Mathematical simulation of drug diffusion is a significant tool for predicting the bio‐transport process. Moreover, the reported models in the literature are based on Fick's approach, which leads to an infinite propagation speed. Consequently, it is essential to construct a mathematical model to represent the diffusion processes for estimating drug concentrations at different sites and throughout the circulation. Thus, in this article, the diffusion process is employed to propose three models for estimating the drug release from multi‐layer cylindrical tablets. A fractional model is presented based on Fick's approach, while classical and fractional Cattaneo models are presented using the relaxed principle. Various numerical methods are used to solve the specified problem. The numerical scheme's stability and convergence are demonstrated. Drug concentration and mass profiles are presented for the tablet and the external medium and compared with the in vivo plasma profiles. The results show the efficiency and precision of the proposed fractional models based on the fourth‐order weighted‐shifted Grünwald–Letnikov difference operator approximation. These models are compatible with the in vivo data compared with the classical Fick's one.

In this paper, we consider the bivariate Hermite interpolation introduced by Bojanov and Xu [SIAM J. Numer. Anal. 39(5) (2002) 1780-1793. The nodes of the interpolation with 2k− , where = 0 or 1, are the intersection points of 2k+1 distinct rays from the origin with a multiset of k+1− concentric circles. Parameters are the values and successive radial derivatives, whenever the corresponding circle is multiple. The poisedness of this interpolation was proved only for the set of equidistant rays [Bojanov and Xu, 2002] and its counterparts with other conic sections [Hakopian and Ismail, East J. Approx. 9 (2003) 251-267]. We show that the poisedness of this (k + 1 − )(2k + 1) dimensional Hermite interpolation problem is equivalent to the poisedness of certain 2k + 1 dimensional Lagrange interpolation problems. Then the poisedness of Bojanov-Xu interpolation for a wide family of sets of rays satisfying some simple conditions is established. Our results hold also with above circles replaced by ellipses, hyperbolas, and pairs of parallel lines.Next a conjecture [Hakopian and Ismail, J. Approx. Theory 116 (2002) 76-99] concerning a poisedness relation between the Bojanov-Xu interpolation, with set of rays symmetric about x-axis, and certain univariate lacunary interpolations is established. At the end the poisedness for a wide class of lacunary interpolations is obtained.

As a generalization of regular rings, we introduce the notion, of m-regular rings, that is for all R a  , there is a fixed positive integer m such that m a is a Von-Neumann regular element. Some characterization and basic properties of these rings will be given. Also, we study the relationship between them and Von-Neumann regular rings, regular rings, reduced rings, locally rings, uniform rings and 2-primal rings.

Let I be a right (left) ideal of a ring R. Then R/I is a right (left) generalized mflat modules (GmFmodule) if and only if for each aI  , there exist bI  and a fixed positive integer m such that