Doubly excited resonance states of two-electron systems in exponential cosine-screened Coulomb potentials

We have made an investigation on the resonance states in two-electron systems interacting with exponential cosine-screened Coulomb potentials (ECSCP) within the framework of stabilization method. Highly correlated wave functions are used to take into account the correlations of the charge particles. Results for resonance energies and widths are reported for Z=1,2 and 3. For the unscreened case, our reported results are in nice agreement with the results so far available in the literature.     

用Origin 6.0绘制波函数平面图形

Origin6．0作为一个数学软件平台,集数据管理与分析、二维及三维函数和离散数据绘图、程序开发等多功能于一体,广泛用于科学研究和工程计算。本文利用Origin6．0软件的函数绘图功能,绘制氢原子径向波函数、径向密度函数、径向分布函数、氢原子轨道及电子云的角度分布、杂化轨道等二维图形。给出了波函数的软件可读化形式和绘图的实际操作过程。与其它函数绘图方法相比,界面更友好,操作更方便。     

On the convergence of Newton-type methods using recurrent functions

We introduce the new idea of recurrent functions to provide a new semilocal convergence analysis for Newton-type methods. It turns out that our sufficient convergence conditions are weaker, and the error bounds are tighter than in earlier studies in many interesting cases [X. Chen, On the convergence of Broyden-like methods for nonlinear equations with nondifferentiable terms, Ann. Inst. Statist. Math. 42 (1990), pp. 387–401; X. Chen and T. Yamamoto, Convergence domains of certain iterative methods for solving nonlinear equations, Numer. Funct. Anal. Optim. 10 (1989), pp. 37–48; Y. Chen and D. Cai, Inexact overlapped block Broyden methods for solving nonlinear equations, Appl. Math. Comput. 136 (2003), pp. 215–228; J.E. Dennis, Toward a unified convergence theory for Newton-like methods, in Nonlinear Functional Analysis and Applications, L.B. Rall, ed., Academic Press, New York, 1971, pp. 425–472; P. Deuflhard, Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms, Springer Series in Computational Mathematics, Vol. 35, Springer-Verlag, Berlin, 2004; P. Deuflhard and G. Heindl, Affine invariant convergence theorems for Newton's method and extensions to related methods, SIAM J. Numer. Anal. 16 (1979), pp. 1–10; Z. Huang, A note of Kantorovich theorem for Newton iteration, J. Comput. Appl. Math. 47 (1993), pp. 211–217; L.V. Kantorovich and G.P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982; D. Li and M. Fukushima, Globally Convergent Broyden-like Methods for Semismooth Equations and Applications to VIP, NCP and MCP, Optimization and Numerical Algebra (Nanjing, 1999), Ann. Oper. Res. 103 (2001), pp. 71–97; C. Ma, A smoothing Broyden-like method for the mixed complementarity problems, Math. Comput. Modelling 41 (2005), pp. 523–538; G.J. Miel, Unified error analysis for Newton-type methods, Numer. Math. 33 (1979), pp. 391–396; G.J. Miel, Majorizing sequences and error bounds for iterative methods, Math. Comp. 34 (1980), pp. 185–202; I. Moret, A note on Newton type iterative methods, Computing 33 (1984), pp. 65–73; F.A. Potra, Sharp error bounds for a class of Newton-like methods, Libertas Math. 5 (1985), pp. 71–84; W.C. Rheinboldt, A unified convergence theory for a class of iterative processes, SIAM J. Numer. Anal. 5 (1968), pp. 42–63; T. Yamamoto, A convergence theorem for Newton-like methods in Banach spaces, Numer. Math. 51 (1987), pp. 545–557; P.P. Zabrejko and D.F. Nguen, The majorant method in the theory of Newton–Kantorovich approximations and the Pták error estimates, Numer. Funct. Anal. Optim. 9 (1987), pp. 671–684; A.I. Zin[cbreve]enko, Some approximate methods of solving equations with non-differentiable operators, (Ukrainian), Dopovidi Akad. Nauk Ukraïn. RSR (1963), pp. 156–161]. Applications and numerical examples, involving a nonlinear integral equation of Chandrasekhar-type, and a differential equation are also provided in this study.     

Computing Green's function of the initial-boundary value problem for the wave equation in a layered cylinder

A new analytical method for the approximate computation of the time-dependent Green's function for the initial-boundary value problem of the three-dimensional wave equation on multi-layered bounded cylinder is suggested in this paper. The method is based on the derivation of eigenvalues and eigenfunctions for an ordinary differential equation with piecewise constant coefficients, and an approximate computation of Green's function in the form of the Fourier series with a finite number of terms relative to the orthogonal set of the derived eigenfunctions. The computational experiment confirms the robustness of the method.     

An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method

In this paper we propose a collocation method for solving some well-known classes of Lane-Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain. They are categorized as singular initial value problems. The proposed approach is based on a Hermite function collocation (HFC) method. To illustrate the reliability of the method, some special cases of the equations are solved as test examples. The new method reduces the solution of a problem to the solution of a system of algebraic equations. Hermite functions have prefect properties that make them useful to achieve this goal. We compare the present work with some well-known results and show that the new method is efficient and applicable.