Generation and removal of apparent singularities in linear ordinary differential equations with polynomial coefficients

We discuss several examples of generating apparent singular points as a result of differentiating particular homogeneous linear ordinary differential equations with polynomial coefficients and formulate two general conjectures on the generation and removal of apparent singularities in arbitrary Fuchsian differential equations with polynomial coefficients. We consider a model problem in polymer physics.     

Non-limit-circle criteria for singular Hamiltonian differential systems

Non-limit-circle criteria for singular Hamiltonian differential expressions with complex coefficients are obtained. The main results are extensions of the previous limit-point criterion due to H. Weyl for second-order differential equations.     

闭光滑流形上的高阶线性微-积分方程

利用积分变换技巧,作者给出了C~n中闭光滑可定向流形上一个新的Bochner-Martinelli型积分的高阶偏导数的奇异积分的Hadamard主值,获得了高阶奇异积分的Plemelj公式和合成公式,还讨论了相应的变系数线性微分积分方程的正则化,证明其可转化为一类等价的Fredholm方程。并且指出其特征方程当给出一组适当的边值条件时,在L~*中存在唯一解。     

ANNOUNCEMENT ON“MAXIMUM PRINCIPLE FOR NON-UNIFORMLY PARABOLIC EQUATIONS AND APPLICATIONS”

In this note we announce the global boundedness for the solutions to a class of possibly degenerate parabolic equations by De-Giorgi’s iteration.In particular,the existence of weak solutions for possibly degenerate stochastic differential equations with singular diffusion coefficients is obtained.     

Some Solutions for a Class of Singular Equations

In this paper we obtain all solutions which depend only on r for a class of partial differential equations of higher order with singular coefficients.     

INTERFACE PROBLEMS FOR ELLIPTIC DIFFERENTIAL EQUATIONS

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Skorohod problem and multivalued stochastic evolution equations in Banach spaces

By solving a deterministic Skorohod problem in the framework of evolutional triple, we prove the existence and uniqueness of solutions to multivalued stochastic evolution equations involving maximal monotone operators. The existence and uniqueness of invariant measures associated with the solutions as Markov processes are also considered in the present paper. Moreover, we apply the results to stochastic differential equations with normal reflecting boundary conditions and with singular drift terms, as well as a class of multivalued nonlinear stochastic partial differential equations with possibly discontinuous coefficients.     

Local study of a family of planar cubic systems

Normal systems of differential equations in the plane are considered. All topological types of their singular points are described. A criterion in terms of coefficients is obtained.     

Irregular points of modulation

The local theory of singular points is extended to a large class of linear, second-order, ordinary differential equations which can be physical Schroedinger equations, or govern the modulation of real oscillators or waves. In addition to Langer's fractional turning points, such equations admit highly irregular points at which the coefficients of the differential equation can be almost arbitrarily multivalued. (Genuine coalescence of singular points, however, is not considered.) A local representation of the solutions is established, which generalizes Frobenius' method of power series, and reveals a remarkable, two-variable structure. Bounds are obtained on the departure of solution structure from the structure characteristic of regular points.     

Problems for elliptic singular equations with a quadratic gradient term

We investigate the homogeneous Dirichlet problem for a class of second-order elliptic partial differential equations with a quadratic gradient term and singular data. In particular, we study the asymptotic behaviour of the solution near the boundary under suitable assumptions on the growth of the coefficients of the equation.     

Approximation and limit theorems for quantum stochastic models with unbounded coefficients

We prove a limit theorem for quantum stochastic differential equations with unbounded coefficients which extends the Trotter-Kato theorem for contraction semigroups. From this theorem, general results on the convergence of approximations and singular perturbations are obtained. The results are illustrated in several examples of physical interest.     

Analysis of the stress singularity for a bi-material V-notch by the boundary element method

A new algorithm coupling the boundary element technique with the characteristic expansion method is proposed for the computation of the singular stress field in the V-notched bi-material structure. After the stress asymptotic expansions are introduced into the linear elasticity equilibrium equations, the governing equations at the small sector dug out from the bi-material V-notch tip region are transformed into the ordinary differential eigen-equations. All the parameters in the asymptotic expansions except the combination coefficients can be achieved by solving the established eigen-equations with the interpolating matrix method. Furthermore, the conventional boundary element method is applied to modeling the remaining structure without the notch tip region. The combination coefficients in the asymptotic expansion forms can be computed by the discretized boundary integral equations. Thus, the singular stress field at the V-notch tip and the generalized stress intensity factors of the bi-material notch are successfully calculated. The accurate singular stress field obtained here is very useful in the evaluation of the fracture property and the fatigue life of the V-notched bi-material structure.     

Zur Verallgemeinerung von Wärmepolynomen

The investigations ofRosenbloom-Widder [13] on expansions in terms of heat polynomials have been generalized for partial differential equations of second order with singular coefficients, equations of higher order etc. In the present paper we introduce a new method for the discussion of such problems. The technique is related to the treatment of the Cauchy problem on the base of the Ovcyannikov theorem. The method is used for the discussion of two classes of partial differential equations of arbitrary order.     

Problems for elliptic singular equations with a gradient term

We investigate the homogeneous Dirichlet problem for a class of second-order nonlinear elliptic partial differential equations with singular data. In particular, we study the asymptotic behaviour of the solution near the boundary up to the second order under various assumptions on the growth of the coefficients of the equation.     

Spline collocation method for linear singular hyperbolic systems

Some classes of singular systems of partial differential equations with variable matrix coefficients and internal hyperbolic structure are considered. The spline collocation method is used to numerically solve such systems. Sufficient conditions for the convergence of the numerical procedure are obtained. Numerical results are presented.     

Interface Problems for Elliptic Systems

Interface problems for elliptic systems of second order partial differential equations are studied. The main result is that the solution in the neighborhood of the singular point can be divided into two parts one of which is a solution to the homogeneous system with constant coefficients, and the other one possesses higher regularity.     

On the strong Feller property for stochastic delay differential equations with singular drift

In this paper, we prove the strong Feller property for stochastic delay (or functional) differential equations with singular drift. We extend an approach of Maslowski and Seidler to derive the strong Feller property of those equations, see Maslowski and Seidler (2000). The argumentation is based on the well-posedness and the strong Feller property of the equations’ drift-free version. To this aim, we investigate a certain convergence of random variables in topological spaces in order to deal with discontinuous drift coefficients.     

一般线性Lie超代数的微分算子表示

对一般线性Lie超代数的Verma模及向量相干态(vector-coherent-states)表示给出了以向量为系数的微分算子实现.对一般线性Lie超代数的单一非典型(singlyatypical)Kac模,我们具体写出了本原权向量,并对这种情形确切写出了用以刻画向量相干态表示的单子表示的微分方程.     

On the instantaneous blow-up of solutions of some quasilinear evolution problems

We study the existence of time-local solutions of higher-order quasilinear evolution partial differential equations and inequalities with singular coefficients and initial conditions. We obtain sufficient conditions for the instantaneous blow-up of solutions and estimate their lifespan. We show that the results cannot be improved in the function class considered.     

On the absence of local solutions of several evolutionary problems

We consider the existence problem for local (with respect to time) solutions of quasilinear evolutionary partial differential equations and inequalities with singular coefficients and initial conditions. We obtain sufficient conditions for instantaneous blow-up of solutions and show that the results thus obtained cannot be improved in the function class under study.