Existence of the Global Smooth Solution to the Period Boundary Value Problem of Fractional Nonlinear Schrodinger Equation

It is well known that Feynman and Hibbs[1] used path integrals over Brownian paths to derive the standard(nonfractional) Schrodinger equation. Recently, Laskin[5, 6] showed that the path integral over the Lévy-like quantum mechanical paths allows to develop the generalization of the quantum mechanics. Namely, if the path integral over Brownian trajectories leads to the well known Schrodinger equation, then the path integral over Lévy trajectories leads to the fractional Schrodinger equation. Laskin[7] showed the Hermiticity of the fractional Hamilton operator and established the parity conservation law. Xiaoyi Guo and Mingyu Xu[4] studied some physical applications of the fractional Schrodinger equation.     

求解第一类积分方程的正则化—小波方法及其数值试验

1 方法的描述 第一类(Fredholm)积分方程是指形如 (1.1)的积分方程,其中核k(x,y)和右端函数f(x)给定,u(x)是未知函数.许多物理、化学、力学和工程应用问题都能导致第一类积分方程.求解第一类积分方程的一个本质性困难是方程的不适定性,即解的存在性、唯一性和稳定性遭到破坏.常用的数值方法有奇异值分解(SVD)方法、Tikhonov正则化方法、投影方法、正则化-样条方法、再生核方法等.本文提出一种新的正则化-小波方法,在第一类积分方程有多个解时,可以求出具有最小范数的数值解;如果原积分方程有唯一解,则所得的数值解收敛于准确解.数值试验表明,该方法是可行的. 我们在L~2[a,b]中考虑第一类(Fredholm)积分方程,即假设方程(1.1)中积分算子K∈L~2([a,b]×[a,b])及右端f(x)∈L~2[a,b]给定.为保证数值求解算法的稳定性,我们先用正则化方法处理该方程,将不适定问题化为泛函极值问题来求解,然后利用多重正交样条小波基构造求解格式.由于我们给出了直接计算低阶的多重正交样条小波基函数的一般公式,使得解法可以在计算机迅速实现.     

Dual integral equations related to the kontorovich-lebedev transform : PMM vol. 38, n 6, 1974, pp. 1090–1097

We study the dual integral equations related to the Kontorovich-Lebedev integral transforms arising in the course of solution of the problems of mathematical physics, in particular of the mixed boundary value problems for the wedge-shaped regions. We show that the solutions of these equations can be expressed in quadratures, using the auxilliary functions satisfying the integral Fredholm equation of second kind with a symmetric kernel.At present, the dual equations investigated in most detail are those connected with the Fourier and Hankel integral transforms. The results obtained and their applications are given in [1–3]. A large number of papers also deal with the theory and applications of the dual integral equations connected with the Mehler-Fock integral transform and its generalizations [4–11]., The dual integral transforms considered in the present paper belong to a more complex class than those listed above, and so far, no effective solution has been obtained for them. The only relevant results known to the authors are those in [12, 13]. In [12] a method of solving the equations (1.2) is given for a single particular value of the parameter γ = π/2, while in [13] the dual equations of the type under consideration are reduced to a solution of an infinite system of linear algebraic equations.     

两平行圆板间径向层流进口段效应分析

本文首先将B.B.方法[1]推广至二平行圆板间的径向扩散流动,由边界层运动方程式同时推导出动量积分方程式和能量积分方程式,而后再用Picard逐次逼近法[2]解能量积分方程式,求得进口段通道长随边界层厚度而改变的二级近似显函数表达式.从而为进口段效应诸系数的直接解析分析提供了可能.特别是当圆板外径小于进口段长度时,更加突出地表现了本方法的优越性.由于采用了能量积分方程式,则压力损失系数的各项才得以从理论上独立地推导出来.本文所提供的压力损失系数计算值,在进口修正雷诺数Re<100时,和文献[3]比较与实验值更为接近.因此在该范围内本文的结果既可靠又简便.     

Self-similar solutions of semilinear wave equation with variable speed of propagation

We investigate the issue of existence of the self-similar solutions of the generalized Tricomi equation in the half-space where the equation is hyperbolic. We look for the self-similar solutions via the Cauchy problem. An integral transformation suggested in [K. Yagdjian, A note on the fundamental solution for the Tricomi-type equation in the hyperbolic domain, J. Differential Equations 206 (2004) 227-252] is used to represent solutions of the Cauchy problem for the linear Tricomi-type equation in terms of fundamental solutions of the classical wave equation. This representation allows us to prove decay estimates for the linear Tricomi-type equation with a source term. Obtained in [K. Yagdjian, The self-similar solutions of the Tricomi-type equations, Z. Angew. Math. Phys., in press, doi:10.1007/s00033-006-5099-2] estimates for the self-similar solutions of the linear Tricomi-type equation are the key tools to prove existence of the self-similar solutions.     

On global existence and attractivity results for nonlinear functional integral equations

In this paper two existence results concerning the global attractivity and global asymptotic attractivity for a certain functional nonlinear integral equation are proved. Our existence results include several existence as well as attractivity results obtained earlier by Banas and Dhage (2008) [1], Hu and Yan (2006) [3], Dhage (2009) [15] and Banas and Rzepka (2003) [7] as special cases under some weaker Lipschitz conditions. A measure theoretic fixed point theorem of Dhage (2008) [6] is used in formulating our main results and the characterizations of solutions are obtained in the space of functions defined, continuous and bounded on unbounded intervals.     

Integral formula for the characteristic Cauchy problem on a curved background

We give a local integral formula, valid on general curved spacetimes, for the characteristic Cauchy problem for the Dirac equation with arbitrary spin. The derivation of the formula is based on the work of Friedlander (1975) [6] for the wave equation. A parametrix for the square of the Dirac operator, which is a spinor wave equation, is built using Friedlander's construction. Deriving the representation formula obtained in function of the characteristic data for this particular wave equation gives an integral formula for the Goursat problem. The results obtained by Penrose (1963) in the flat case in [21] are recovered directly.     

A boundary integral equation method for the transmission eigenvalue problem

We propose a new integral equation formulation to characterize and compute transmission eigenvalues for constant refractive index that play an important role in inverse scattering problems for penetrable media. As opposed to the recently developed approach by Cossonnière and Haddar [1,2] which relies on a two by two system of boundary integral equations our analysis is based on only one integral equation in terms of Dirichlet-to-Neumann or Robin-to-Dirichlet operators which results in a noticeable reduction of computational costs. We establish Fredholm properties of the integral operators and their analytic dependence on the wave number. Further we employ the numerical algorithm for analytic non-linear eigenvalue problems that was recently proposed by Beyn [3] for the numerical computation of transmission eigenvalues via this new integral equation.     

刚性基础上的弹性层在表面垂直集中简谐载荷作用下的动力响应

本文首次将文献[1]所提出的线载荷积分方程法应用于求解弹性动力学问题。导出了刚性基础上的弹性层在表面垂直集中简谐载荷作用下动力响应问题的一维非奇异积分方程组,并求得了数值解。     

Monotonic positive solution of a nonlinear quadratic functional integral equation

The existence of positive monotonic solutions, in the class of continuous functions, for some nonlinear quadratic integral equation have been studied in [4], [5], [6], [7] and [8]. Here we are concerning with a nonlinear quadratic integral equation of Volterra type and we shall prove the existence of at least one L₁-positive monotonic solution for that equation under Carathèodory condition.     

推广的KdV方程特征问题解的存在唯一性

推广的KdV方程u_t+αuu_x+μu_x3+εu_x5=0[1]是典型的可积方程.它先后在研究冷等离子体中磁声波的传播[2],传输线中孤立波[3]和分层流体中界面孤立波[4]时导出.本文对推广的KdV方程的特征问题,在Riemann函数的基础上,设计一恰当结构,并由此化待征问题为一与之等价的积分微分方程.而该积分微分方程对应的映射E是列自身的映射[5],依不动点原理,积分微分方程有唯一的正则解,即推广的KdV方程的特征问题有唯一解,且由积分微分方程序列所得的迭代解于Ω上一致收敛.     

一类非线性悬臂梁方程正解的存在性与多解性

研究了非线性四阶常微分方程u(4)(t)=f(t,u(t),u'(t)),t ∈[0,1]\E在边界条件u(0)=u'(0)=u"(1)=u"'(1)=0下的正解,其中E(∩)[0,1]是一个零测度的闭集,而非线性项,(t,u,u)可以在t∈E时奇异.通过构造适当的积分方程并利用锥上的不动点定理证明了这个方程在满足与n有关的条件下存在n个正解,其中n是某个自然数.     

Differential operator related to the generalized superradiance integral equation

In this paper we consider a class of problems which are generalized versions of the three-dimensional superradiance integral equation. A commuting differential operator will be found for this generalized problem. For the three-dimensional superradiance problem an alternative set of complete eigenfunctions will also be provided. The kernel for the superradiance problem when restricted to one-dimension is the same as appeared in the works of Slepian, Landau and Pollak (cf. Slepian and Pollak (1961) [1], Landau and Pollak (1961, 1962) [2] and [3], Slepian (1964, 1978) [4] and [5]). The uniqueness of the differential operator commuting with that kernel is indicated.     

Instability in the critical case of pair of pure imaginary roots for a class of systems with aftereffect

The stability of a system described by Volterra integrodifferential equations is investigated in the critical case when the characteristic equation has a pair of pure imaginery roots. Conditions for instability, analogous to the well-known conditions from the theory of differential equations [1], are derived. (A similar result was established previously in [2] for integrodifferential equations of simpler structure with integral kernels of exponential-polynomial type). For the proof, several manipulations are used to simplify the original equation and, in particular, to reduce the linearized equation to the form of a differential equation with constant diagonal matrix. (An analogous approach was used to analyse instability for Volterra integrodifferential equations in the critical case of zero root in [3, 4]). As an example, the sign of the Lyapunov constant in the problem of the rotational motion of a rigid body with viscoelastic supports is calculated.     

Ito型随机积分方程依参数收敛性定理

本文在一定假设条件下，给出了一类　型随机积分方程依参数收敛性问题的一个定理，推广了［２］［３］的相应结果。     

On Taylor-series expansion methods for the second kind integral equations

In this paper, we comment on the recent papers by Yuhe Ren et al. (1999) [1] and Maleknejad et al. (2006) [7] concerning the use of the Taylor series to approximate a solution of the Fredholm integral equation of the second kind as well as a solution of a system of Fredholm equations. The technique presented in Yuhe Ren et al. (1999) [1] takes advantage of a rapidly decaying convolution kernel k(|s−t|) as |s−t| increases. However, it does not apply to equations having other types of kernels. We present in this paper a more general Taylor expansion method which can be applied to approximate a solution of the Fredholm equation having a smooth kernel. Also, it is shown that when the new method is applied to the Fredholm equation with a rapidly decaying kernel, it provides more accurate results than the method in Yuhe Ren et al. (1999) [1]. We also discuss an application of the new Taylor-series method to a system of Fredholm integral equations of the second kind.     

奇异积分方程解的稳定性

该文在[1]的基础上，进一步探讨了区间[－1，1]上带Cauchy核奇异积分方程权函数的分解及其解的稳定性和在Chebyshev模下的稳定性条件．证明了扰动方程的可解性及原方程的解对于已知函数的连续依赖性．     

Weak-Type (1-1) Inequality for the Monge-Ampère SIO's

The purpose of this article is to prove a weak-type (1-1) estimate for a class of singular integral operators related to the real analysis of the Monge-Ampère equation. The method uses the theory of homogeneous spaces. It should be noted that the class of operators has been widened since [3] by replacing the Lipschitz type condition with a weaker Hölder-a\alpha type condition ($0 < \alpha \leq 1$0 < \alpha \leq 1).     

A connection-theoretic approach to reduction of second-order dynamical systems with symmetry

We deal with reduction of Lagrangian systems that are invariant under the action of the symmetry group. Unlike the bulk of the literature we do not rely on methods coming from the calculus of variations. Our method is based on the geometrical analysis of regular Lagrangian systems, where solutions of the Euler-Lagrange equations are interpreted as integral curves of the associated second-order differential equation field. In particular, we explain so-called Lagrange-Poincaré reduction [1] and Routh reduction [3] from the viewpoint of that vector field. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence of Positive Solutions for Delay Differential Equations with Impulses

A powerful comparison result and some effective sufficient conditions for existence of positive solutions of the delay differential equation with impulses[formula]are obtained by establishing the equivalence of existence of positive solutions of (∗) and solutions of an integral equation.