Finite-velocity diffusion on a comb

A Cattaneo equation for a comb structure is considered. We present a rigorous analysis of the obtained fractional diffusion equation, and corresponding solutions for the probability distribution function are obtained in the form of the Fox $H$-function and its infinite series. The mean square displacement along the backbone is obtained as well in terms of the infinite series of the Fox $H$-function. The obtained solutions describe the transition from normal diffusion to subdiffusion, which results from the comb geometry.Comment: 7 page

Heat equations beyond Fourier: from heat waves to thermal metamaterials

In the past decades, numerous heat conduction models beyond Fourier have been developed to account for the large gradients, fast phenomena, wave propagation, or heterogeneous material structure, such as being typical for biological systems, superlattices, or thermal metamaterials. It became a challenge to orient among the models, mainly due to their various thermodynamic backgrounds and possible compatibility issues. Additionally, in light of the recent findings on the field of non-Fourier heat conduction, it is not even straightforward how to interpret and utilize a non-Fourier heat equation, primarily when one aims to thermally design the material structure to construct the new generation of thermal metamaterials. Adding that numerous modeling strategies can be found in the literature accompanying different interpretations even for the same heat equation makes it even more difficult to orient ourselves and find a comprehensive picture of this field of research. Therefore, this review aims to ease the orientation among advanced heat equations beyond Fourier by discussing properties concerning their possible practical applications in light of experiments. We start from the simplest model with basic principles and notions, then proceed toward the more complex models related to coupled phenomena such as ballistic heat conduction. We do not enter the often complicated technical details of each thermodynamic framework but do not aim to compare each approach. However, we still briefly present their background to highlight their origin and the limitations acting on the models. Additionally, the field of non-Fourier heat conduction has become quite segmented, and that paper also aims to provide a common ground, a comprehensive mutual understanding of the basics of each model, together with what phenomenon they can be applied to

Solutions of Fractional Diffusion Equations and Cattaneo-Hristov Diffusion Model

The analytical solutions of the fractional diffusion equations in one and two-dimensional spaces have been proposed. The analytical solution of the Cattaneo-Hristov diffusion model with the particular boundary conditions has been suggested. In general, the numerical methods have been used to solve the fractional diffusion equations and the Cattaneo-Hristov diffusion model. The Laplace and the Fourier sine transforms have been used to get the analytical solutions. The analytical solutions of the classical diffusion equations and the Cattaneo-Hristov diffusion model obtained when the order of the fractional derivative converges to 1 have been recalled. The graphical representations of the analytical solutions of the fractional diffusion equations and the Cattaneo-Hristov diffusion model have been provided

Probabilistic foundation of nonlocal diffusion and formulation and analysis for elliptic problems on uncertain domains

2011 Summer.Includes bibliographical references.In the first part of this dissertation, we study the nonlocal diffusion equation with so-called Lévy measure ν as the master equation for a pure-jump Lévy process. In the case ν ∈ L1(R), a relationship to fractional diffusion is established in a limit of vanishing nonlocality, which implies the convergence of a compound Poisson process to a stable process. In the case ν ∉ L1(R), the smoothing of the nonlocal operator is shown to correspond precisely to the activity of the underlying Lévy process and the variation of its sample paths. We introduce volume-constrained nonlocal diffusion equations and demonstrate that they are the master equations for Lévy processes restricted to a bounded domain. The ensuing variational formulation and conforming finite element method provide a powerful tool for studying both Lévy processes and fractional diffusion on bounded, non-simple geometries with volume constraints. In the second part of this dissertation, we consider the problem of estimating the distribution of a quantity of interest computed from the solution of an elliptic partial differential equation posed on a domain Ω(θ) ⊂ R2 with a randomly perturbed boundary, where (θ) is a random vector with given probability structure. We construct a piecewise smooth transformation from a partition of Ω(θ) to a reference domain Ω in order to avoid the complications associated with solving the problems on Ω(θ). The domain decomposition formulation is exploited by localizing the effect of the randomness to boundary elements in order to achieve a computationally efficient Monte Carlo sampling procedure. An a posteriori error analysis for the approximate distribution, which includes a deterministic error for each sample and a stochastic error from the effect of sampling, is also presented. We thus provide an efficient means to estimate the distribution of a quantity of interest via a Monte Carlo sampling procedure while also providing a posteriori error estimates for each sample

On fractional differential equations: the generalised Cattaneo equations

The aim of this dissertation is to determine numerical solutions to fractional di usion and fractional Cattaneo equations using nite di erence formula and other de ned schemes. The spatial derivatives and time derivatives of integer order are approximated by a nite di erence approximation. Spatial derivatives of fractional order are approximated using the Gr unwald formula. Fractional time derivatives are approximated using the Gr unwald-Letnikov de nition of the Riemann-Liouville fractional derivative. The resulting di erence schemes are evaluated using Mathematica. The results obtained show that the fractional Cattaneo equaions have propagation and di usive properties. When the fractional exponent is 0:1 with the di usivity coe cient being greater than 0:1 one obtains numerical results that are unstable and display oscillatory behaviour. For other combinations of values, numerical results are stable and consistent with di usive behaviour

Optimal Perturbation Iteration Method for Solving Fractional Model of Damped Burgers’ Equation

The newly constructed optimal perturbation iteration procedure with Laplace transform is applied to obtain the new approximate semi-analytical solutions of the fractional type of damped Burgers’ equation. The classical damped Burgers’ equation is remodeled to fractional differential form via the Atangana–Baleanu fractional derivatives described with the help of the Mittag–Leffler function. To display the efficiency of the proposed optimal perturbation iteration technique, an extended example is deeply analyzed.This work was supported in part by the Basque Government, through project IT1207-19

Experimental and Novel Analytic Results for Couplings in Ordered Submicroscopic Systems: from Optomechanics to Thermomechanics

Theoretical modelling of challenging multiscale problems arising in complex (and sometimes bioinspired) solids are presented. Such activities are supported by analytical, numerical and experimental studies. For instance, this is the case for studying the response of hierarchical and nano-composites, nanostructured solid/semi-fluid membranes, polymeric nanocomposites, to electromagnetic, mechanical, thermal, and sometimes biological, electrical, and chemical agents. Such actions are notoriously important for sensors, polymeric films, artificial muscles, cell membranes, metamaterials, hierarchical composite interfaces and other novel class of materials. The main purpose of this project is to make significant advancements in the study of such composites, with a focus on the electromagnetic and mechanical performances of the mentioned structures, with particular regards to novel concept devices for sensing. These latter ones have been studied with different configuration, from 3D colloidal to 2D quasi-hemispherical micro voids elastomeric grating as strain sensors. Exhibited time-rate dependent behavior and structural phenomena induced by the nano/micro-structure and their adaptation to the applied actions, have been explored. Such, and similar, ordered submicroscopic systems undergoing thermal and mechanical stimuli often exhibit an anomalous response. Indeed, they neither follow Fourier’s law for heat transport nor their mechanical time-dependent behavior exhibiting classical hereditariness. Such features are known both for natural and artificial materials, such as bone, lipid membranes, metallic and polymeric “spongy” composites (like foams) and many others. Strong efforts have been made in the last years to scale-up the thermal, mechanical and micro-fluidic properties of such solids, to the extent of understanding their effective bulk and interface features. The analysis of the physical grounds highlighted above has led to findings that allow the describing of those materials’ effective characteristics through their fractional-order response. Fractional-order frameworks have also been employed in analyzing heat transfer to the extent of generalizing the classical Fourier and Cattaneo transport equations and also for studying consolidation phenomenon. Overall, the research outcomes have fulfilled all the research objectives of this thesis thanks to the strong interconnection between several disciplines, ranging from mechanics to physics, from structural health monitoring to chemistry, both from an analytical and numerical point of view to the experimental one

Special Functions: Fractional Calculus and the Pathway for Entropy

Historically, the notion of entropy emerged in conceptually very distinct contexts. This book deals with the connection between entropy, probability, and fractional dynamics as they appeared, for example, in solar neutrino astrophysics since the 1970's (Mathai and Rathie 1975, Mathai and Pederzoli 1977, Mathai and Saxena 1978, Mathai, Saxena, and Haubold 2010). The original solar neutrino problem, experimentally and theoretically, was resolved through the discovery of neutrino oscillations and was recently enriched by neutrino entanglement entropy. To reconsider possible new physics of solar neutrinos, diffusion entropy analysis, utilizing Boltzmann entropy, and standard deviation analysis was undertaken with Super-Kamiokande solar neutrino data. This analysis revealed a non-Gaussian signal with harmonic content. The Hurst exponent is different from the scaling exponent of the probability density function and both Hurst exponent and scaling exponent of the Super-Kamiokande data deviate considerably from the value of ½, which indicates that the statistics of the underlying phenomenon is anomalous. Here experiment may provide guidance about the generalization of theory of Boltzmann statistical mechanics. Arguments in the so-called Boltzmann-Planck-Einstein discussion related to Planck's discovery of the black-body radiation law are recapitulated mathematically and statistically and emphasize from this discussion is pursued that a meaningful implementation of the complex ‘entropy-probability-dynamics’ may offer two ways for explaining the results of diffusion entropy analysis and standard deviation analysis. One way is to consider an anomalous diffusion process that needs to use the fractional space-time diffusion equation (Gorenflo and Mainardi) and the other way is to consider a generalized Boltzmann entropy by assuming a power law probability density function. Here new mathematical framework, invented by sheer thought, may provide guidance for the generalization of Boltzmann statistical mechanics. In this book Boltzmann entropy, generalized by Tsallis and Mathai, is considered. The second one contains a varying parameter that is used to construct an entropic pathway covering generalized type-1 beta, type-2 beta, and gamma families of densities. Similarly, pathways for respective distributions and differential equations can be developed. Mathai's entropy is optimized under various conditions reproducing the well-known Boltzmann distribution, Raleigh distribution, and other distributions used in physics. Properties of the entropy measure for the generalized entropy are examined. In this process the role of special functions of mathematical physics, particularly the H-function, is highlighted

Recent Trends in Coatings and Thin Film–Modeling and Application

Over the past four decades, there has been increased attention given to the research of fluid mechanics due to its wide application in industry and phycology. Major advances in the modeling of key topics such Newtonian and non-Newtonian fluids and thin film flows have been made and finally published in the Special Issue of coatings. This is an attempt to edit the Special Issue into a book. Although this book is not a formal textbook, it will definitely be useful for university teachers, research students, industrial researchers and in overcoming the difficulties occurring in the said topic, while dealing with the nonlinear governing equations. For such types of equations, it is often more difficult to find an analytical solution or even a numerical one. This book has successfully handled this challenging job with the latest techniques. In addition, the findings of the simulation are logically realistic and meet the standard of sufficient scientific value