Multistep methods for differential algebraic equations

Multistep methods for the differential/algebraic equations (DAEs) in the form of

Parameter choice by discrepancy principles for ill-posed problems leading to optimal convergence rates

Schock (Ref. 1) considered a general a posteriori parameter choice strategy for the Tikhonov regularization of the ill-posed operator equationTx=y which provides nearly the optimal rate of convergence if the minimal-norm least-squares solution belongs to the range of the operator (T ^* T) ^v , o<v1. Recently, Nair (Ref. 2) improved the result of Schock and also provided the optimal rate ifv=1. In this note, we further improve the result and show in particular that the optimal rate can be achieved for 1/2v1.The final version of this work was written while M. T. Nair was a Visiting Fellow at the Centre for Mathematics and Its Applications, Australian National University, Canberra, Australia. The work of S. George was supported by a Senior Research Fellowship from CSIR, India.     

An energy-transport model for semiconductors derived from the Boltzmann equation

An energy-transport model is rigorously derived from the Boltzmann transport equation of semiconductors under the hypothesis that the energy gain or loss of the electrons by the phonon collisions is weak. Retaining at leading order electron-electron collisions and elastic collisions (i.e., impurity scattering and the elastic part of phonon collisions), a rigorous diffusion limit of the Boltzmann equation can be carried over, which leads to a set of diffusion equations for the electron density and temperature. The derivation is given in both the degenerate and nondegenerate cases.     

Existence and asymptotic results for a system of integro-partial differential equations

A non-Fourier phase field model is considered. A global existence result for a Dirichlet, or generalized Neumann, initial-boundary value problem is obtained, followed by a discussion of the regularity and asymptotic properties of solutions ast.This research was supported in part by the National Science Foundation under Grant DMS 91-11794 and in part by the Italian M.U.R.S.T. project Problemi non lineari...Part of this author's work was done while visiting Ohio University.     

WKB-type approximations for second-order differential equations in -algebras

Rigorous asymptotic approximations of the WKB (or Liouville-Green) type are obtained for a basis of solutions to in the framework of -algebras. Both cases and are included, thus generalizing the classical theory for scalar equations developed by F.W.J. Olver to matrix as well as to infinite-dimensional equations.

     

An application of the generalized alternating polynomials to the numerical solution of Fredholm integral equations

Generalized alternating polynomials have been introduced by the author earlier. In the present paper their indirect analogue is constructed for numerical solution of the Fredholm linear integral equations. Although the proposed method is a particular case of the general projection scheme, its valuable feature is the presence of a sequence of parameters, which, for sufficiently smooth kernels and inhomogeneous terms, serves as an indicator of the quality of approximation.     

Efficiency of composite vs. conically shaped chromatographic columns

The theory of composite chromatographic columns has been re-examined and general equations relating their efficiency to those of individual columns were derived. The derivations were based on elementary principles without reference to the principle of variance additivity. Derived also, were general equations for the efficiency of columns with continuously varying diameters, HETP and other properties from which formulae for special cases of practical interest were deduced. An example comparing the efficiencies of two columns, one conical and the other composite was given. The calculations show clearly that the conical column is superior in efficiency for the same total length and total retention volume.     

Global optimization and stochastic differential equations

Let ⁿ be then-dimensional real Euclidean space,x=(x ₁,x ₂, ...,x _n)^{T n} , and letf: ⁿ R be a real-valued function. We consider the problem of finding the global minimizers off. A new method to compute numerically the global minimizers by following the paths of a system of stochastic differential equations is proposed. This method is motivated by quantum mechanics. Some numerical experience on a set of test problems is presented. The method compares favorably with other existing methods for global optimization.This research has been supported by the European Research Office of the US Army under Contract No. DAJA-37-81-C-0740.The third author gratefully acknowledges Prof. A. Rinnooy Kan for bringing to his attention Ref. 4.     

The Kissinger law and isokinetic effect: Part II. Experimental analysis

On the basis of copper sulphate pentahydrate thermal dissociation, for analyzed reactions I to IV, 6 thermokinetic equations was discussed. Arrhenius law parameters were determined and the isokinetic effect (IE) and Kissinger law appearing was analyzed. It was found that only dependence resulting from isokinetic effect, in the form k _m=q/T _m, relates to the suitable thermokinetic Eq. (2) and Kissinger law in modified form (14). The confirmation was made that the possibility of determining the averaged activation energy from thermokinetic equations using suitable correction coefficients exists.This revised version was published online in November 2005 with corrections to the Cover Date.     

Deformation behavior of thin,compliant layers under tensile loading conditions

In this article, a connection is made between the behavior of thin layers of Newtonian liquids under tensile loading conditions and the behavior of highly deformable elastic or viscoelastic solids, which are more commonly used as adhesives. The behavior of Newtonian liquids is understood in the most quantitative detail and serves as a starting point for understanding the origins of fingering and cavitation instabilities that appear when the tensile deformation rates applied to these layers are sufficiently large. Similar instabilities appear in solid systems and can be attributed to common features of the stress distribution for incompressible liquids and solids. A unifying treatment is presented that can be used to understand the overall deformation behavior and adhesive performance of a wide variety of solid and liquid systems that are typically applied as thin layers. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 4023–4043, 2004