Staff

Abstract:
It is well known that a tensor Stieltjes function f represents an effective transport coefficient q of an inhomogeneous medium consisting of two isotropic components. In this paper, we investigate multipoint matrix Padé approximants to matrix expansions of f. We prove that matrix Padé ones to f estimate f from the top and below. Consequently the Padé approximants to q form upper and lower bounds on q. The inequalities for matrix Padé bounds on f and q are established. They reduce to the inequalities for scalar Padé ones Tokarzewski (ZAMP 61:773–780, 2010). As an illustrative example, matrix Padé estimates of an effective conductivity of a specially laminated two-phase medium are computed.

Keywords:
Matrix Stieltjes functions, Matrix continued fractions, Effective transport coefficients, Composites, Porous materials, Bounds

Abstract:
In this paper we establish the general inequalities for diagonal and subdiagonal multipoint Pad´e approximants to a Stieltjes function f in terms of power expansion of f on the real line. The inequalities derived produce the best upper and lower bounds on f with respect to the given coefficients of Stieltjes series. As an example of applications sequences of upper and lower Pad´e bounds converging to the effective dielectric constant of a random array of spheres are evaluated.

Keywords:
N -point Pad´e approximants, Stieltjes functions, continued fractions

Abstract:
The effective transport coefficient q of two-phase media is uniquely expressed by a special Stieltjes function f. In this paper, we incorporate into bounds on q, the power expansions of f available at a number of real points. To this end we apply the S-continued fraction technique developed in Baker (Essentials of Padé Approximants, Academic Press, New York, 1975) and Tokarzewski (IFTR Rep 4:3–171, 2005). Moreover, we establish the fundamental inequalities for S-bounds on q. As an example of an application the sequences of upper and lower S-bounds on the effective conductivity of a face-centered cubic lattice of spheres are evaluated.

Keywords:
Stieltjes functions, S-continued fractions, Effective transport coefficients, Composites

Abstract:
Upper and lower estimates of Stieltjes function by N-point Padé approximants can be obtained using the new general inequality reported by Tokarzewski et al. (Arch. Mech. 54 (2002) 141–153) and rigorously proved in the present paper. In addition, we prove that the multipoint Padé approximants to Stieltjes function are symmetric with respect to the order of choice of the considered points.

Keywords:
N-point Padé approximants, Stieltjes functions

Abstract:
A Stieltjes function is expanded in mixed T- and S-continued fraction. The relations between approximants of this continued fraction and two-point Padé approximants are established. The method used by Gilewicz and Magnus (J. Comput. Appl. Math. 49 (1993) 79; Integral Transforms Special Functions 1 (1993) 9) has been adapted to obtain the exact relations between the errors of the contiguous two-point Padé approximants in the whole cut complex plane.

Keywords:
Two-point Padé approximation, Stieltjes functions

Abstract:
The aim of this contribution is to examine the S-continued fraction method of obtaining bounds on the effective dielectric constant εe of a two-phase composite for the case where the dielectric coefficients ε1and ε2 of both components are either complex or real. The starting point for our study is a power expansion of εe(z) at(z)=0 (z)=ε2/ε1-1. The obtained S-continued fraction bounds have an interesting mathematical structure convenient for theoretical and numerical investigations of εe. They also agree with the earlier estimations reported by Bergman and Milton. Specific examples of calculation of bounds on εe by theS-continued fraction method are also provided.

Keywords:
composite medium, bounds, dielectric constants, Stieltjes function