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## 2007年论文摘要2007年论文摘要 Error analysis of spectral method on a triangle Ben-yu Guo and Li-lian Wang Advances in Comp. Math., 26(2007),473-496. Abstract In this paper, the orthogonal polynomial approximation on triangle proposed by Dubiner [2], is studied. Some approximation results are established in certain non-uniformly Jacobi-weighted Sobolev space, which play important role in numerical analysis of spectral and triangle spectral element methods for differential equations on complex geometries. As an example, a model problem is considered. Spectral method for differential equations of degenerate type on unbounded domains by using generalized Laguerre functions Guo Ben-yu and Zhang Xiao-yong Appl. Numer. Math., 57(2007),455-471. Abstract In this paper, we develop the orthogonal approximation by using generalized Laguerre functions. Some basic results on this approximation are established, which serve as the mathematical foundation of spectral methods for various differential equations on unbounded domains. As an example of applications, we propose a spectral method for a partial differential equation of degenerate type, which plays an important role in financial mathematics and other fields. The convergence of proposed scheme is proved. Numerical results show its spectral accuracy in space. Mixed Jacobi-spherical harmonic spectral method for Navier-Stokes equations Guo Ben-yu and Huang Wei Appl. Numer. Math., 57(2007),939-961. Abstract Mixed Jacobi-spherical harmonic spectral method is proposed for the Navier-Stokes equations in a ball. Its stability and convergence are proved. Numerical results demonstrate the efficiency of this approach. Some results on the mixed Jacobi-spherical harmonic approximation are established, which play important role in numerical analysis of spectral method in spherical geometry. Numerical integration based on Laguerre-Gauss interpolation Guo Ben-yu and Wang Zhong-qing Comput. Meth. Appl. Mech. Engrg., 196(2007), 3726-3741. Abstract In this paper, we propose two efficient numerical integrators for ordinary differential equations based on modified Laguerre-Gauss interpolations. The global convergence of proposed algorithms is proved. Numerical results demonstrate the spectral accuracy of these new schemes and agree well with the theoretical analysis. Periodic and almost periodic solutions of nonlinear discrete Volterra equations with unbounded delay Yihong Song and Hongjiong Tian J. Comp. Appl. Math. , 205(2007), 859 –870. Abstract The existence of periodic and almost periodic solutions of nonlinear discrete Volterra equations with unbounded delay is obtained by using stability properties of a bounded solution. Numerical methods for singularly perturbed delay differential equations Yeguo Sun, Dongyue Zhang and Hongjiong Tian J. Syst. Simu., 19(2007), 3943-3944，3992. Abstract This paper is concerned with uniformly convergent numerical methods for singularly perturbed delay differential equations. Two uniformly convergent numerical schemes which is based on the exponential fitting technique for linear and nonlinear problems are examined for singular perturbation problems with after-effect. Numerical examples are given to testify our theoretical results. A simple proof of inequalities of integrals of composite functions Zhenglu Jiang, Xiaoyong Fu and Hongjiong Tian J. Math. Anal. Appl., 332(2007), 1307-1312. Abstract In this paper we give a simple proof of inequalities of integrals of functions which are the composition of nonnegative continuous convex functions on a vector space Rmand vector-valued functions in a weakly compact subset of a Banach vector space generated by m -spaces for . Also, the same inequalities hold if these vector-valued functions are in a weakly* compact subset of a Banach vector space generated by m -spaces instead. Numerical dissipativity of multistep methods for delay differential equations Hongjiong Tian, Liqiang Fan and Jiaxiang Xiang Appl. Math. Comp.,188(2007), 934-941. Abstract Dissipative differential equations have frequently appeared in the fields of physics, engineering, and biology. In this paper we investigate numerical dissipativity of linear multistep and one-leg methods applied to a class of dissipative delay differential equations. We show that for such class of dissipative systems these numerical methods are dissipative if and only if they are A-stable for ordinary differential equations. One numerical experiment is given to illustrate our result. GPLm—stability of block Theta method for delay differential equation Cong Yuhao, Li Shundao and Tan Xiuli J. of System Simulation, 19(2007), 3937-3039. Abstract The stability behavior of numerical solution for delay differential equations with many delays was studied. The conditions of GPm-stability and GPLm-stability of block theta method for delay differential equations with many delays were discussed .By Lagrange Interpolation, it is shown that block theta method is GPm-stable if and only if it is A-stable, block theta method is GPLm—stable if and only if theta=1. A numerical method for a Cauchy problem for elliptic partial differential equations W. Han, J. Huang, K. Kazmi, and Y. Chen Inverse Problems, 23(2007), 2401-2415. Abstract The Cauchy problem for an elliptic partial differential equation is ill-posed. In this paper, we study a numerical method for solving the Cauchy problem. The numerical method is based on a reformulation of the Cauchy problem through an optimal control approach coupled with a regularization term which is included to treat the severe ill-conditioning of the corresponding discretized formulation. We prove convergence of the numerical method and present theoretical results for the limiting behaviors of the numerical solution as the regularization parameter approaches zero. Results from some numerical examples are reported. A finite element method for general elastic multi-structures J. Huang, L. Guo and Z. Shi Computers Math. with Appl., 53(2007), 1867-1895. Abstract A finite element method is proposed for the general elastic multi-structure problem, where displacements on bodies, longitudinal displacements on plates, longitudinal displacements and rotational angles on rods are discretized using conforming linear elements, transverse displacements on plates and rods are discretized respectively using TRUNC elements and Hermite elements of third order, and the discrete generalized displacement fields in individual elastic members are coupled together by some feasible interface conditions. The unique solvability of the method is verified by the Lax-Milgram lemma after deriving generalized Korn's inequalities in some nonconforming element spaces on elastic multi-structures. The quasi-optimal error estimate in the energy norm is also established. Some numerical results are presented in the end. Uniform a priori estimates for elliptic and static Maxwell interface problems J. Huang and J. Zou Discrete and Continuous Dynamical Systems Ser. B, 7(2007), 145-170. Abstract We present some new a priori estimates of the solutions to three-dimensional elliptic interface problems and static Maxwell interface system with variable coefficients. Different from the classical a priori estimates, the physical coefficient functions of the interface problems appear in these new estimates explicitly. Spectral-domain decomposition method and its applications in finance Xu Chenglong Recent Progress in Scientific Computing, Science Press, 2007,367-381. Abstract The modified Laguerre spectral-finite difference schemes are proposed for a class of degenerate PDEs arising from finance with discontinuous coefficient. The domain-decomposition technique is used. Error estimation of the schemes is obtained. Numerical results are given which show the efficiency and the convergence of the schemes. The extrapolation of Numerov’s scheme for nonlinear two-point boundary value problems Yuan-Ming Wang Appl. Numer. Math., 57(2007), 253-269. Abstract This paper is concerned with the extrapolation algorithm of Numerov's scheme for semilinear and strongly nonlinear two-point boundary value problems. The asymptotic error expansion of the solution of Numerov's scheme is obtained. Based on the asymptotic error expansion, Richardson's extrapolation is constructed, and so the accuracy of the numerical solution is greatly increased. Numerical results are presented to demonstrate the efficiency of the extrapolation algorithm. Monotone iterative technique for numerical solutions of fourth-order nonlinear elliptic boundary value problems Yuan-Ming Wang Applied Numerical Mathematics, 57(2007), 1081-1096. Abstract This paper is concerned with finite difference solutions of a class of fourth-order nonlinear elliptic boundary value problems. The nonlinear function is not necessarily monotone. A new monotone iterative technique is developed, and three basic monotone iterative processes for the finite difference system are constructed. Several theoretical comparison results among the various monotone sequences are given. A simple and easily verified condition is obtained to guarantee a geometric convergence of the iterations. Numerical results for a model problem with known analytical solution are given. Error and stability of monotone method for numerical solutions of fourth-order semilinear elliptic boundary value problems Yuan-Ming Wang J. Comp. Appl. Math., 200(2007), 503-519. Abstract This paper is concerned with the error and stability analysis of the monotone method for numerical solutions of fourth-order semilinear elliptic boundary value problems. A comparison result among the various monotone sequences is given. The global error is analyzed, and some sufficient conditions are formulated to guarantee a geometric rate of convergence. The stability of the monotone method is proved. Some numerical results are presented. Asymptotic behavior of solutions for a class of predator-prey reaction-diffusion systems with time delays Yuan-Ming Wang J. Math. Anal. Appl.，328(2007), 137-150. Abstract The aim of this paper is to investigate the asymptotic behavior of solutions for a class of three-species predator-prey reaction-diffusion systems with time delays under homogeneous Neumann boundary condition. Some simple and easily verifiable conditions are given to the rate constants of the reaction functions to ensure the convergence of the time-dependent solution to a constant steady-state solution. The conditions for the convergence are independent of diffusion coefficients and time delays, and the conclusions are directly applicable to the corresponding parabolic-ordinary differential system and to the corresponding system without time delays. Strong Tractability of Quasi-Monte Carlo Quadrature Using Nets for Certain Banach Spaces R. X. Yue and F. J. Hickernell SIAM J. Numer. Anal., 44(2006), 2559-2583 Abstract We consider multivariate integration in the weighted spaces of functions with mixed first derivatives bounded in norms and the weighted coefficients introduced via norms, where . The integration domain may be bounded or unbounded. The worst-case error and randomized error are investigated for quasi-Monte Carlo quadrature rules. For the worst-case setting the quadrature rule uses deterministic -sequences in base , and for the randomized setting the quadrature rule uses randomly scrambled digital -nets in base . Sufficient conditions are found under which multivariate integration is strongly tractable in the worst-case and randomized settings, respectively. Similar results hold for the Banach spaces of finite-order weights. Results presented in this article extend and improve upon those found previously. Numerical differentiation and its applications J. Cheng, X. Z. Jia and Y. B. Wang Inverse Problems in Science and Engineering, 15(2007), 339–357 Abstract Differentiation is one of the most important concepts in calculus, which has been used almost everywhere in many fields of mathematics and applied mathematics. It is natural that numerical differentiation should be an important technique for the engineers. However, since it is ill-posed in Hadamard’s sense, which means that any small error in the measurements will be enlarged, it is very difficult for the engineers to use this technique. In this article, we propose a new simple numerical method to reconstruct the original function and its derivatives from scattered input data and show that our method is effective and can be realized easily. Return |