微分周形式与稀疏微分结式

代数周(Chow)形式和代数结式是代数几何的基本概念,同时还是消去理论的强大工具.一个自然的想法是在微分代数几何中发展相应的周形式和结式理论.但是由于微分结构的复杂性,在本文的研究工作之前,微分结式只有部分结果,而微分周形式与稀疏微分结式理论一直没有得到发展.本文的主要结果包括:第一,发展一般(generic)情形的微分相交理论,作为应用,证明一般情形的微分维数猜想.第二,初步建立微分周形式理论.对不可约微分代数簇定义微分周形式并证明其基本性质,特别地,给出微分周形式的Poisson分解公式,引入微分代数簇的主微分次数这一不变量并证明一类微分代数闭链的周簇和周坐标的存在性.作为应用,首次严格定义微分结式,证明其基本性质.第三,初步建立稀疏微分结式理论.引入Laurent微分本性系统的概念,定义稀疏微分结式,证明其基本性质,特别地,引入微分环面簇的概念,给出稀疏微分结式阶数和次数界的估计,并基于此给出计算稀疏微分结式的单指数时间算法.     

[3, 3] Padé approximation method for solving space fractional Fokker–Planck equations

The fractional Fokker–Planck equation has been used in various areas of engineering and physics. In this paper, we proposed a novel numerical scheme for solving the space fractional Fokker–Planck equation with the help of the [3, 3] Padé approximation. It is proved that the numerical method is unconditionally stable in view of the matrix analysis method. Finally, a numerical example is proposed to prove the effectiveness of the numerical scheme.     

Existence of solutions of initial value problems for nonlinear fractional differential equations

In this paper, by using the Schauder fixed point theorem, we study the existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations and obtain some new results.     

基于Bell多项式求解非线性发展方程Hirota双线性形式的新算法

基于Bell多项式,构造了获得非线性发展方程双线性形式的一个新算法,并且开发了相应的程序包.非线性发展方程与其双线性形式之间的变换可由该程序包自动推导,同时给出了一些具有代表性的实例验证了该程序包的有效性与可靠性.     

On the stability of two new two-step explicit methods for the numerical integration of second order initial value problem on a variable mesh

We present two novel two-step explicit methods for the numerical solution of the second order initial value problem on a variable mesh. In the case of a constant mesh the method is superstable in the sense of Chawla (1985). Numerical experimentation is provided to verify the stability analysis.     

EXISTENCE AND DECAY OF WEAK SOLUTIONS TO DEGENERATE DAVEY-STEWARTSON EQUATIONS

In this paper, the authors consider a system of degenerate Davey-Stewartson equations. They prove the global existence of weak solutions in some weighted function spaces and the decay of weak solutions in some anisotropic spaces for appropriate initial data.     

New approach to a generalized fractional integral

The paper presents a new fractional integration, which generalizes the Riemann-Liouville and Hadamard fractional integrals into a single form. Conditions are given for such a fractional integration operator to be bounded in an extended Lebesgue measurable space. Semigroup property for the above operator is also proved. We give a general definition of the fractional derivatives and give some examples.     

An explicit algorithm for monotone variational inequalities

We introduce a fully explicit method for solving monotone variational inequalities in Hilbert spaces, where orthogonal projections onto the feasible set are replaced by projections onto suitable hyperplanes. We prove weak convergence of the whole generated sequence to a solution of the problem, under only the assumptions of continuity and monotonicity of the operator and existence of solutions.     

Exact WKB analysis of a Schrödinger equation with a merging triplet of two simple poles and one simple turning point,II — Its relevance to the Mathieu equation and the Legendre equation

We develop the exact WKB analysis of an M2P1T (merging two simple poles and one simple turning point) Schrödinger equation. In Part II, using a WKB-theoretic transformation to the algebraic Mathieu equation constructed in Part I, we calculate the alien derivative of its Borel transformed WKB solutions at each fixed singular point relevant to the simple poles through the analysis of Borel transformed WKB solutions of the Legendre equations. In the course of the calculation of the alien derivative we make full use of microdifferential operators whose symbols are given by the infinite series that appear in the coefficients of the algebraic Mathieu equation and the Legendre equation.     

On the hyperbolic order

We define the hyperbolic order of any locally injective holomorphic function between arbitrary hyperbolic domains of the complex plane and study the relation between the hyperbolic order and the Schwarzian derivative for locally injective holomorphic functions from the unit disk into itself.