Fractional Differential Equations

Annals of physics

The flow of homogeneous fluids through porous media: analogies with other physical problems

Mathematica--A System for Doing Mathematics by Computer.

A new collection of real world applications of fractional calculus in science and engineering

1. Aquatic plants are a key component of spatial heterogeneity in a waterscape, contributing to habitat complexity and helping determine diversity at various spatial scales. Theoretically, the more complex a habitat, the higher the number of species present. 2. Few empirical data are available to test the hypothesis that complexity increases diversity in aquatic communities (e.g. Jeffries, 1993). Fractal dimension has become widely applied in ecology as a tool to quantify the degree of complexity at different scales. 3. We investigated the hypothesis that complexity in vegetated habitat in two tropical lagoons mediates littoral invertebrate number of taxa (S) and density (N). Aquatic macrophyte habitat complexity was defined using a fractal dimension and a gradient of natural plant complexities. We also considered plant area, plant identity and, only for S, invertebrate density as additional explanatory variables. 4. Our results indicate that habitat complexity provided by the different architectures of aquatic plants, significantly affects both S and total N. However, number of individuals (as a result of passive sampling) also helps to account for S and, together with plant identity and area, contributes to the determination of N. We suggest that measurements of structural complexity, measured through fractal geometry, should be included in studies aimed at explaining attributes of attached invertebrates at small (e.g. plant or leaf) scales.

Influence of aquatic macrophyte habitat complexity on invertebrate abundance and richness in tropical lagoons

Variational calculus theory of elasticity 1 - fundamentals theory of elasticity II - applications molecular models subsurface deformations in nematics.

An Elementary Course on the Continuum Theory for Nematic Liquid Crystals

NEMATIC LIQUID CRYSTALS: FUNDAMENTALS AND BULK PROPERTIES Nematic Order Parameters Maier-Saupe Approach for the Nematic-Isotropic Phase Transition Continuum Description of the Nematic Phase External Field Contribution to the Elastic Energy Density Equilbrium Configuration in Strong and Weak Anchoring Situations NEMATIC LIQUID CRYSTALS: SURFACE PROPERTIES Anchoring The Anchoring Energy Function Periodic Deformations in Nematic Liquid Crystals Spontaneous Periodic Distortions in Nematic Liquid Crystals: Dependence on the Tilt Angle Theoretical Models for the Surface Free Energy Stochastic Contribution to the Anchoring Energy: Deviation from the Rapini-Papolar Expression Validity of the Elastic Model for Nematic Surface Anchoring Energy Contribution of the Smectic-Nematic to the Surface Energy TEMPERATURE DEPENDENCE OF THE SURFACE FREE ENERGY Thermal Relaxation Model of Surface Director Gliding in Lyotropic Liquid Crystals Thermal Renormalization of the Anchoring Energy Temperature Induced Surface Transition in Nematic Liquid Crystals Oriented by Evaporated SiOx A Local Self-Consistent Approach to the Phase Transition at the Nematic Liquid Crystal - Wall Interface SURFACE ADSORPTION OF PARTICLES: GENERAL CHARACTERISTICS Introduction Statement of the Problem: Langmuir's Isotherm Surface Adsorption and Anchoring Transition Photo-Manipulation of the Anchoring Strength of Photochromic Nematic Liquid Crystal Adsorption of Magnetic Grains in Magnetic Fluids Selective Ionic Adsorption in Liquid Crystals SURFACE ELECTRIC FIELD-INDUCED INSTABILITIES IN NEMATIC LIQUID CRYSTALS SAMPLE The General Electrostatic Model Analysis I: Variation Solution of the Eigenvalue Problem Analysis II: Series Solution of the Eigenvalue Problem Role of the Elastic Term Linear in the Spatial Derivatives of the Nematic Director in a 1D Geometry Conclusions IONS AND NEMATIC SURFACE ENERGY: BEYOND THE EXPONENTIAL APPROXIMATION FOR THE ELECTRIC FIELD OF IONIC ORIGIN Introduction Theoretical Model: Non-Linear Poisson-Boltzman Equation External Electric Field Effect in the Nematic Anchoring Energy Contribution to the Surface Energy of Dielectric Origin Conclusions ADSORPTION PHENOMENON AND EXTERNAL FIELD EFFECT ON AN ISOTROPIC LIQUID CONTAINING IMPURITIES Introduction The Model The Charge and Field Distributions Ionic Adsorption in the Absence of External Equilibrium Distribution of Charges Extensions of the Model Influence of the Adsorption Energy in the Dieletric Contribution to the Anchoring Energy FERMILIKE DESCRIPTION OF THE ADSORPTION PHENOMENON Introduction The Fermi-Like Model Application: Ion Adsorption in Isotropic Fluids Limits Numerical Results Conclusions

Adsorption Phenomena and Anchoring Energy in Nematic Liquid Crystals

Anomalous diffusion has been detected in a wide variety of scenarios, from fractal media, systems with memory, transport processes in porous media, to fluctuations of financial markets, tumour growth, and complex fluids Providing a contemporary treatment of this process, this book examines the recent literature on anomalous diffusion and covers a rich class of problems in which surface effects are important, offering detailed mathematical tools of usual and fractional calculus for a wide audience of scientists and graduate students in physics, mathematics, chemistry and engineering Including the basic mathematical tools needed to understand the rules for operating with the fractional derivatives and fractional differential equations, this self-contained text presents the possibility of using fractional diffusion equations with anomalous diffusion phenomena to propose powerful mathematical models for a large variety of fundamental and practical problems in a fast-growing field of research

Fractional Diffusion Equations and Anomalous Diffusion

Several indices have been created to measure diversity, and the most frequently used are the Shannon-Wiener (H) and Simpson (D) indices along with the number of species (S) and evenness (E). Controversies about which index should be used are common in literature. However, a generalized entropy (Tsallis entropy) has the potential to solve part of these problems. Here we explore a family of diversity indices (S q ; where q is the Tsallis index) and evenness (E q ), based on Tsallis entropy that incorporates the most used indices. It approaches S when q = 0, H when q → 1 and gives D when q = 2. In general, varying the value of the Tsallis index (q), S q varies from emphasis on species richness (q 1). Similarly, E q also works as a tool to investigate diversity. In particular, for a given community, its minimum value represents the maximum deviation from homogeneity (E q = 1) for a particular q (herein named q*). It is remarkable that our analysis indicates that q* and its corresponding evenness, E q *, are negatively affected by S when using simulated data. They may represent an index related to species rarity. Furthermore, S q* (i.e. the value of S q for a specific q*) is positively affected by richness that is an important property of any diversity index. In general, our findings indicate that the indices H, D, S, S q* , E and E q* are only part of a whole set of possibilities. In addition, the ecological properties of E q* and S q* , proposed here for the first time, show promise in ecology.

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Luiz Roberto Evangelista

Papers

Influence of aquatic macrophyte habitat complexity on invertebrate abundance and richness in tropical lagoons

An Elementary Course on the Continuum Theory for Nematic Liquid Crystals

Adsorption Phenomena and Anchoring Energy in Nematic Liquid Crystals

Fractional Diffusion Equations and Anomalous Diffusion

A unified index to measure ecological diversity and species rarity