Singular Limits for Reaction-Diffusion Equations with Fractional Laplacian and Local or Nonlocal Nonlinearity

We perform an asymptotic analysis of models of population dynamics with a fractional Laplacian and local or nonlocal reaction terms. The first part of the paper is devoted to the long time/long range rescaling of the fractional Fisher-KPP equation. This rescaling is based on the exponential speed of propagation of the population. In particular we show that the only role of the fractional Laplacian in determining this speed is at the initial layer where it determines the thickness of the tails of the solutions. Next, we show that such rescaling is also possible for models with non-local reaction terms, as selection-mutation models. However, to obtain a more relevant qualitative behavior for this second case, we introduce, in the second part of the paper, a second rescaling where we assume that the diffusion steps are small. In this way, using a WKB ansatz, we obtain a Hamilton-Jacobi equation in the limit which describes the asymptotic dynamics of the solutions, similarly to the case of selection-mutation models with a classical Laplace term or an integral kernel with thin tails. However, the rescaling introduced here is very different from the latter cases. We extend these results to the multidimensional case.     

分数阶奇异微分方程初值问题正解的存在性

分数阶微分方程解的存在性问题近年来引起了许多人的关注,本文考察了一个非线性分数阶奇异微分方程初值问题正解的存在性,我们的结果削弱了Babakhani, Varsha的假设条件,并要求其非线性项具有一定程度的奇异性．     

非线性算子方程的泰勒展式算法

本文的目的是给出一种解Ｈｉｌｂｅｒｔ空间中非线性方程的ｋ阶泰勒展式算法（ｋ１）．标准Ｇａｌｅｒｋｉｎ方法可以看作１阶泰勒展式算法，而最优非线性Ｇａｌｅｒｋｉｎ方法可视为２阶泰勒展式算法．我们应用这种算法于定常的Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程的数值逼近．在一定情景下，最优非线性Ｇａｌｅｒｋｉｎ方法提供比标准Ｇａｌｅｒｋｉｎ方法和非线性Ｇａｌｅｒｋｉｎ方法更高阶的收敛速度．     

Extensions of spaces with cylindrical measures and supports of measures determined by the Lévy Laplacian

Extensions of noncountably additive (cylindrical) measures are described, and examples of Hilbert supports of the Lévy-Gauss measure are given.Translated fromMatematicheskie Zametki, Vol. 64, No. 4, pp. 483–492, October, 1998.L. Accardi acknowledges partial support of the Russian Foundation for Basic Research under grant No. 96-01-00030.     

一类分数阶非线性波方程的精确解及应用

基于分离变量的思想构造了分数阶非线性波方程含常系数的解的形式.在用待定系数法求解时,根据原方程确定假设解中的待定参数,得到具体解的表达式.利用该方法求解了3个非线性波方程,即分数阶CH(Camassa-Holm)方程、时间分数阶空间五阶Kdv-like方程、分数阶广义Ostrovsky方程.比较简便地得到了这些方程的精确解.文献中关于整数阶非线性波方程的结果成为本文结果的特例.通过数值模拟给出了部分解的图像.对能够通过待定系数法求出精确解的分数阶微分方程所应满足的条件进行了阐述.     

A New Generalized Gronwall Inequality with a Double Singularity and Its Applications to Fractional Stochastic Differential Equations

Abstract

In many cases, the existence and uniqueness of the solution of a differential equation are proved using fixed point theory. In this paper, we utilize the theory of operators and ingenious techniques to investigate the well-posedness of mild solution to semilinear fractional stochastic differential equations. We first discuss some properties of a class of Volterra integral operators and then establish a new generalized Gronwall integral inequality with a double singularity. Finally, we use the properties and integral inequality to study the well-posedness of mild solution to the semilinear fractional stochastic differential equations. One sees that it is concise and effectiveness using the previous results to investigate the well-posedness of the mild solution.     

Classical Method of Moments for Partially and Discretely Observed Ergodic Models

We discuss the method of moments for a partially and discretely observed model driven by a time-homogeneous Lévy process. We suppose that the unobserved process is an -Markov process and that the data, which comes from another process, are available only at regularly spaced time points. Stochastic differential equations are particularly treated among many other possible models. Some illustrative examples are presented with simulations.     

非Lipschitz系数下随机泛函微分方程适度解的存在唯一性及其渐近性态(英文)

本文主要运用Picard迭代和算子分数次幂方法,讨论了随机时滞偏微分方程适度解的存在性与唯一性,并对解的渐近性态进行了研究.这里方程的系数不满足Lipschitz条件,时滞r>0为有限的.最后给出了一个非Lipschitz条件的例子.     

A note on the reflected backward stochastic differential equations driven by a Lévy process with stochastic Lipschitz condition

In this note, we prove the existence and uniqueness of the solution for a class of reflected backward stochastic differential equations (RBSDEs in short) related to the subdifferential operator of a lower semi-continuous convex function, driven by Teugels martingales associated with a Lévy process. Some known results are generalized and improved.     

具p-Laplacian算子分数阶微分方程边值问题正解的存在唯一性

本文运用非紧性测度,给出了一类具p-Laplacian算子分数阶微分方程边值问题正解的存在性与唯一性.最后,用一个例子阐述我们的主要结果.     

Analyticity of a Nonlinear Operator Associated to the Conformal Representation in Schauder Spaces. An Integral Equation Approach

We consider the Riemann map g_ζ,w of the complex unit disk to the plane domain 𝕀[ζ] enclosed by the Jordan curve ζ and normalized by the conditions g_ζ,w(0) = w, g′_ζ,w(0) > 0, where w is a point of 𝕀[ζ], and we present a nonlinear singular integral equation approach to prove that the nonlinear operator which takes the pair (ζ, w) to the map g^(–1)_ζ,w ○ ζ is real analytic in Schauder spaces.     

概率收缩偶与非阿基米德Menger概率赋范空间中非线性方程组的解*

本文在非阿基米德Menger概率赋范空间中引入了概率收缩偶的概念,研究了非阿基米德Menger概率赋范空间中具概率收缩偶的非线性方程组的解的存在性与唯一性.发展和改进了引文[1～5]的相应结果.     

一类具有P-Laplacian算子的非线性分数阶微分方程解的存在性研究

通过构造Green函数的性质,利用Banach不动点定理,研究了一类带有P-Lapalcian算子Caputo型非线性分数阶微分方程的正解问题;并通过构造第二变元的Lipschitz条件,在P-Lapalcian算子参数取不同值的范围下,分别证明了边值问题正解的存在性与唯一性.     

Fractional stochastic Burgers‐type equation in Hölder space—Wellposedness and approximations

In this work, we use the spectral Galerkin method to prove the existence of a pathwise unique mild solution of a fractional stochastic partial differential equation of Burgers type in a Hölder space. We get the temporal regularity, and using a combination of Galerkin and exponential‐Euler methods, we obtain a full discretization scheme of the solution. Moreover, we calculate the rates of convergence for both approximations (Galerkin and full discretization) with respect to time and to space.     

一类非线性微分方程组中心和焦点判定的简便方法

本文讨论非线性微分方程组dxdt=-y +x F( x,y) ,dydt=x +y F( x,y)( 1 )在奇点 ( 0 ,0 )邻近积分曲线的结构 .得到了判定原点 ( 0 ,0 )是微分方程组 ( 1 )的焦点或中心的简便方法 .     

Numerical solution for a class of SPDEs over bounded domains

Parabolic stochastic partial differential Equations (SPDEs) with multiplicative noise play a central rôle in nonlinear filtering. More precisely, the conditional distribution of a partially observed diffusion solves the normalized version of an equation of this type. We show that one can approximate the solution of the SPDE by the (unweighted) empirical measure of a finite system of interacting particle for the case when the diffusion evolves in a compact state space with reflecting boundary. This approximation differs from existing approximations where the particles are weighted and the particle interaction arises through the choice of the weights and not at the level of the particles' motion as it is the case in this work. The system of stochastic differential equations modelling the trajectories of the particles is approximated by the recursive projection scheme introduced by Pettersson [Stoch. Process. Appl. 59(2) (1995), pp. 295–308].     

关于一类非线性微分方程组的边值问题的渐近解（Ⅰ）

应用新的方法 ,研究一类非线性微分方程组　　u″ =v,εv″ f(x ,u ,u′)v′ -g(x ,u ,u′)v =0　　 (0 <ε 1) ,的边值问题的解的渐近性质·　构造出解的渐近展开式 ,和估计了余项 ,改进并拓广了前人的工作·     

Analysis of positive solution and Hyers‐Ulam stability for a class of singular fractional differential equations with p‐Laplacian in Banach space

This paper deals with 2 core aspects of fractional calculus including existence of positive solution and Hyers‐Ulam stability for a class of singular fractional differential equations with nonlinear p‐Laplacian operator in Caputo sense. For these aims, the suggested problem is converted into an integral equation via Green function , for ε∈(n−1,n], where n≥4. Then, the Green function is examined whether it is increasing or decreasing and positive or negative function. After these properties, some classical fixed‐point theorems are used for the existence of positive solution. Hyers‐Ulam stability of the proposed problem is also considered. For the application of the results, an expressive example is included.     

多重尺度法的计算机计算——一类非线性方程组的狄立克雷问题

应用多重尺度的边界层方法和计算机符号运算研究一类非线性方程组的边值问题解的渐近性质,构造出解的渐近展开式和估计了余项.并分析一个实例.为多重尺度方法的应用提供新的前景.     

一类四阶奇异微分方程正解存在的充分必要条件

通过Green函数给出了一对具体的上下解,对一类奇异微分方程边值问题做了研究,得到正解存在的充分必要条件.