Localization of Fréchet Frames and Expansion of Generalized Functions.

Matrix-type operators with the off-diagonal decay of polynomial or sub-exponential types are revisited with weaker assumptions concerning row or column estimates, still giving the continuity results for the frame type operators. Such results are extended from Banach to Fréchet spaces. Moreover, the localization of Fréchet frames is used for the frame expansions of tempered distributions and a class of Beurling ultradistributions.

On the thermodynamical restrictions in isothermal deformations of fractional Burgers model.

We investigate, in the distributional setting, the restrictions on the constitutive equation for a fractional Burgers model of viscoelastic fluid that follow from the weak form of the entropy inequality under isothermal conditions. The results are generalized, from the Burgers model, to an arbitrary class of linear constitutive equations with fractional derivatives. Our results show that the restrictions obtained here on the coefficients of constitutive equations are weaker when compared with the restrictions obtained by Bagley-Torvik method. We show the precise relation between restrictions derived here and those derived by Bagley-Torvik. We deal with the creep test, for the case when Bagley-Torvik conditions are violated, and new conditions obtained in this work are satisfied. The results show a qualitative difference in the form of creep function. This article is part of the theme issue 'Advanced materials modelling via fractional calculus: challenges and perspectives'.

A Method for Prediction of Femoral Component of Hip Prosthesis Durability due to Aseptic Loosening by Using Coffin/Manson Fatigue Model.

The purpose of this work is to develop a new model estimate of the fatigue life of a hip prosthesis due to aseptic loosening as a multifactorial phenomenon. The formula developed here is a three-parameter model based on Basquin's law for fatigue, eccentric compression formula for the compressive stress and torsion in the prosthesis due to the horizontal components of the contact force. With our model, we can accurately predict the durability of a hip prosthesis due to the following four parameters: body weight, femoral offset, duration, and intensity of daily physical activities of a patient. The agreement of the prediction with the real life of the prosthesis, observed on 15 patients, is found to be adequate. Based on the formula derived for a particular implant, there was a high degree of concurrence between the model-predicted and actual values of aseptic loosening (durability) proved by the Mann-Whitney U test. By virtue of the validated model, it is possible to predict, quantitatively, the influence of various factors on the hip life. For example, we can conclude that a 10% decrease of a patient's body mass, with all other conditions being the same, causes 5% increase of the hip fatigue life.

A nonlinear two compartmental fractional derivative model.

This study presents a new nonlinear two compartmental model and its application to the evaluation of valproic acid (VPA) pharmacokinetics in human volunteers after oral administration. We have used literature VPA concentrations. In the model, the integer order derivatives are replaced by derivatives of real order often called fractional order derivatives. Physically that means that the history (memory) of a biological process, realized as a transfer from one compartment to another, is taken into account with the mass balance conservation observed. Our contribution is the analysis of a specific nonlinear two compartmental model with the application in evaluation of VPA pharmacokinetics. The agreement of the values predicted by the proposed model with the values obtained through experiments is shown to be good. Thus, pharmacokinetics of VPA after oral application can be described well by a nonlinear two compartmental model with fractional derivatives of the same order proposed here. Parameters in the model are determined by the least-squares method and the particle swarm optimization (PSO) numerical procedure is used. The results show that the nonlinear fractional order two compartmental model for VPA pharmacokinetics is superior in comparison to the classical (integer order) linear two compartmental model and to the linear fractional order two compartmental model.

A new approach to the compartmental analysis in pharmacokinetics: fractional time evolution of diclofenac.

This study presents a new two compartmental model and its application to the evaluation of diclofenac pharmacokinetics in a small number of healthy adults, during a bioequivalence trial. In the model the integer order derivatives are replaced by derivatives of real order often called fractional order derivatives. Physically that means that a history (memory) of a biological process, realized as a transfer from one compartment to another one with the mass balance conservation, is taken into account. This kind of investigations in pharmacokinetics is founded by Dokoumetzidis and Macheras through the one compartmental models while our contribution is the analysis of multi-dimensional compartmental models with the applications of the two compartmental model in evaluation of diclofenac pharmacokinetics. Two experiments were preformed with 12 healthy volunteers with two slow release 100 mg diclofenac tablet formulations. The agreement of the values predicted by the proposed model with the values obtained through experiments is shown to be good. Thus, pharmacokinetics of slow release diclofenac can be described well by a specific two compartmental model with fractional derivatives of the same order. Parameters in the model are determined by the least-squares method and the Particle Swarm Optimization (PSO) numerical procedure is used. The results show that the fractional order two compartmental model for diclofenac is superior in comparison to the classical two compartmental model. Actually this is true in general case since the classical one is a special case of the fractional one.