Error analysis of spectral method on a triangle
Ben-yu Guo and Li-lian Wang
Advances in Comp. Math., 26(2007),473-496.
In this paper, the orthogonal polynomial approximation on triangle proposed by Dubiner , 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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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
Recent Progress in Scientific Computing, Science Press, 2007,367-381.
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
Appl. Numer. Math., 57(2007), 253-269.
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
Applied Numerical Mathematics, 57(2007), 1081-1096.
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
J. Comp. Appl. Math., 200(2007), 503-519.
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
J. Math. Anal. Appl.，328(2007), 137-150.
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
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
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.
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