics of the capillary motions of surfactant solutions with a charged free surface

Abastract A dispersion relation is derived and analyzed for the spectrum of capillary motions on the charged plane surface of a liquid in which a surfactant is dissolved. It is shown that two additional wave motions are generated in this kind of system by bulk diffusion and surface diffusion of the surfactant and are sensitive to the diffusion coefficients and elastic properties of the surfactant films and to the viscosity of the solution and the presence of a surface charge. In solutions of inactive surfactants the growth rate of Tonks-Frenkel instability increases as the surfactant concentration increases. Zh. Tekh. Fiz. 68, 22–29 (February 1998)     

Exact and numerical solutions to the perturbed sine-Gordon equation

We present numerical and analytic solutions to the perturbed sine-Gordon equation, which models long Josephson tunnel junctions. We make comparisons between numerical results and results obtained from perturbational methods. We present unstable, analytic kink solutions to the equation and further a solution, which is an array of kinks, corresponding to a solution, where the current through the junction is larger than the critical current.     

Transformation of sound-pressure level from the free field to the eardrum presented in numerical form

In an earlier work [E. A. G. Shaw, J. Acoust. Soc. Am. 56, 1848-1861 (1974)], self-consistent families of curves which best fitted the experimental data from 12 studies were presented to show the average sound-pressure transformation from the free field to the human eardrum in the horizontal plane. Tabulated values of azimuthal dependence at 15-deg intervals in azimuth and transformation to the eardrum at 0 degree azimuth are given at 43 selected frequencies between 0.2 and 12 kHz. These values are self-consistent within 0.1 dB.     

Accurate non-perturbative solution of eigenvalue problems with application to anharmonic oscillator

A method for eigenvalue problems is presented. As an example, we have obtained very accurate eigenvalues and eigenfunctions of the quartic anharmonic oscillator.The method is non-perturbative and involves the use of an appropriately scaled set of basis functions for the determination of each eigenvalue. The claimed accuracy for all eigenvalues is 15 significant figures. The method does not deteriorate for higher eigenvalues.     

A new class of stationary solutions to the five-dimensional Kaluza-Klein field equations

Using the five-dimensional potential formalism, a set of six new stationary axisymmetric solutions to the Kaluza-Klein field equations are constructed from the one dimensional subspace of the potential space. All the solutions have scalar potential and magnetic field, two of them possess magnetic monopoles and the other describes a magnetic dipole.     

Local solutions to high-frequency 2D scattering problems

We consider the solution of high-frequency scattering problems in two dimensions, modeled by an integral equation on the boundary of a smooth scattering object. We devise a numerical method to obtain solutions on only parts of the boundary with little computational effort. The method incorporates asymptotic properties of the solution and can therefore attain particularly good results for high frequencies. We show that the integral equation in this approach reduces to an ordinary differential equation.     

Effect of plasticizer on surface of free films prepared from aqueous solutions of salts of cationic polymers with different plasticizers

Acquisition of a more detailed understanding of all technological processes is currently a relevant tendency in pharmaceutical technology and hence in industry. A knowledge of film formation from dispersion of polymers is very important during the coating of solid dosage forms. This process and the structure of the film can be influenced by different additives. In the present study, taste-masking films were prepared from aqueous citric acid solutions of a cationic polymer (Eudragit^® E PO) with various hydrophilic plasticizers (glycerol, propylene glycol and different poly(ethylene glycols)). The mechanical properties, film thickness, wetting properties and surface free energy of the free films were studied. The aim was to evaluate the properties of surface of free films to predict the arrangement of macromolecules in films formed from aqueous solutions of salts of cationic polymers. A high molecular weight of the plasticizer decreased the work of deformation. The surface free energy and the polarity were highest for the film without plasticizer; the hydrophilic additives decreased these parameters. The direction of the change in polarity (a hydrophilic component caused a decrease in the polarity) was unexpected. It can be explained by the change in orientation of the macromolecules, a hydrophobic surface being formed. Examination of the mechanical properties and film thickness can furnish additional results towards a knowledge of film formation by this not frequently applied type of polymer from aqueous solution.     

Number of elastic scatterings on free stationary nuclei to slow down a neutron

We report the formulation of the number of elastic scatterings required to slow down a neutron. By establishing its analytical expression, we show that this number displays a discontinuity and an oscillatory transient that progressively dampens when the neutron energy decreases. This result does not apply to neutrons with energies lower than a few eV, as we restrict our study to scatterings on free stationary nuclei.