Applied Numerical Analysis & Computational Mathematics最新文献

In this work, the radial time-independent Schrödinger equation of a screened Coulomb potential system at the zero energy limit is first converted to a weighted eigenvalue problem of an ordinary differential operator. Then, by using an appropriate coordinate transformation, the differential equation is transformed into a form whose first and second order derivative related terms become same as the Extended Jacobi Polynomials' differential equation's corresponding terms. Only difference is the appearance of a multiplicative operator which can be considered as an effective potential. Work focuses on the point whether the solution is obtained easily depending on the structure of this potential. In this direction a screened Coulomb potential with a specific rational screening function is considered. The analytical solutions for the critical values of the screening parameter and the form of the wave function at the threshold of the continous spectrum are obtained. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Adaptive multilevel methods are methods for solving partial differential equations that combine adaptive grid refinement with multigrid solution techniques. These methods have been shown to be very effective on sequential computers. Recently, a technique for parallelizing these methods for cluster computers has been developed. This paper presents an overview of a particular adaptive multilevel method and the parallelization of that method via the full domain partition. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Despite the intensive work done on developing numerical schemes for stochastic differential equations, stability analysis of these schemes when applied to stochastic differential systems is still under premature stage. Motivated by the work of Saito and Mitsui in [11], we investigate mean square stability for a class of weak second order Runge-Kutta schemes applied to 2-dimensional stochastic differential systems. Stability criteria will be established, and numerical examples in support of the theoretical analysis will be provided. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The main purpose of this work is to obtain the general structure of a product type of multivariate function when the values of the function are given randomly at the nodes of a hyperprism. When the dimensionality of multivariate interpolation and the number of the data sets increase unboundedly, many problems can be encountered in the standard numerical methods. Factorized High Dimensional Model Representation (FHDMR) is used to obtain the general structure of this given multivariate function. The components of FHDMR are determined by using the components of the Generalized High Dimensional Model Representation (GHDMR).The given random discrete data is partitioned by GHDMR method. This partitioned data produces a structure for the multivariate function by using single variable Lagrange interpolation formula including only the constant term and the univariate terms of the GHDMR components. Finally, a general structure is obtained via FHMDR by using these components. By this way, the multidimensionality of a multivariate interpolation is approximated by lower dimensional interpolations like univariate ones. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

In recent years, model order reduction scheme for reducing the dimension of linear systems has become very popular for computer aided design of the systems. In this paper, we analyze the existent methods of the model order reduction techniques used in Very Large Scale Integrated (VLSI) circuit interconnection modelling in terms of numerical stability, computational speed and accuracy. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

An explicit hybrid symmetric four-step method of algebraic order six is developed in this paper. Numerical comparative results from the application of the new method to well known periodic orbital problems, demonstrate the efficiency of the method presented here. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

A set of discrete data (x_k, f (x_k)) (k = 1, 2, …, m) may be fitted in any l_p norm by a nonlinear form derived from a function g (L) of a linear form L = L(x). Such a nonlinear approximation problem may under appropriate conditions be (asymptotically) replaced by the fitting of g^–1 (f) by L in any l_p norm with respect to a weight function w = g′ (g^–1 (f)). In practice this “direct method” can yield very good results, sometimes coming close to a best approximation. However, to ensure a near-best approximation, by using an iterative procedure based on fitting L, two algorithms are proposed in the l₂ norm - one already established by Mason and Upton (1989) and one completely new, based on minimising the two algorithms and multiplicative combinations of errors, respectively. For a general g we prove they converge locally and linearly with small constants. Moreover it is established that they converge to different (nonlinear) “Galerkin type” approximations, the first based on making the explicit error ϵ ≡ f – g (L) orthogonal to a set of functions forming a basis for L, and the second based on making the implicit error ϵ* ≡ w(g^–1 (f) – L) orthogonal to such a basis. Finally, and mainly for comparison purposes, the well known Gauss-Newton algorithm is adopted for the determination of a best (nonlinear) approximation. Illustrative problems are tackled and numerical results show how effective all of the algorithms can be. To add a further novel feature, L is here chosen throughout to be a radial basis function (RBF), and, as far as we are aware, this is one of the first successful uses of a (nonlinear) function of an RBF as an approximation form in data fitting. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

This work deals with the optimal control of one dimensional quantum harmonic oscillator under an external field characterized by a linear dipole function. The penalty term is taken as kinetic energy. The objective operator whose expectation value is desired to get a prescribed target value is taken as the square of the position operator. The dipole function hypothesis in the external field is valid only for weak fields otherwise hyperpolarizability terms which contain powers of the field amplitude higher than 1 should be considered. The weak field assumption enables us to develop a first order perturbation approach to get approximate solutions to the wave and costate equations. These solutions contain the field amplitude and another unknown, so-called deviation constant, through some certain integrals. By inserting these expressions into the connection equation which functionally relates the field amplitude to the wave and costate function it is possible to produce an integral equation. Same manipulations on the objective equation results in an algebraic equation to determine the deviation parameter. The algebraic deviation equation produces incompatibility which can be relaxed by including second order perturbative term of wave function. The integral and algebraic equations mentioned above are asymptotically solved. Their global solutions are left to future works since the main purpose of this work is to obtain the so-called field and deviation equations. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

This paper compares the performances of a least squares finite element method (LSFEM), a mixed Galerkin finite element method (MGFEM), and an implicit finite difference method (IFDM), when they are applied to an acoustic scattering problem modelled by the one-dimensional Helmholtz equation. First, the boundary value problem is written as a first order system of differential equations, and variational formulations for the LSFEM and MGFEM are derived. Then, by using appropriate basis functions, the stiffness matrices and load vectors which define the corresponding linear algebraic systems are obtained. In all cases, the stiffness matrix is banded and for the LSFEM it is also Hermitian. Numerical tests show that all these methods have quadratic convergence to the exact solution. However, their efficiency assessed in terms of the number of nodes and computing time required to reach quadratic convergence varies. It is observed that LSFEM requires many more nodes and employs much more time than MGFEM and IFDM to reach the quadratic convergence for high frequencies. It is also found that MGFEM has less numerical dispersion and, as a consequence, performs better than IFDM and LSFEM for high frequencies. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)