A STOCHASTIC GALERKIN METHOD FOR MAXWELL EQUATIONS WITH UNCERTAINTY

In this article, we investigate a stochastic Galerkin method for the Maxwell equations with random inputs. The generalized Polynomial Chaos(gPC) expansion technique is used to obtain a deterministic system of the gPC expansion coefficients. The regularity of the solution with respect to the random is analyzed. On the basis of the regularity results,the optimal convergence rate of the stochastic Galerkin approach for Maxwell equations with random inputs is proved. Numerical examples are presented to support the theoretical analysis.     

Singularly Perturbed Markov Chains with Two Small Parameters: A Matched Asymptotic Expansion

This work is concerned with asymptotic properties of solutions to forward equations for singularly perturbed Markov chains with two small parameters. It is motivated by the model of a cost-minimizing firm involving production planning and capacity expansion and a two-level hierarchical decomposition. Our effort focuses on obtaining asymptotic expansions of the solutions to the forward equation. Different from previous work on singularly perturbed Markov chains, the inner expansion terms are constructed by solving certain partial differential equations. The methods of undetermined coefficients are used. The error bound is obtained.     

A perturbation-boundary integral equation method for transient dynamics of nonlinear elastic materials

This study presents a formulation for solving the transient dynamics of nonlinear elastic materials. By using a perturbation expansion to linearize the basic equations and applying the Laplace transform to the subsequent perturbation equations, the boundary value problem of the transformed equations is further reduced to various boundary integral equations. After discretization of the integral equations, these are solved numerically, completing the solution in the Laplace transform space. Performing a numerical inversion of the Laplace transform yields the solution of the problem in the time domain.     

A new generalized compound Riccati equations rational expansion method to construct many new exact complexiton solutions of nonlinear evolution equations with symbolic computation

Based on symbolic computation and the idea of rational expansion method, a new generalized compound Riccati equations rational expansion method (GCRERE) is suggested to construct a series of exact complexiton solutions for nonlinear evolution equations. Compared with most existing rational expansion methods and other sophisticated methods, the proposed method not only recover some known solutions, but also find some new and general complexiton solutions. The validity and reliability of the method is tested by its application to the (2+1)-dimensional Burgers equation. It is shown that more complexiton solutions can be found by this new method.     

一类摄动超越方程的渐近解(英文)

用直接展开法得到了一类摄动超越方程的渐近解     

一类二维对偶积分方程的解及其应用

二维对偶积分方程的理论与方法,在数学上尚未建立,因而完全的分析解不可能得到,从而使一些力学、物理与工程问题无法求解．利用双重展开和边界配置方法,得到了在数学和物理学上有着广泛应用的一类二维对偶积分方程的解答．把二维对偶积分方程化简成无限代数方程组,此方法的精确度取决于计算点的配置（即所谓边界配置）．通过对固体力学中某些复杂的初值-边值问题的应用说明此是方法有效的．     

A Note on the Numerical Solution of Quasi-linear Parabolic Partial Differential Equations with Error Estimates

A method is considered for the numerical solution of quasi-linearpartial differential equations. The partial differential equationis reduced to a set of ordinary differential equations usinga Chebyshev series expansion. The exact solution of this setof ordinary differential equations is shown to be the solutionof a perturbed form of the original equation. This enables errorestimates to be found for linear and mildly non-linear problems.     

A Small Noise Asymptotic Expansion for Young SDE Driven by Fractional Brownian Motion: A Sharp Error Estimate With Malliavin Calculus

This article shows an analytically tractable small noise asymptotic expansion with a sharp error estimate for the expectation of the solution to Young’s pathwise stochastic differential equations (SDEs) driven by fractional Brownian motions with the Hurst index H > 1/2. In particular, our asymptotic expansion can be regarded as small noise and small time asymptotics by the error estimate with Malliavin culculus. As an application, we give an expansion formula in one-dimensional general Young SDE driven by fractional Brownian motion. We show the validity of the expansion through numerical experiments.     

A formulation to obtain semi-analytical planetary theories using true anomalies as temporal variables

The main methods used to obtain analytical theories of perturbed motion in celestial mechanics are based on the expansion of the disturbing function in trigonometric series of the mean anomalies (or longitudes). In this paper a new method based on the double Fourier series expansion using the true anomalies (or longitudes) is developed. The method involves a semi-analytical technique to allow the expansion of the inverse of the distance with great accuracy, and a new integration technique using a linear combination of the true anomalies based on an iterative method to integrate each term of the expansion of the Lagrange planetary equations.     

一类具非线性边值条件的非线性方程的奇摄动问题

研究了一类具非线性边值条件的非线性方程的奇摄动问题,运用合成展开法构造了问题的形式渐近解,并用微分不等式理论证明了所得渐近解的一致有效性.     

平面不可压缩的NAVIER-STOKES方程新五模类LORENZ方程组的混沌行为

本文研究了平面正方形区域上不可压缩的Navier-Stokes方程五模类Lorenz方程组的混沌行为问题.利用傅立叶展开方法对Navier-Stokes方程进行模式截断,获得了新五模类Lorenz方程组,给出了该方程组定常解及其稳定性的讨论,证明了该方程组吸引子的存在性,并对其全局稳定性进行了分析和讨论.     

一类积分方程的五解定理

利用H.Amann的一个不动点定理及锥拉伸锥压缩不动点定理讨论了一类Hammerstein型积分方程的正解,得到了一个五解定理.     

A time domain analysis of wave motions in chiral materials

A time domain analysis of the equations governing wave propagation in an unbounded chiral medium is presented by a spectral family approach and by a direct eigenfunction expansion using Beltrami–Moses fields.     

A note on the polynomial solution of a class of dual integral equations arising in mixed boundary value problems of elasticity

Summary The present paper is concerned with finding an effective polynomial solution to a class of dual integral equations which arise in many mixed boundary value problems in the theory of elasticity. The dual integral equations are first transformed into a Fredholm integration equation of the second kind via an auxiliary function, which is next reduced to an infinite system of linear algebraic equations by representing the unknown auxiliary function in the form of an infinite series of Jacobi polynomials. The approximate solution of this infinite system of equations can be obtained by a suitable truncation. It is shown that the unknown function involving the dual integral equations can also be expressed in the form of an infinite series of Jacobi polynomials with the same expansion coefficients with no numerical integration involved. The main advantage of the present approach is that the solution of the dual integral equations thus obtained is numerically more stable than that obtained by reducing themdirectly into an infinite system of equations, insofar as the expansion coefficients are determined essentially by solving asecond kind integral equation.     

获得非线性演化方程Backlund变换的一种新的途径

本文给出一种求非线性演化方程Backlund变换的方法,应用于非线性演化方程时,得到了与WTC方法一致的Backlund变换,避开了WTC方法涉及到的递推关系和截尾的讨论.     

A new semi-analytical collocation method for solving multi-term fractional partial differential equations with time variable coefficients

The aim of this paper is to present a new numerical method for solving a wide class of fractional partial differential equations (FPDEs) such as wave-diffusion equations, modified anomalous fractional sub-diffusion equations, time-fractional telegraph equations. The proposed method is based on the Fourier series expansion along the spatial coordinate which transforms the original equation into a sequence of multi-term fractional ordinary differential equations (ODEs). These fractional equations are solved by the use of a new efficient numerical technique – the backward substitution method. The numerical examples confirm the high accuracy and efficiency of the proposed numerical scheme in solving FPDEs with variable in time coefficients.     

A problem of dynamics of a thin inhomogeneous bar of the Kelvin-Foight material

A complete asymptotic expansion of a three-dimensional problem of the linear viscoelasticity determined in a domain being a thin inhomogeneous bar (the bar cross section diameter and the typical size of inhomogeneity are small parameters of the same order). Homogenized equations for oscillations of a bar are deduced. It is shown that those equations contain integral terms of convolution type.     

A new extended Riccati equation rational expansion method and its application

In this paper, a new extended Riccati equation rational expansion method is suggested to constructing multiple exact solutions for nonlinear evolution equations. The validity and reliability of the method is tested by its application to the dispersive long wave system and the Broer–Kaup–Kupershmidt system. The method can be applied to other nonlinear evolution equations in mathematical physics.     

A new periodic solution to Jacobi elliptic functions of MKdV equation and BBM equation

Based on the homogeneous balance method,the Jacobi elliptic expansion method and the auxiliary equation method,the first elliptic function equation is used to get a new kind of solutions of nonlinear evolution equations.New exact solutions to the Jacobi elliptic function of MKdV equations and Benjamin-Bona-Mahoney (BBM) equations are obtained with the aid of computer algebraic system Maple.The method is also valid for other (1+1)-dimensional and higher dimensional systems.     

A generalized inverse method for asymptotic linear programming

Consider a linear program in which the entries of the coefficient matrix vary linearly with time. To study the behavior of optimal solutions as time goes to infinity, it is convenient to express the inverse of the basis matrix as a series expansion of powers of the time parameter. We show that an algorithm of Wilkinson (1982) for solving singular differential equations can be used to obtain such an expansion efficiently. The resolvent expansions of dynamic programming are a special case of this method.