WEAK APPROXIMATION OF OBLIQUELY REFLECTED DIFFUSIONS IN TIME-DEPENDENT DOMAINS

In an earlier paper, we proved the existence of solutions to the Skorohod problem with oblique reflection in time-dependent domains and, subsequently, applied this result to the problem of constructing solutions, in time-dependent domains, to stochastic differential equations with oblique reflection. In this paper we use these results to construct weak approximations of solutions to stochastic differential equations with oblique reflection, in time-dependent domains in Rd, by means of a projected Euler scheme. We prove that the constructed method has, as is the case for normal reflection and time-independent domains, an order of convergence equal to 1/2 and we evaluate the method empirically by means of two numerical examples. Furthermore, using a well-known extension of the Feynman-Kac formula, to stochastic differential equations with reflection, our method gives, in addition, a Monte Carlo method for solving second order parabolic partial differential equations with Robin boundary conditions in time-dependent domains.     

FINITE ELEMENT AND DISCONTINUOUS GALERKIN METHOD FOR STOCHASTIC HELMHOLTZ EQUATION IN TWO- AND THREE-DIMENSIONS

In this paper, we consider the finite element method and discontinuous Galerkin method for the stochastic Helmholtz equation in R^d （d = 2, 3）. Convergence analysis and error estimates are presented for the numerical solutions. The effects of the noises on the accuracy of the approximations are illustrated. Numerical experiments are carried out to verify our theoretical results.     

EFFICIENT NUMERICAL ALGORITHMS FOR THREE-DIMENSIONAL FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS

This paper detailedly discusses the locally one-dimensional numerical methods for ef- ficiently solving the three-dimensional fractional partial differential equations, including fractional advection diffusion equation and Riesz fractional diffusion equation. The second order finite difference scheme is used to discretize the space fractional derivative and the Crank-Nicolson procedure to the time derivative. We theoretically prove and numerically verify that the presented numerical methods are unconditionally stable and second order convergent in both space and time directions. In particular, for the Riesz fractional dif- fusion equation, the idea of reducing the splitting error is used to further improve the algorithm, and the unconditional stability and convergency are also strictly proved and numerically verified for the improved scheme.     

REDUCED BASIS METHOD FOR PARAMETRIZED ELLIPTIC ADVECTION-REACTION PROBLEMS

In this work we consider the Reduced Basis method for the solution of parametrized advection-reaction partial differential equations. For the generation of the basis we adopt a stabilized finite element method and we define the Reduced Basis method in the ＂primal- dual＂ formulation for this stabilized problem. We provide a priori Reduced Basis error estimates and we discuss the effects of the finite element approximation on the Reduced Basis error. We propose an adaptive algorithm, based on the a posteriori Reduced Basis error estimate, for the selection of the sample sets upon which the basis are built; the idea leading this algorithm is the minimization of the computational costs associated with the solution of the Reduced Basis problem. Numerical tests demonstrate the efficiency, in terms of computational costs, of the ＂primal-dual＂ Reduced Basis approach with respect to an ＂only primal＂ one. Parametrized advection-reaction partial differential equations, Reduced Basis method, ＂primal-dual＂ reduced basis approach, Stabilized finite element method, a posteriori error estimation.     

RESERVOIR DESCRIPTION BY USING A PIECEWISE CONSTANT LEVEL SET METHOD

We consider the permeability estimation problem in two-phase porous media flow. We try to identify the permeability field by utilizing both the production data from wells as well as inverted seismic data. The permeability field is assumed to be piecewise constant, or can be approximated well by a piecewise constant function. A variant of the level set method, called Piecewise Constant Level Set Method is used to represent the interfaces between the regions with different permeability levels. The inverse problem is solved by minimizing a functional, and TV norm regularization is used to deal with the ill-posedness. We also use the operator-splitting technique to decompose the constraint term from the fidelity term. This gives us more flexibility to deal with the constraint and helps to stabilize the algorithm.     

Existence and Uniqueness of Renormalized Solution of Nonlinear Degenerated Elliptic Problems

In this paper, We study a general class of nonlinear degenerated elliptic problems associated with the differential inclusion β(u)-div(a(x, Du)+ F(u)) ■ f in Ω, where f ∈ L1 Ω. A vector field a(·,·) is a Carath′eodory function. Using truncation techniques and the generalized monotonicity method in the functional spaces we prove the existence of renormalized solutions for general L1-data. Under an additional strict monotonicity assumption uniqueness of the renormalized solution is established.     

NODAL O（h^4）-SUPERCONVERGENCE IN 3D BY AVERAGING PIECEWISE LINEAR,BILINEAR, AND TRILINEAR FE APPROXIMATIONS

We construct and analyse a nodal O（h^4）-superconvergent FE scheme for approximating the Poisson equation with homogeneous boundary conditions in three-dimensional domains by means of piecewise trilinear functions. The scheme is based on averaging the equations that arise from FE approximations on uniform cubic, tetrahedral, and prismatic partitions. This approach presents a three-dimensional generalization of a two-dimensional averaging of linear and bilinear elements which also exhibits nodal O（h^4）-superconvergence （ultracon- vergence）. The obtained superconvergence result is illustrated by two numerical examples.     

Carleson-Type Maximal Operators with Variable Kernels

The author considers the L^p boundedness for two kinds of Carleson-type maximal operators with variable kernels Ω（x,y＇）/｜y｜^n,where Ω（x,y＇）∈L^∞（R^n）×W2^s（S^n-1）for some s〉0.     

Schatten p -class Toeplitz operators with unbounded symbols on pluriharmonic Bergman space

In this paper, we construct the function u in L2（Bn, dA） which is unbounded on any neighborhood of each boundary point of Bn such that Tu is the Schatten p-class （0 〈 p 〈 ∞） operator on pluriharmonic Bergman space h2（Bn, dA） for several complex variables. In addition, we also discuss the compactness of Toeplitz operators with L1 symbols.     

关于线性偏微分方程的Cauchy问题的唯一性

<正> Cauchy问题的唯一性是偏微分方程的基本问题之一.经典的Cauchy-Kowalewski定理断言,解析方程或方程组的解析解是唯一的.1901年,Holmgren证明了,线性的解析方程或方程组的光滑解的唯一性.在取消关于系数的解析性的假设这个方向上的第一个结果是由Carleman在1939年给出的,他证明了两个自变量的相应结果,其中假设方程的主部的系数是实的,以及特征根是单重的,因而特征根的虚部如果不恒为零则总不为零.     

Boundedness for Hardy Type Operators on Herz Spaces with Variable Exponents

In this paper, we will prove the boundedness of Hardy type operators Hβ(x) and H*β(x) of variable order β(x) on Herz spaces Kα(·)p(·),q and Kα(·)p(·),q' where α(·) and p(·)are both variable.     

抽象算子在偏微分方程中的应用(Ⅱ)

Ｔｈｅｏｒｅｍ１　Ｌｅｔｍ∈Ｎａｎｄｍ１,Ｐ（ｘ）ｂｅａｎａｒｂｉｔｒａｒｙｏｒｄｅｒｐａｒｔｉａｌｄｉｆｆｅｒｅｎｔｉａｌｏｐｅｒａ－ｔｏｒ．Ｔｈｅｎｆ（ｘ,ｔ）,φｊ（ｘ）∈Ｊ．（ＷｈｅｒｅＪｓｔａｎｄｆｏｒｔｈｅｓｅｔｏｆａｎａｌｙｔｉｃｆｕｎｃｔｉｏｎｉｎｔｈｉｓｐａｐｅｒ）ｔ＋Ｐ（ｘ）ｍｕ＝ｆ（ｘ,ｔ）ｊｕｔｊｔ＝０＝φｊ（ｘ）　　ｘ∈Ｒｎ,ｔ∈Ｒ１ｊ＝０,１,２,…,ｍ－１．ｕ（ｘ,ｔ）＝∫ｔ０（ｔ－τ）ｍ－１（ｍ－１）！ｅ－（ｔ－τ）Ｐ（ｘ）ｆ（ｘ,τ）ｄτ＋ｅ－…     

Operator Equations and Duality Mappings in Sobolev Spaces with Variable Exponents

After studying in a previous work the smoothness of the space UΓ0={u∈W1,p(·)(Ω);u=0 on Γ0 Γ=Ω},where dΓ-measΓ0>0,with p(·)∈C(Ω)and p(x)>1 for all x∈Ω,the authors study in this paper the strict and uniform convexity as well as some special properties of duality mappings defined on the same space.The results obtained in this direction are used for proving existence results for operator equations having the form Ju=Nfu,where J is a duality mapping on UΓ0 corresponding to the gauge function,and Nf is the Nemytskij operator generated by a Carath′eodory function f satisfying an appropriate growth condition ensuring that Nf may be viewed as acting from UΓ0 into its dual.     

Discretizations of Linear Elliptic Partial Differential Inclusions

Differential inclusions provide a suitable framework for modelling choice and uncertainty. In finite dimensions, the theory of ordinary differential inclusions and their numerical approximations is well-developed, whereas little is known for partial differential inclusions, which are the deterministic counterparts of stochastic partial differential equations.

The aim of this article is to analyze strategies for the numerical approximation of the solution set of a linear elliptic partial differential inclusion. The geometry of its solution set is studied, numerical methods are proposed, and error estimates are provided.     

Necessary Maximum Principle of Stochastic Optimal Control with Delay and Jump Diffusion

In this paper, we have studied the necessary maximum principle of stochastic optimal control problem with delay and jump diffusion.     

Local solvability of partial differential operators on the Heisenberg group

In this article the following class of partial differential operators is examined for local solvability: Let P(X, Y) be a homogeneous polynomial of degree n ≥ 2 in the non-commuting variables X and Y. Suppose that the complex polynomial P(iz, 1) has distinct roots and that P(z, 0) = zⁿ. The operators which we investigate are of the form P(X, Y) where X = δ_x and Y = δ_y + xδ_w for variables (x, y, w) ∈ ?³. We find that the operators P (X, Y) are locally solvable if and only if the kernels of the ordinary differential operators P(iδ_x, ± x)* contain no Schwartz-class functions other than the zero function. The proof of this theorem involves the construction of a parametrix along with invariance properties of Heisenberg group operators and the application of Sobolev-space inequalities by Hörmander as necessary conditions for local solvability.     

A SPECTRAL METHOD FOR PANTOGRAPH-TYPE DELAY DIFFERENTIAL EQUATIONS AND ITS CONVERGENCE ANALYSIS

We propose a novel numerical approach for delay differential equations with vanishing proportional delays based on spectral methods. A Legendre-collocation method is employed to obtain highly accurate numerical approximations to the exact solution. It is proved theoretically and demonstrated numerically that the proposed method converges exponentially provided that the data in the are smooth. given pantograph delay differential equation     

A LIL and limit distributions for trimmed sums of random vectors attracted to operator semi-stable laws

Let θ∈ Rdbe a unit vector and let X,X1,X2,...be a sequence of i.i.d.Rd-valued random vectors attracted to operator semi-stable laws.For each integer n ≥ 1,let X1,n ≤···≤ Xn,n denote the order statistics of X1,X2,...,Xn according to priority of index,namely | X1,n,θ | ≥···≥ | Xn,n,θ |,where ·,· is an inner product on Rd.For all integers r ≥ 0,define by(r)Sn = n-ri=1Xi,n the trimmed sum.In this paper we investigate a law of the iterated logarithm and limit distributions for trimmed sums(r)Sn.Our results give information about the maximal growth rate of sample paths for partial sums of X when r extreme terms are excluded.A stochastically compactness of(r)Sn is obtained.     

Solutions to Boundary Value Problem of Nonlinear Impulsive Differential Equation of Fractional Order

This paper is devoted to study the existence and uniqueness of solutions to a boundary value problem of nonlinear fractional differential equation with impulsive effects.The arguments are based upon Schauder and Banach fixed-point theorems. We improve and generalize the results presented in[B.Ahmad,S.Sivasundaram, Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations,Nonlinear Analysis：Hybrid Systems,3（2009）,251- 258].     

Coerciveness inequalities for nonelliptic systems of partial differential equations

Annali di Matematica Pura ed Applicata (1923 -) - This paper deals with first-order matrix partial differential operators of the form (1) $$L = - i \mathop \sum \limits_{j = 1}^n L_j (x)D_i + L_o...