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1

Pseudo almost periodic solutions of infinite class under the light of measure theory

Votsia Djokata, Zabsonre Issa

Silesian Journal of Pure and Applied Mathematics

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2020

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vol. 10, iss. 1

15--43

EN

The aim of this work is to present new approach to study weighted pseudo almost periodic functions with infinite delay using the measure theory. We present a new concept of weighted ergodic functions which is more general than the classical one. Then we establish many interesting results on the functional space of such functions. We study the existence and uniqueness of (μ, ν)-pseudo almost periodic solutions of infinite class for some neutral partial functional differential equations in a Banach space when the delay is distributed on ]−∞, 0] using the spectra decomposition of the phase space developed in Adimy and co-authors.

2

Computer-based simulation of plasma concentration time-proﬁles of drug in nonlinear two-compartment model

Piekarski S., Kiełczyński P., Szalewski M., Rewekant M.

Computer Assisted Methods in Engineering and Science

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2013

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Vol. 20, no. 4

279--288

EN

The main interest of pharmacokinetics is the study of the fate of drugs in the living organism. This work proposes the system of the conservation laws that describes time-dependent concentrations of a drug, after a single intravenous administration. Compared with others, the proposed model considers both free and protein-bound drug concentrations at the same time. Plasma protein binding captured in the model enters the nonlinearity arising from the Guldberg-Waage law. According to our best knowledge, the analytical solution for our system does not exist. Our model allows the calculation of the free and bound-drug protein concentrations at any time point and at any dose after single intravenous bolus dose administration. In order to compare the empirical with simulated data, a numerical approach has been proposed. On the basis of published experimental data the model validation has been carried out. The goodness of ﬁt was satisfactory (R 2 = 0.99) and the experimental and simulated AUC (area under the curve) values, as the measure of the bioavailability of drug, were similar (150 M/hxh−1). The preliminary assessment of the model credibility was positive and encouraged further studies.

3

Some analytical properties of dissolving operators related with the Cauchy problem for a class of nonautonomous partial differential equations. Part. 1

Pytel-Kudela M., Prykarpatsky A.K.

Opuscula Mathematica

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2006

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Vol. 26, no. 1

131-136

EN

The analytical properties of dissolving operators related with the Cauchy problem for a class of nonautonomous partial differential equations in Hilbert spaces are studied using theory of bilinear forms in respectively rigged Hilbert spaces triples. Theorems specifying the existence of a dissolving operator for a class of adiabatically perturbed nonautonomous partial differential equations are stated. Some applications of the results obtained are discussed.

4

Gradient formulation in coupled damage-plasticity

Voyiadjis G. Z., Dorgan R. J.

Archives of Mechanics

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2001

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Vol. 53, nr 4-5

565-597

EN

This work provides a consistent and systematic framework for the gradient approach in coupled damage-plasticity that enables one to better understand the effects of material inhomogeneity on the macroscopic behavior and the material instabilities. The idea of multiple scale effects is made more general and complete by introducing damage and plasticity internal state variables and the corresponding gradients at both the macro and mesoscale levels. The mesoscale gradient approach allows one to obtain more precise characterization of the nonlinearity in the damage distribution; to address issues such as lack of statistical homogeneous state variables at the macroscale level such as debonding of fibers in composite materials, crack, voids, etc., and to address nonlocal influences associated with crack interaction. The macroscale gradients allow one to address non-local behavior of materials and interpret the collective behavior of defects such as dislocations and cracks. The development of evolution equations for plasticity and damage is treated in a similar mathematical approach and formulation since both address defects such as dislocations for the former and cracks/voids for the latter. Computational issues of the gradient approach are introduced in a form that can be applied using the finite element approach.

5

The existence of moments of solutions to transport equations with inelastic scattering and thie applications in the asymptotic analysis

Banasiak J.

Journal of Applied Analysis

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2000

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Vol. 6, nr 2

187-211

EN

In this paper we prove the existence of all moments of the solutions to the time-dependent spatially homogeneous transport equation describing elastic and inelastic scattering of particles. The proof uses the theory of resolvent positive operators and Desch's perturbation theorem. As an application we carry out the asymptotic analysis of the full transport equation with dominant elastic scattering and, using the results of the first part of the paper, we show that its solution can be approximated in the L1-norm by the solution of the limit equation obtained by formal asymptotic expansion.