Strong stability of impulsive systems

The strong stability of the zero solution of impulsive systems with impulses at fixed moments of time is investigated. It is proved that the existence of piecewise continuous functions with certain properties is a necessary and sufficient condition for the strong stability of the zero solution of such systems. By using differential inequalities for piecewise continuous functions, sufficient conditions for the strong stability of the zero solution are found.     

Response functions for crystals and surfaces,with applications to surface scattering

A general solution of the equations of forced motion of a harmonic crystal or other vibrating system with arbitrary time-dependent forces acting on the atoms is given. The solution is given in terms of dynamical “response functions”, for which expressions in terms of the normal mode frequencies and eigenvectors (polarization vectors) are given. Numerical calculations of the response functions are described for (111) and (100) surfaces of face-centered cubic crystals interacting with Lennard-Jones 6–12 potentials, and the qualitative features of the surface and bulk response functions are discussed. The use of these functions in problems of atomic scattering from surfaces is outline, and conveneint parameterized forms for this application are given.     

Explicit spheroidal wave functions of the hydrogen atom

A simple and straightforward calculating scheme is suggested for finding wave functions of the hydrogen atom in prolate spheroidal coordinates. The wave functions are found in an explicit form by the direct solution of appropriate one-dimensional equations. The suggested calculating scheme allows us to carry out simple calculations and to obtain spheroidal wave functions in principle for arbitrary eigenstates of the hydrogen atom. Expansions are found for the obtained spheroidal wave functions over a spherical basis.     

Effects of Boundary Conditions and Relative Dimensions of a Solution System on Scaling Laws

A two-dimensional hydrodynamic model is employed to analyze the characteristics of crystal growth from solution with only the variation of the solution density caused by the temperature change taken into account. In that case all the characteristics of the solution system only depend on three numbers: the Rayleigh number Ra, the Prandtl number Pr and the Schemidt number Sc. In certain regions of the parameter (Ra, Pr and Sc) spaces, some scaling laws are generated: the scales of the temperature distribution index S_θ, the concentration distribution index S_ø (see text), the fluid velocity and the growth rate of crystal are given by power functions of Ra, Pr and Sc. The effects of the geometrical shape and the boundary condition of the solution system on the scaling laws are studied. When the ratio X of the height to the length of the solution system changes, the scaling laws are still valid and only the coefficients of power functions are changed, which are also power functions of A. The scaling laws are valid both under the isothermal temperature boundary condition and the adiabatic boundary condition at the surfaces of the top and the bottom sides of the solution system. The only difference is that the ratio of S_θ to Ra is greater for the latter than for the former. In certain ranges of Ra, there are no differences between the other power functions for the two cases.     

Periodic folded waves for a （2＋1）-dimensional modified dispersive water wave equation

A general solution, including three arbitrary functions, is obtained for a (2+1)-dimensional modified dispersive water-wave (MDWW) equation by means of the WTC truncation method. Introducing proper multiple valued functions and Jacobi elliptic functions in the seed solution, special types of periodic folded waves are derived. In the long wave limit these periodic folded wave patterns may degenerate into single localized folded solitary wave excitations. The interactions of the periodic folded waves and the degenerated single folded solitary waves are investigated graphically and found to be completely elastic.     

System of matrix-vector equations in the multiconfiguration Hartree-Fock method

A method for determining radial parts of one-electron functions entering the manyelectron basis is proposed within the multiconfiguration Hartree-Fock procedure. The solution of the Hartree-Fock equations is reduced to the solution of a system of matrix-vector equations. Rules of construction of these equations are formulated, and a stable numerical scheme of their solution is found.     

Non-singlet splitting functions in QED

Iterative solution of QED evolution equations for non-singlet electron structure functions is considered. Analytical expressions in the fourth and fifth orders are presented in terms of splitting functions. Relation to the existing exponentiated solution is discussed.     

Periodic Folded Wave Patterns for (2+1)-Dimensional Higher-Order Broer-Kaup Equation

A general solution including three arbitrary functions is obtained for the (2+1)-dimensional higher-order Broer-Kaup equation by means of WTC truncation method. Introducing proper multiple valued functions and Jacobi elliptic functions in the seed solution, special types of periodic folded waves are derived. In long wave limit these periodic folded wave patterns may degenerate into single localized folded solitary wave excitations. The interactions of the periodic folded waves and their degenerated single folded solitary waves are investigated graphically and are found to be completely elastic.     

The phase-integral method for radiative transfer problems with highly-peaked phase functions

Complete solutions to the radiative transfer equation, including both azimuth and depth dependence are provided by the discrete ordinate method of Chandrasekhar, but these solutions are often limited because of large computer requirements. This paper presents a “phase-integral” method which greatly reduces the number of discrete ordinates needed in the solution for highly-peaked phase functions. A composite quadrature method is shown to be effective in further reducing the number of discrete ordinates required for highly anisotropic phase functions. Examples are given to indicate convergence requirements and expected accuracy in the complete solution for Henyey-Greenstein and cloud-type phase functions.     

Novel loop-like solitons for generalized Vakhnenko equation

A nontraveling wave solution of a generalized Vakhnenko equation arising from the high-frequent wave motion in a relaxing medium is derived via the extended Riccati mapping method. The solution includes an arbitrary function of an independent variable. Based on the solution, two hyperbolic functions are chosen to construct new solitons. Novel single-loop-like and double-loop-like solitons are found for the equation.     

A DIRECT PERTURBATION THEORY OF THE NONLINEAR SCHR?DINGER EQUATION WITH CORRECTIONS

An exact direct perturbation theory of nonlinear Schrodinger equation with corrections is developed under the condition that the initial value of the perturbed solution is equal to the value of an exact multisoliton solution at a particular time. After showing the squared Jost functions are the eigenfunctions of the linearized operator with a vanishing eigenvalue,suitable definitions of adjoint functions and inner product are introduced. Orthogonal relations are derived and the expansion of the unity in terms of the squared Jost functions is naturally implied. The completeness of the squared Jost functions is shown by the generalized Marchenko equation. As an example,the evolution of a Raman loss compensated soliton in an optical fiber is treated.     

Novel loop-like solitons for the generalized Vakhnenko equation

A non-traveling wave solution of a generalized Vakhnenko equation arising from the high-frequent wave motion in a relaxing medium is derived via the extended Riccati mapping method.The solution includes an arbitrary function of an independent variable.Based on the solution,two hyperbolic functions are chosen to construct new solitons.Novel single-loop-like and double-loop-like solitons are found for the equation.     

General solutions of plane elasticity of one-dimensional orthorhombic quasicrystals with piezoelectric effect

By introducing four potential functions, the governing equations of plane problems in 1D orthorhombic quasicrystals with piezoelectric effect are composed of four second-order partial differential equations, in which the quasi-harmonic functions are the essential unknowns. The general solution of these equations is further established, and all expressions are expressed in terms of the potential functions. As an application of the general solution, the closed-form solutions are obtained for wedge problems or half-plane problems of 1D orthorhombic piezoelectric quasicrystals.     

Acoustics of a flanged cylindrical pipe using singular basis functions

The problem of acoustic radiation from a cylindrical pipe with an infinite flange has been discussed in a number of papers. The most common approach is to decompose the field inside the pipe over a basis of Bessel functions. A very large number of basis functions is usually required, with a large degree of ripple appearing as an artifact in the solution. In this paper it is shown that a close analysis of the velocity field near the corner yields a new family of functions, which are called "edge functions." Using this set of functions as test functions and applying the moment method on the boundary between the waveguide and free space, a solution is obtained with greatly improved convergence properties and no ripple.     

Spectral solvers for spherical elliptic problems

A new family of direct spectral solvers for the 3D Helmholtz equation in a spherical gap and inside a sphere for nonaxisymmetric problems is presented. A variational formulation (no collocation) is adopted, based on the Fourier expansion and the associated Legendre functions to represent the angular dependence over the sphere and using basis functions generated by Legendre or Jacobi polynomials to represent the radial structure of the solution. In the present method, boundary conditions on the polar axis and at the sphere center are not required and never mentioned, by construction. The spectral solution of the vector Dirichlet problem is also considered, by employing a transformation that uncouples the spherical components of the Fourier modes and that is implemented here for the first time. The condition numbers of the matrices involved in the scalar solvers are computed and the spectral convergence of all the proposed solution algorithms is verified by numerical tests.     

(2+1)维非线性Burgers方程变量分离解和新型孤波结构

利用变量分离方法，获得了(2+1)维非线性Burgers方程的变量分离解.由于在Bcklund变换和变量分离步骤中引入了作为种子解的任意函数, 因而精确解中含有三个任意函数(其中一个为条件函数),适当地选择任意函数，可以获得多种形状的扭状孤波解、周期性孤子解和格子型孤波解. 关键词： 变量分离解 非线性波方程 （2+1）维     

Mathematical Models for the Propagation of Stress Waves in Elastic Rods: Exact Solutions and Numerical Simulation

In this work, the Bishop and Love models for longitudinal vibrations are adopted to study the dynamics of isotropic rods with conical and exponential cross-sections. Exact solutions of both models are derived, using appropriate transformations. The analytical solutions of these two models are obtained in terms of generalised hypergeometric functions and Legendre spherical functions respectively. The exact solution of Love model for a rod with exponential cross-section is expressed as a sum of Gauss hypergeometric functions. The models are solved numerically by using the method of lines to reduce the original PDE to a system of ODEs. The accuracy of the numerical approximations is studied in the case of special solutions.     

Multiscale modeling of alloy solidification using a database approach

A two-scale model based on a database approach is presented to investigate alloy solidification. Appropriate assumptions are introduced to describe the behavior of macroscopic temperature, macroscopic concentration, liquid volume fraction and microstructure features. These assumptions lead to a macroscale model with two unknown functions: liquid volume fraction and microstructure features. These functions are computed using information from microscale solutions of selected problems. This work addresses the selection of sample problems relevant to the interested problem and the utilization of data from the microscale solution of the selected sample problems. A computationally efficient model, which is different from the microscale and macroscale models, is utilized to find relevant sample problems. In this work, the computationally efficient model is a sharp interface solidification model of a pure material. Similarities between the sample problems and the problem of interest are explored by assuming that the liquid volume fraction and microstructure features are functions of solution features extracted from the solution of the computationally efficient model. The solution features of the computationally efficient model are selected as the interface velocity and thermal gradient in the liquid at the time the sharp solid–liquid interface passes through. An analytical solution of the computationally efficient model is utilized to select sample problems relevant to solution features obtained at any location of the domain of the problem of interest. The microscale solution of selected sample problems is then utilized to evaluate the two unknown functions (liquid volume fraction and microstructure features) in the macroscale model. The temperature solution of the macroscale model is further used to improve the estimation of the liquid volume fraction and microstructure features. Interpolation is utilized in the feature space to greatly reduce the number of required sample problems. The efficiency of the proposed multiscale framework is demonstrated with numerical examples that consider a large number of crystals. A computationally intensive fully-resolved microscale analysis is also performed to evaluate the accuracy of the multiscale framework.     

New Results for Directed Vesicles and Chains near an Attractive Wall

In this paper we present new exact results for single fully directed walks and fully directed vesicles near an attractive wall. This involves a novel method of solution for these types of problems. The major advantage of this method is that it, unlike many other single-walker methods, generalizes to an arbitrary number of walkers. The method of solution involves solving a set of partial difference equations with a Bethe Ansatz. The solution is expressed as a “constant-term” formula which evaluates to sums of products of binomial coefficients. The vesicle critical temperature is found at which a binding transition takes place, and the asymptotic forms of the associated partition functions are found to have three different entropic exponents depending on whether the temperature is above, below, or at its critical value. The expected number of monomers adsorbed onto the surface is found to become proportional to the vesicle length at temperatures below critical. Scaling functions near the critical point are determined.     

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.