针对一维空间分数阶扩散方程的一个新的最大模近似公式

Based on the maximum principle,the difference formula defined on a non-integral node is given to approximate the fractional Riemann-Liouville derivative and the finite difference scheme for solving one-dimensional space fractional diffusion equations(FDEs) with variable coefficients is presented.Furthermore,using the maximum principle the scheme is proved unconditionally stable and secondorder accuracy in spatial grid size.Several numerical examples are given to verify the efficiency of the scheme.     

Iterative methods for a forward-backward heat equation in two-dimension

A finite difference method is introduced to solve the forward-backward heat equation in two space dimensions. In this procedure, the backward and forward difference scheme in two subdomains and a coarse-mesh second-order central difference scheme at the middle interface are used. Maximum norm error estimate for the procedure is derived. Then an iterative method based on domain decomposition is presented for the numerical scheme and the convergence of the given method is established. Then numerical experiments are presented to support the theoretical analysis.     

Analysis of an Implicit Finite Difference Scheme for Time Fractional Diffusion Equation

Time fractional diffusion equation is usually used to describe the problems involving non-Markovian random walks. This kind of equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α∈(0, 1). In this paper, an implicit finite difference scheme for solving the time fractional diffusion equation with source term is presented and analyzed, where the fractional derivative is described in the Caputo sense. Stability and convergence of this scheme are rigorously established by a Fourier analysis. And using numerical experiments illustrates the accuracy and effectiveness of the scheme mentioned in this paper.     

带分数阶Robin边界条件的时间-空间分数阶扩散方程的有限差分方法（英文）

In this paper, an efficient numerical method is proposed to solve the Caputo-Riesz fractional diffusion equation with fractional Robin boundary conditions. We approximate the Riesz space fractional derivatives using the fractional central difference scheme with second-order accurate. A priori estimation of the solution of the numerical scheme is given, and the stability and convergence of the numerical scheme are analyzed.Finally, a numerical example is used to verify the accuracy and efficiency...     

A FAST AND HIGH ACCURACY NUMERICAL SIMULATION FOR A FRACTIONAL BLACK-SCHOLES MODEL ON TWO ASSETS

In this paper, a two dimensional(2D) fractional Black-Scholes(FBS) model on two assets following independent geometric Lévy processes is solved numerically. A high order convergent implicit difference scheme is constructed and detailed numerical analysis is established. The fractional derivative is a quasidifferential operator, whose nonlocal nature yields a dense lower Hessenberg block coefficient matrix. In order to speed up calculation and save storage space, a fast bi-conjugate gradient stabilized(FBi-CGSTAB) method is proposed to solve the resultant linear system. Finally, one example with a known exact solution is provided to assess the effectiveness and efficiency of the presented fast numerical technique. The pricing of a European Call-on-Min option is showed in the other example, in which the influence of fractional derivative order and volatility on the 2D FBS model is revealed by comparing with the classical 2D B-S model.     

A numerical algorithm for determination of a control parameter in two-dimensional parabolic inverse problems

Numerical solution of the parabolic partial differential equations with an unknown parameter play a very important role in engineering applications. In this study we present a high order scheme for determining unknown control parameter and unknown solution of two-dimensional parabolic inverse problem with overspecialization at a point in the spatial domain. In this approach, a compact fourth-order scheme is used to discretize spatial derivatives of equation and reduces the problem to a system of ordinary differential equations(ODEs).Then we apply a fourth order boundary value method to the solution of resulting system of ODEs. So the proposed method has fourth order of accuracy in both space and time components and is unconditionally stable due to the favorable stability property of boundary value methods. The results of numerical experiments are presented and some comparisons are made with several well-known finite difference schemes in the literature.Also we will investigate the effect of noise in data on the approximate solutions.     

A conservative numerical method for the fractional nonlinear Schr¨odinger equation in two dimensions

This paper proposes and analyzes an efficient finite difference scheme for the two-dimensional nonlinear Schr?dinger(NLS) equation involving fractional Laplacian. The scheme is based on a weighted and shifted Grünwald-Letnikov difference(WSGD) operator for the spatial fractional Laplacian. We prove that the proposed method preserves the mass and energy conservation laws in semi-discrete formulations. By introducing the differentiation matrices, the semi-discrete fractional nonlinear Schr?dinger(FNLS) equation can be rewritten as a system of nonlinear ordinary differential equations(ODEs) in matrix formulations. Two kinds of time discretization methods are proposed for the semi-discrete formulation. One is based on the Crank-Nicolson(CN) method which can be proved to preserve the fully discrete mass and energy conservation. The other one is the compact implicit integration factor(c IIF) method which demands much less computational effort. It can be shown that the cIIF scheme can approximate CN scheme with the error O(τ~2). Finally numerical results are presented to demonstrate the method's conservation, accuracy, efficiency and the capability of capturing blow-up.     

差商算子的多尺度分解与B样条级数的非均匀细分算法

This paper presents a decomposition scheme of the m-th order divided difference operator on non-uniform knots. Through it we introduce a subdivision algorithm of arbitrary B-spline series. Two numerical experiments on function approximation and curve fitting are given to show the efficiency of our algorithm.     

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.     

NUMERICAL SOLUTION OF THE SPACE FRACTIONAL DIFFERENTIAL EQUATION

In this paper, a space fractional differential equation is considered. The equation is obtained from the parabolic equation containing advection, diffusion and reaction terms by replacing the second order derivative in space by a fractional derivative in space of order. An implicit finite difference approximation for this equation is presented. The stability and convergence of the finite difference approximation are proved. A fractional-order method of lines is also presented. Finally, some numerical results are given.     

New Sobolev-Weinstein Spaces and Applications

In this paper, we consider the generalized Weinstein operator $\Delta_{W}^{d,\alpha,n}$, we introduce new Sobolev-Weinstein spaces denoted $\mathscr H_{\alpha,d,n}^{s}(\mathbb{R}_{+}^{d+1}),$ $s\in\mathbb{R},$ associated with the generalized Weinstein operator and we investigate their properties. Next, as application, we study the extremal functions on the spaces $\mathscr H_{\alpha,d,n}^{s}(\mathbb{R}_{+}^{d+1})$ using the theory of reproducing kernels.     

A numerical method for solving KdV equation with multilevel B-spline quasi-interpolation

In this article, we use a multilevel quartic spline quasi-interpolation scheme to solve the one-dimensional nonlinear Korteweg–de Vries (KdV) equation which exhibits a large number of physical phenomena. The presented scheme is obtained by using the second-order central divided difference of the spatial derivative to approximate the third-order spatial derivative, and the forward divided difference to approximate the temporal derivative, where the spatial derivative is approximated by the proposed quasi-interpolation operator. Compared to other numerical methods, the main advantages of our scheme are the higher accuracy and lower computational complexity. Meanwhile, the algorithm is very simple and easy to implement. Numerical experiments in this article also show that our scheme is feasible and valid.     

Numerical solution of Burgers-Fisher equation by cubic B-spline quasi-interpolation

In this paper, numerical solution of Burgers-Fisher equation is presented based on the cubic B-spline quasi-interpolation. At first, the generalized Burgers-Fisher equation and the cubic B-spline quasi-interpolation are introduced. Moreover, the numerical scheme is presented, by using the derivative of the quasi-interpolation to approximate the spatial derivative of the dependent variable and a low order forward difference to approximate the time derivative of the dependent variable. Moreover, the stability of this method is studied. At last, the numerical results obtained by this way have been compared with the exact solution to show the efficiency of the method. The main advantage of the resulting scheme is that the algorithm is very simple, so it is very easy to implement.     

Gorenstein子范畴与相对奇点范畴

令A是阿贝尔范畴, T是A的一个自正交子范畴, 且T中每个对象均有有限投射维数和内射维数. 假设左Gorenstein子范畴lG(T)等于T的右正交类,且右Gorenstein子范畴rG(T)等于T的左正交类,我们证明了Gorenstein子范畴$G(T)$等于T的左正交类与T的右正交类之交,并且证明了它们的稳定范畴三角等价于A关于T的相对奇点范畴.作为应用,令$R$是有有限左自内射维数的左诺特环, $_RC_s$是半对偶化双模,且所有内射左$R$-模的平坦维数的上确界有限, 我们证明了 若$\mbox{}_RC$有有限内射（平坦）维数且$C$的右正交类包含$R$,则存在从$C$-Gorenstein投射模与关于$C$的Bass类的交到关于$C$-投射模的相对奇点范畴间的三角等价,推广了某些经典的结果.     

单圈图的极小能量

For a simple graph G, the energy E(G) is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let Undenote the set of all connected unicyclic graphs with order n, and Ur n= {G ∈ Un| d(x) = r for any vertex x ∈ V(Cl)}, where r ≥ 2 and Cl is the unique cycle in G. Every unicyclic graph in Ur nis said to be a cycle-r-regular graph.In this paper, we completely characterize that C39(2, 2, 2) ο Sn-8is the unique graph having minimal energy in U4 n. Moreover, the graph with minimal energy is uniquely determined in Ur nfor r = 3, 4.     

Bergman空间上Cowen-Douglas算子的换位

令H~∞(D)表示单位圆盘D上的有界解析函数全体构成的代数,对于■∈H~∞(D),在某种条件下,证明T=M■是一个指标为n的Cowen-Douglaus算子,并且给出了■(T)/rad■(T)可交换的一个充分条件.当n=1时,刻画了T的换位.     

Multiquadric quasi-interpolation method for fractional integral-differential equations

In this paper, Multiquadric quasi-interpolation method is used to approximate fractional integral equations and fractional differential equations. Firstly, we construct two operators for approximating the Hadamard integral-differential equation based on quasi interpolators, and verify their properties and order of convergence. Secondly, we obtain that the approximation order of the integral scheme is 3, and the approximation order of the differential scheme is $3-\mu$ for $\mu(0<\mu<1)$ order fractional Hadamard derivative. Finally, The results of numerical experiments show that the numerical results are in greement with the theoretical analysis.     

对称广义中心对称矩阵模型修正的矩阵逼近法及其扰动性

X,B是实测的位移矩阵和载荷矩阵,C是有限元方法得到的估计矩阵,给出了AX=B的对称广义中心对称矩阵解集合ζ的表达式,对于逼近问题||C-A||F=min A∈ζ||C-A||F的解A,给出了它的表达式并分析了解A的扰动性,数值结果表明方法是行之有效的.     

粘合、倾斜同调维数与高维Auslander-Reiten理论

本文主要研究阿贝尔范畴粘合$(\mathscr{A}, \mathscr{B}, \mathscr{C})$中$\mathscr{A}$, $\mathscr{B}$与$\mathscr{C}$之间的倾斜同调维数关系. 特别地,对遗传的阿贝尔范畴$\mathscr{B}$,给出了粘合$(\mathscr{A}, \mathscr{B}, \mathscr{C})$中的范畴之间的$n$-几乎可裂序列间的联系.     

Janowski星形和凸函数族相关的非线性积分算子性质

In this paper, we consider a general nonlinear integral operator H_(αi),β_i(f_1,..., f_n;g_1,..., g_n)(z). Some results including coefficient problems, univalency condition and radius of convexity for this integral operator are given. Furthermore, we discuss the mapping properties between H_(αi),β_i(f_1,..., f_n; g_1,..., g_n)(z) and subclasses of analytic functions with bounded boundary rotation. The same subjects for some corresponding classes are shown upon specializing the parameters in our main results.