Applied Numerical Analysis & Computational Mathematics最新文献

This paper presents an approach of eigenvalue perturbation theory, which frequently arises in engineering and physical science. In particular, the problem of interest is an eigenvalue problem of the form (A + εB)φ(ε) = λ(ε)φ(ε) where A and B are n × n matrices, ε is a parameter, λ(ε) is an eigenvalue, and φ(ε) is the corresponding eigenvector. In working with perturbation theory, we assume that the eigenvalue λ(ε) has a power series expansion. As such, a large effort presented in this paper involves the derivation of formulas for the power series coefficients, which are used to approximate λ(ε). In the process, the analysis requires some basic background of complex function theory. The rest of this paper presents an application of this approach to a common problem in engineering, namely, the vibration of a square membrane under the effect of a small perturbation, which results in a shape of a trapezoid. The displacement of the membrane of this particular shape is described by the differential equation u_tt = c²Δu with a fixed boundary Γ and is subjected to the boundary condition u = 0 on Γ. While the solution of the unperturbed hyperbolic problem of this type is well known and easy to find, it becomes quite difficult when the domain is perturbed, giving rise to a slightly different shape other than the original standard shapes, such as squares, rectangles, or circles. This paper addresses one of these aspects in which the domain results in a shape of a trapezoid. The approach should apply to other shapes as well. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The main subject of this paper is the characterization of a modified family of Newton-type iterative processes of R-order at least three to approximate a solution of a nonlinear equation in Banach spaces. We weaken the usual semilocal convergence conditions, for this type of iterative processes, assuming that the second derivative is ω-conditioned. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Two different linearized schemes are applied to a parametric finite-difference scheme concerning the numerical solution of the Boussinesq equation. At the first linearized scheme the nonlinear term of the equation is substituted by an appropriate value, while at the second scheme we use Taylor's expansion. Both schemes are analyzed for local truncation error, stability and convergence. The results of the experiments are examined for their accuracy for the single and the double-soliton waves to known from the bibliography numerical schemes. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

A mixed-stress formulation presented here combines the methodologies of phase field and fluid-structure interactions in order to simulate phase transformations involving moving and rotating solid bodies interacting with fluids. The Navier-Stokes equations are solved everywhere, with an additional strain tensor field overlaid on the solid, such that elastic stress can be incorporated into the equation of motion. An additional term in the equation for strain evolution provides rotation due to local vorticity. The results exhibit a small amount of artificial erosion of the solid due to nonzero normal velocity within the diffuse interface at separation points; this is discussed here as a phenomenon common to all diffuse interface formulations incorporating convection. Two-dimensional simulations presented here demonstrate oscillations due to feedback between surface tension and elastic stress, and rigid body motion with particle collision and agglomeration which involves a change in the topology of the fluid-solid interface. The formulation and simulations presented here use isotropic Cahn-Hilliard free energy, and extension of the formulation to three-dimensional and anisotropic systems is straightforward. Likewise, these simulations describe linear elastic solids with Poisson ratio of 1 (equivalent to ½ in three dimensions), and extension to more complex mechanical constitutive behavior is discussed. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Robust control of dynamic linear systems with model uncertainties involves modified Riccati equations (MRE) i.e. Riccati equations with additional terms depending on the uncertainties and on specific upper bounding functions. Robust controller design is based on the existence and computation of positive definite solutions of MRE. In the present paper MRE are investigated and computational techniques are presented. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

A procedure to match a Taylor series with a series of exponentials allows an adequate solution for a category of nonlinear ordinary differential equations. We deal with examples describing the motion of a metal ball in a fluid in order to determine the viscosity. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The most widely used and accepted method of studying wind generated waves is an examination of the spectra at a single point. This approach is based mostly on the assumption that recorded time series of the surface elevation, pressure or velocities are the results of linear superposition of small amplitude oscillations regardless of their directions of propagation. In this work, the vertical and horizontal accelerations of a floating buoy have been used in order to calculate the directional wave spectra in coastal zones. The method used is a parametric one, which incorporates two waves propagating in different directions. The resulting splitting of the incoming waves into two main directions is used for 24-hours wave height prediction with an adaptive neural network. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

V-cycle, F-cycle and W-cycle multigrid algorithms for interior penalty methods for second order elliptic boundary value problems are studied in this paper. It is shown that these algorithms converge uniformly with respect to all grid levels if the number of smoothing steps is sufficiently large, and that the contraction numbers decrease as the number of smoothing steps increases, at a rate determined by the elliptic regularity of the problem. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Differential qd (dqd) algorithm with shifts is probably the fastest known algorithm which computes eigenvalues of symmetric tridiagonal matrices with high relative accuracy.

In this paper we will construct a similar algorithm for computing eigenvalues of skew-symmetric matrices, which is based on implicit usage of both the QR and the symplectic QR factorizations. If we apply this algorithm to tridiagonal skew-symmetric matrices, we obtain the skew-symmetric dqd algorithm. This algorithm also enjoys high relative stability. However, incorporation of shifts is much harder then in the symmetric case, and yet to be implemented.

Finally, the standard algorithm for computing the eigenvalues of tridiagonal skew-symmetric matrices can also be interpreted in the context of the skew-symmetric dqd algorithm. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

In this work, a new representation is developed for generalized hypergeometric functions of type _pF_p. To this end a first order vector differential equation is constructed in a way such that the derivative of the unknown vector has unit matrix coefficient. The solution vector differential equation is rewritten in the form of a series expansion. An infinite process of factor extractions and annihilations is employed to construct finally a vector differential equation that can be easily solved. Truncation of this scheme can be used to get approximations to hypergeometric functions of type _pF_p. A simple, yet meaningful, implementation seems to give quite promising results. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)