Asymptotic stability of numerical methods for neutral delay differential equations with multiple delays
Y. Cong, L. Xu and J. Kuang
J. System Simulation 18(2006), 3387-3389,3406.
The sufficient condition which analytic solution of neutral delay differential equations with multiple delays is asymptotically stable was given; the asymptotic stability of linear multistep methods for the numerical solution of neutral delay differential equations with multiple delays was discussed. By Lagrange Interpolation, it is shown that linear multistep methods are asymptotically stable if and only if it is A-stable.
Modified Laguerre pseudospectral method refined by multidomain Legendre pseudospectral approximation
B. Guo and L. Wang
J. Comp. Appl. Math. 190(2006), 304-324.
A modified Laguerre pseudospectral method is proposed for differential equations on the half line. The numerical solutions are refined by multidomain Legendre pseudospectral approximation. Numerical results show the spectral accuracy of this approach. Some approximation results on the modified Laguerre and Legendre interpolations are established. The convergence of proposed method is proved.
Optimal spectral-Galerkin methods using
generalized Jacobi polynomials
B. Guo, J. Shen and L. Wang
J. Sci. Comp. 27(2006), 305-322.
We extend the definition of the classical Jacobi polynomials with indexes to allow and to be negative integers. We show that the generalized Jacobi polynomials, with indexes corresponding to the number of boundary conditions in a given partial differential equation, are the natural basis functions for the spectral approximation of this partial differential equation. Moreover, the use of generalized Jacobi polynomials leads to much simplified analysis, more precise error estimates and well conditioned algorithms.
Generalized Laguerre interpolation and pseudospectral method for unbounded domains
B. Guo, L. Wang and Z. Wang
SIAM J. Numer. Anal. 43(2006), 2567-2589.
In this paper, error estimates for generalized Laguerre-Gauss-type interpolations are derived in non-uniformly Sobolev spaces weighted with Generalized Laguerre pseudospectral methods are analyzed and implemented. Two model problems are considered. The proposed schemes keep spectral accuracy, and with suitable choice of basis functions, lead to sparse and symmetric linear systems.
Mixed Leguerre-Hermite spectral method
for heat transfer in infinite plate
B. Guo and T. Wang
Comp. Math. Appl. 51(2006), 751-768.
Non-isotropic heat transfer in an infinite plate is considered. A weak formulation is derived, which is appropriate for its numerical simulation. Mixed Legendre-Hermite spectral method is proposed for this problem. Its convergence is proved. Numerical results show the efficiency of this new approach.
Remarks on error estimates for the TRUNC plate element
L. Guo, J. Huang and Z. Shi
J. Comp. Math. 24(2006), 103-112.
This paper provides a simplified derivation for error estimates of the TRUNC plate element. The error analysis for the problem with mixed boundary conditions is also discussed.
Finite element analysis for general elastic multi-structures
J. Huang, Z. Shi and Y. Xu
Science in China 49(2006), 109-129.
A finite element method is introduced to solve the general elastic multi-structure problem, in which the displacements on bodies, the longitudinal displacements on plates and the longitudinal displacements on beams are discretized using conforming linear elements, the rotational angles on beams are discretized using conforming elements of second order, the transverse displacements on plates and beams are discretized by the Morley elements and the Hermite elements of third order, respectively. The generalized Korn's inequality is established on related nonconforming element spaces, which implies the unique solvability of the finite element method. Finally, the optimal error estimate in the energy norm is derived for the method.
The unique determination of the primary current by MEG and EEG
L. Peng, J. Cheng and J. Lu
Phys. Med. Biol. 51(2006), 5565-5580.
In this paper, we use a more realistic head model with ovoid geometry for approximation of a human head. By inverting the Geselowitz equation, some analytic results on the inverse MEG problem are presented in homogenous ovoid geometry. On one hand, some information about the components of primary current is shown by the decomposition of primary current in different coordinates in the case of a special MEG sensor position. On the other hand, in the general case, using decomposition of the primary current in spherical coordinates, we show that two scalar functions which specify the tangential part of the primary current can be uniquely determined with the assumption that two scalar functions are conjugate and harmonic in terms of two variables. Hence, the tangential part of the current can be completely known from the two scalar functions. Moreover, we obtain the unique determination of the primary current by combining MEG and EEG.
Spurious numerical solutions of delay differential equations
H. Tian, L. Fan, Z. Yuan and J. Xiang
J. Comp. Math. 24(2006), 181-192.
This paper deals with the relationship between asymptotic behavior of the numerical solution and that of the true solution itself for fixed step-sizes. The numerical solution is viewed as a dynamicalsystem in which the step-size acts as a parameter. We present a unified approach to look for bifurcations from the steady solutions into spurious solutions as step-size varies.
Jacobi pseudospectral method for fourth order problems
Z. Wan, B. Guo and Z. Wang
J. Comp. Math. 24(2006), 481-500.
In this paper, we investigate Jacobi pseudospectral method for fourth order problems. We establish some basic results on the Jacobi-Gauss-type interpolations in non-uniformly weighted Sobolev spaces, which serve as important tools in analysis of numerical quadratures, and numerical methods of differential and integral equations. Then we propose Jacobi pseudospectral schemes for several singular problems and multiple-dimensional problems of fourth order. Numerical results demonstrate the spectral accuracy of these schemes, and coincide well with theoretical analysis.
Stair Laguerre pseudospectral method for differential
equations on the half line
L. Wang and B. Guo
Adv. Comp. Math. 25(2006), 305-322.
A stair Laguerre pseudospectral method is proposed for numerical solutions of differential equations on the half line. Some approximation results are established. A stair Laguerre pseudospcetral scheme is constructed for a model problem. The convergence is proved. The numerical results show that this new method provides much more accurate numerical results than the standard Laguerre spectral method.
Mixed Fourier-Jacobi spectral method
L. Wang and B. Guo
J. Math. Anal. Appl. 315(2006), 8-28.
This paper is for mixed Fourier-Jacobi approximation and its applications to numerical solutions of semi-periodic singular problems, semi-periodic problems on unbounded domains and axisymmetric domains, and exterior problems. The stability and convergence of proposed spectral schemes are proved. Numerical results demonstrate the efficiency of this new approach.
Reconstruction of high order derivatives from input data
Y. Wang, Y. Hon and J. Cheng
J. Inv. Ill-Posed Problems 14(2006), 205-218.
This paper gives a numerical method for reconstructing the original function and its derivatives from discrete input data. It is well known that this problem is ill-posed in the sense of Hadamard. The solution for the first order derivative has been proposed by  and , using the Tikhonov regularization technique. In this paper, under an assumption that the original function has a square integrable k-th order derivative, we propose a reconstruction method for the j-th order derivative where . A convergence rate estimate is obtained by taking a new choice of the Tikhonov parameter. Numerical example is given to verify the effectiveness and accuracy of the proposed method.
Asymptotic behavior of solutions for a Cooperation-Diffusion model with a Saturating interaction
Comp. Math. Appl. 52(2006), 339-350.
This paper is concerned with a Lotka-Volterra cooperation-diffusion model with a saturating interaction term for one species. The goal of the paper is to investigate the asymptotic behavior of the time-dependent solution in relation to the corresponding steady-state solutions under homogeneous Neumann boundary condition. Some simple and easily verifiable conditions are given to the rate constants so that for every nontrivial nonnegative initial function the corresponding time-dependent solution converges to one of the nonnegative constant steady-state solutions as time tends to infinity. This convergence result leads to the existence and uniqueness of a positive (or nonnegative) steady-state solution and the global asymptotic stability of a given nonnegative constant steady-state solution. In terms of ecological dynamics, it also gives some coexistence, permanence and extinction results for the model.
Convergence analysis of a monotone method for fourth-order semilinear elliptic boundary value problems
Appl. Math. Lett. 19(2006), 332-339.
This work is concerned with the convergence of a monotone method for fourth-order semilinear elliptic boundary value problems. A comparison result for the rate of convergence is given. The global error is analyzed, and some sufficient conditions are formulated for guaranteeing a geometric rate of convergence.
Time-delayed finite difference Reaction-Diffusion systems with nonquasimonotone functions
Y. Wang and C.V. Pao
Numer. Math. 103(2006), 332-339.
This paper is concerned with finite difference solutions of a system of reaction-diffusion equations with coupled nonlinear boundary conditions and time delays. The reaction functions and the boundary functions are not necessarily quasimonotone, and the time delays may appear in the reaction functions as well as in the boundary functions. The investigation is devoted to the finite difference system for both the time-dependent problem and its corresponding steady-state problem. Some monotone iteration processes for the finite difference systems are given, and the asymptotic behavior of the time-dependent solution in relation to the steady-state solution is discussed. The asymptotic behavior result leads to some local and global attractors of the time-dependent problem, including the convergence of the time-dependent solution to a unique steady-state solution. An application and some numerical results to an enzyme-substrate reaction-diffusion problem are given. All the results are directly applicable to parabolic-ordinary systems and to reaction-diffusion systems without time delays.
A mixed spectral method for incompressible viscous fluid flow in an
Z. Wang and B. Guo
Comp. Appl. Math. 27(2005),343-364.
This paper considers the numerical simulation of incompressible viscous fluid flow in an infinite strip. A mixed spectral method is proposed using the Legendre approximation in one direction and the Legendre rational approximation in another direction. Numerical results demonstrate the efficiency of this approach. Some results on the mixed Legendre-Legendre rational approximation are established, from which the stability and convergence of proposed method follow.
Error analysis on scrambled Quasi-Monte Carlo quadrature rules using Sobol points
Frontiers and Contemporary Applied Mathematics, CAM 6, 2005, 254-267.
We study the worst-case error and random-case of scrambled Quasi-Monte Carlo quadrature rules using Sobol points. The function spaces considered in this article are the weighted Hilbert spaces generated by Haar wavelets with weights and a parameter which reflects the smoothness of the spaces. Conditions are found under which multivariate integration using the scrambled Sobol points is strongly tractable in the worst-case and random-case settings, respectively. The worst-case results improve upon those of Wang (2003). The random-case results give weaker conditions for strong tractability than in the worst-case setting.
Spherical harmonic-generalized Laguerre spectral method
for exterior problems
X. Zhang and B. Guo
J. Sci. Comp. 27(2006), 523-537.
In this paper, we propose the mixed spherical harmonic-generalized Laguerre spectral method for three-dimensional exterior problems. Some approximation results are established. As an example, a model problem is considered. The convergence of proposed scheme is proved. Numerical results demonstrate the efficiency of this approach.
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