Non-integer Dimensional Space Research Articles

Non-Integer Dimensional Analysis of Ultrasonic Wave Propagation in Fractal Porous Media

This paper explores the acoustics of porous media characterized by fractal, or self-similar, structures. Employing a fractal approach, we use differential operators in non-integer dimensional spaces to address the fundamental equations of acoustics in such media. The primary aim is to examine the transmission of ultrasonic waves within a fractal porous medium. Our findings reveal that the fractal dimension significantly influences wave transmission. In fractal porous materials, waves travel along more complex and intricate paths, resulting in increased tortuosity and attenuation. We introduce the concept of an effective path length leff , which is dependent on the fractal dimension Dx , to describe the actual trajectory of wave propagation. Additionally, we define an effective tortuosity αDx , directly proportional to leff2 , to quantify the additional tortuosity brought about by the fractal structure. The insights gained from this study are crucial, as they enhance our understanding of wave behavior in self-similar porous media, which are prevalent in various natural settings and have multiple practical applications, including sound insulation and the design of acoustic materials. Furthermore, understanding the impact of fractal dimensions on wave behavior is vital for developing more efficient acoustic solutions. This research also sets the stage for further theoretical and experimental work on applying fractal geometry to analyze wave propagation in porous structures.

Spiral waves in fractal dimensions and their elimination in λ − ω systems with less damaging intervention

The theory of spiral waves in excitable media is considered an important topic in several dynamical systems. They have been observed in various physicochemical and biological systems such as the Belousov-Zhabotinsky chemical reactions, lethal arrhythmia in the heart, amoebas Dictyostelium discoideium, disinhibited mammalian neocortex retinal spreading depression, and seizures in the brain. In this study, spiral waves in an excitable system are analytically studied in fractal dimensions based on the theory of integration and differentiation for a non-integer dimensional space. We consider the generic oscillatory model λ−ω system characterized by a rotational symmetry where spiral waves are expected to occur. It was observed that for lower fractal dimensions with respect to unity, spiral waves may be suppressed in an excitable media without destroying the propagation of normal waves. This may be considered as an alternative approach to experimental and numerical approaches found in the literature. Their eliminations are important since they forbid the transition to a chaotic state in excitable media. Our approach could contribute to an improved therapy of clinical conditions such as atrial fibrillation and suppressing arrhythmias in cardiac models.

The Dimensionality of Genetic Information

This paper investigates the dimensionality of genetic information from the perspective of optimal representation. Recently it has been shown that optimal coding of information is in terms of the noninteger dimension of e, which is accompanied by the property of scale invariance. Since Nature is optimal, we should see this dimension reflected in the organization of the genetic code. With this as background, this paper investigates the problem of the logic behind the nature of the assignment of codons to amino acids, for they take different values that range from 1 to 6. It is shown that the non-uniformity of this assignment, which goes against mathematical coding theory that demands a near uniform assignment, is consistent with noninteger dimensions. The reason why the codon assignment for different amino acids varies is because uniformity is a requirement for optimality only in a standard vector space, and is not so in the noninteger dimensional space. It is noteworthy that there are 20 different covering regions in an e-dimensional information space, which is equal to the number of amino acids. The problem of the visualization of data that originates in an e-dimensional space but examined in a 3-dimensional vector space is also discussed. It is shown that the assignment of the codons to the amino acids is fractal-like that is well modeled by the Zipf distribution which is a power law. It is remarkable that the Zipf distribution that holds for the letter frequencies of words in a natural language also applies to the rank order of triplets in the code for amino acids.

Modeling thermal diffusion flames with fractal dimensions

In this study, we have discussed the implications of fractal dimensions in flame diffusion theory based on the notion of non-integer dimensional space for complex fractal media. We have selected the problem of round laminar diffusion flames and we have modeled it in fractal dimension. We have considered the particular case of spatially-varying densities and viscosities which arise in various combustion processes and turbulent jet flames used in power generation and manufacturing. It was observed that the length of the flame is affected not only by the unit velocity but also by the numerical value of the fractal dimension. A lower fractal dimension (with respect to unity) results in an increase of the flame length scale and hence in the turbulent flame speed and turbulence intensity.

Modulational instability in lossless left-handed metamaterials in nonlinear Schrödinger equation with non-integer dimensional space

Apply style for article title, author, affiliation and email as per stylesheet. Several decades ago, antennas had simple shapes that were described in Euclidean geometry. Nowadays, scientists try to make the structure of fractal geometry for applications in the field of electromagnetism, which has led to the development of new innovative antenna devices. Non-integer dimensional space (NDS) is useful to describe the concept of fractional space in fractal structure for real phenomenon of electromagnetic wave propagation. In this work, we investigate effects of NDS and normalized frequency on modulational instability (MI) gain in lossless left-handed metamaterials (LHM). We derive the nonlinear Schrödindiger equation (NLSE) with non-integer transverse laplacian. By means of linear stability analysis method, MI gain expression is also determined. Different forms of figures are obtained due to the signs of group velocity dispersion (GVD) and defocusing/focusing nonlinearity. We show how the increasing value of the normalized frequency enhances the amplitude as well as the bandwidth of MI gain, and waves are more unstable due to non-integer dimension. The obtained results are new and have a relatively newer application in telecommunication by constructing the fractal-shaped antennas operating in multi-frequency bands.

A Generalization of Poiseuille’s Law for the Flow of a Self-Similar (Fractal) Fluid through a Tube Having a Fractal Rough Surface

In this paper, a generalization of Poiseuille’s law for a self-similar fluid flow through a tube having a rough surface is proposed. The originality of this work is to consider, simultaneously, the self-similarity structure of the fluid and the roughness of the tube surface. This study can have a wide range of applications, for example, for fractal fluid dynamics in hydrology. The roughness of the tube surface presents a fractal structure that can be described by the surface fractal noninteger dimensions. Complex fluids that are invariant to changes in scale (self-similar) are modeled as a continuous medium in noninteger dimensional spaces. In this work, the analytical solution of the Navier–Stokes equations for the case of a self-similar fluid flow through a rough “fractal” tube is presented. New expressions of the velocity profiles, the fluid discharge, and the friction factor are determined analytically and plotted numerically. These expressions contain fractal dimensions describing the effects of the fractal aspect of the fluid and of that of the tube surface. This approach reveals some very important results. For the velocity profile to represent a physical solution, the fractal dimension of the fluid ranges between 0.5 and 1. This study also qualitatively demonstrates that self-similar fluids have shear thickening-like behavior. The fractal (self-similarity) nature of the fluid and the roughness of the surface both have a huge impact on the dynamics of the flow. The fractal dimension of the fluid affects the amplitude and the shape of the velocity profile, which loses its parabolic shape for some values of the fluid fractal dimension. By contrast, the roughness of the surface affects only the amplitude of the velocity profile. Nevertheless, both the fluid’s fractal dimension and the surface roughness have a major influence on the behavior of the fluid, and should not be neglected.

Flow of a Self-Similar Non-Newtonian Fluid Using Fractal Dimensions

In this paper, the study of the fully developed flow of a self-similar (fractal) power-law fluid is presented. The rheological way of behaving of the fluid is modeled utilizing the Ostwald–de Waele relationship (covering shear-thinning, Newtonian and shear-thickening fluids). A self-similar (fractal) fluid is depicted as a continuum in a noninteger dimensional space. Involving vector calculus for the instance of a noninteger dimensional space, we determine an analytical solution of the Cauchy equation for the instance of a non-Newtonian self-similar fluid flow in a cylindrical pipe. The plot of the velocity profile obtained shows that the rheological behavior of a non-Newtonian power-law fluid is essentially impacted by its self-similar structure. A self-similar shear thinning fluid and a self-similar Newtonian fluid take on a shear-thickening way of behaving, and a self-similar shear-thickening fluid becomes more shear thickening. This approach has many useful applications in industry, for the investigation of blood flow and fractal fluid hydrology.

Evaluation of Electric Field for a Dielectric Cylinder Placed in Fractional Space

The problem related to the dielectric cylinder placed in non-integer dimensional space (FD space) isthoroughly investigated in this paper. The FD space describes complex phenomena of physics and electromagnetism. We have solved Laplacian equation in FD space to obtain the solution of a dielectric cylinder in low frequency. The problem is solved by the method of separation of variables analytically. The classical solution of the problem can be easily recovered from the derived solution in non-integer dimensional space.

A UNIFIED THREE-DIMENSIONAL EXTENDED FRACTIONAL ANALYTICAL SOLUTION FOR AIR POLLUTANTS DISPERSION

A novel technique is presented to analytically solve the fractional diffusion equation for non-reactive air pollutants emitted from an elevated continuous source into the air. A generalized methodical solution combining first-order Wentzel–Kramers–Brillouin (WKB) approximation theory and the Sturm–Liouville problem is used to solve the air pollutants’ fractional dispersion equation. Drawing insight from previous analysis, we expanded the initial issue assuming the turbulent flow characteristics appearing in the diffusion process in a non-integer dimensional space. We solved the transformed problem and compared the solutions against data from real experiment. Physical consequences connecting to the conventional generalized diffusion equations are presented. The results indicate that the present solutions are in accordance with those obtained in literature. This report demonstrates that fractional equations can be applied in a practical prediction of pollutant dissemination in a turbulent atmosphere.

Response of Homogeneous Conducting Sphere in Non-Integer Dimensional Space

In this paper, we have investigated electric potential and field analytically for homogeneous conducting sphere by solving the Laplacian equation in fractional dimensional space. The laplacian equation in fractional space describes complex phenomena of physics. The separation variable method is used to solve the Laplace differential equation. The mathematical formulae governing the interaction of a low-frequency source of electric current with a spherical anomaly are derived in fractional dimensional space. These formulae are used to determine the apparent resistivity and induced-polarization response. The potential due to the current point source in fractional space is derived using Gegenbauer polynomials. The electric field inrensity of the homogeneous conducting sphere is calculated using the electric potential due to a current point source outside the sphere. The results are compared analytically with classical results by setting the fractional parameter α=3.

Micropolar mechanics of product fractal media

Motivated by the abundance of fractals in the natural world, this paper further develops continuum-type models for product-like fractals. The theory is based on a version of the non-integer dimensional space approach, in which global balance laws written for fractal media are expressed in terms of conventional (integer-order) integrals. Key relations of calculus for finite strain kinematics of fractal media are obtained, especially clarifying the fractal Jacobian and the fractal Reynolds transport theorem. The local forms are then written in terms of partial differential equations with derivatives of integer order. Hence, fractal versions of local continuity, linear and angular momenta and energy balance are derived. The angular momentum balance implies that the approximating continuum is micropolar rather than classical. Accordingly, the Cauchy postulate, lemma and theorem for Cauchy force-stress and couple-stress are re-formulated. The corresponding partial differential equations for finite as well as infinitesimal elasticity are given explicitly in both the displacement and stress formulations. The invariance of the stress field in planar fractal elastic media is shown to hold just like the one in planar micropolar elasticity.

Reflection and transmission of transient ultrasonic wave in fractal porous material: Application of fractional calculus

This paper provides a time domain model for the propagation of transient ultrasonic waves in a self-similar porous material having a rigid frame. This model is based on the formalism of Stillinger–Palmer–Stavrinou, which consists in modeling the fractal material as a porous medium with a non-integer dimensional space. This paper is devoted to the time-domain analytical calculus of the reflection and transmission operators that are expressed in terms of Mittag-Leffler functions. A sensitivity numerical study using ultrasonic reflected and transmitted waves is performed, highlighting the effect of the material’s physical parameters (fractal dimension, tortuosity, viscous characteristic length and porosity) on the waveforms.

A three-dimensional fractional solution for air contaminants dispersal in the planetary boundary layer

In this study, we investigated some closed-form solutions for solving atmospheric dispersion issues under variable atmospherical hypothesis, in a vertically positioned non-homogeneous planetary boundary-layer. In our context, a nonidentical expansion for the solution of the fractional advection-diffusion equation in a non-integer dimensional-space was examined. In a nutshell, a Sturm-Liouville eigenvalue problem with more reliable information concerning the initial value problem is discussed. The performance of the model was estimated by presenting numerical simulations against experimental data. Under these meteorological conditions, fractional-order models performed far better than those of the classical integer-order ones.

Nonlocal quantum system with fractal distribution of states

In this paper, we generalize simple model of quantum mechanics and statistics to take into account power-law nonlocality by using fractional differential equations, and fractal distribution of states. These equations with derivatives of non-integer orders are widely used in describing such complex properties of systems as power-law spatial nonlocality and spatial dispersion, long-range interactions, as well as the openness of quantum systems (interactions with the environment). We suggest a generalization of quantum statistical model of an ideal gas of particles for the case of spatial nonlocality, long-range interactions and fractal distribution of permitted states. To take into account the power-law nonlocality and long-range interactions, we use the fractional Laplacian in the Riesz–Trujillo form in the Schrodinger equation. To take into account the fractal distribution of states, we use non-integer dimensional space approach. Expression for the fractal density of states for fractal distribution of permitted quantum states is proposed. The equation to calculate energy and total number of particles for quantum systems with power-law non-locality and fractal distribution of quantum states are derived. Examples of these calculations for degenerate quantum system are suggested in the framework of the proposed quantum statistical model.

The time-dependent Schrödinger equation in non-integer dimensions for constrained quantum motion

We propose a theoretical model, based on a generalized Schrödinger equation, to study the behavior of a constrained quantum system in non-integer, lower than two-dimensional space. The non-integer dimensional space is formed as a product space X×Y, comprising x-coordinate with a Hausdorff measure of dimension α1=D−1 (1<D<2) and y-coordinate with the Lebesgue measure of dimension of length (α2=1). Geometric constraints are set at y=0. Two different approaches to find the Green's function are employed, both giving the same form in terms of the Fox H-function. For D=2, the solution for two-dimensional quantum motion on a comb is recovered.

Plane-wave electromagnetic scattering from a PEC strip placed at the interface of non-integer dimensional spaces

In this paper, scattering from a perfectly conducting strip placed at planar interface of non-integer dimensional (NID) dielectric media is formulated and investigated using the Kobayashi Potential (KP) method. The KP method is a semi analytical technique to address scattering problems. According to this method, longitudinal component of the unknown scattered field is assumed in terms of unknown weighting functions. Use of the related boundary conditions leads to the formation of algebraic equations and dual integral equations (DIEs). Integrands of the DIEs are expanded in terms of the characteristic functions with unknown expansion coefficients which must satisfy, simultaneously, the required edge and boundary conditions. The expressions derived from expansion of the integrands are combined with algebraic equations in order to express the unknown weighting function in terms of unknown expansion coefficients. Moreover, the projection treatment is applied using properties of the Jacobi polynomials, yielding matrix equation for unknown expansion coefficients. Matrix equation has been solved numerically. The far zone scattering width is investigated with respect to different parameters of the geometry, e.g., size of the strip and dimension of the NID space. Significant effects of different parameters on the scattering width are noted.

On fractional and fractal formulations of gradient linear and nonlinear elasticity

In this paper, we consider extensions of the gradient elasticity models proposed earlier by the second author to describe materials with fractional non-locality and fractality using the techniques developed recently by the first author. We derive a generalization of three-dimensional continuum gradient elasticity theory, starting from integral relations and assuming a weak non-locality of power-law type that gives constitutive relations with fractional Laplacian terms, by utilizing the fractional Taylor series in wave-vector space. In the sequel, we consider more general field equations with fractional derivatives of non-integer order to describe nonlinear elastic effects for gradient materials with power-law long-range interactions in the framework of weak non-locality approximation. The special constitutive relation that we elaborate upon can form the basis for developing a fractional extension of deformation theory of gradient plasticity. Using the perturbation method, we obtain corrections to the constitutive relations of linear fractional gradient elasticity, when the perturbations are caused by weak deviations from linear elasticity or by fractional gradient non-locality. Finally, we discuss fractal materials described by continuum models in non-integer dimensional spaces. Using a recently suggested vector calculus for non-integer dimensional spaces, we consider problems of fractal gradient elasticity.

Radiations from line source buried beneath a non-integer dimensional planar dielectric slab

Radiation due to an electric line source placed under a planar dielectric slab of non-integer dimensional space is determined. It has been assumed that the non-integer dimensional dielectric slab is sandwiched between two integer dimensional dielectric mediums. Radiated fields are written in terms of unknown spectrums of propagating waves. The unknown spectrum functions are determined using the boundary conditions. Special cases have also been discussed to verify the derived results. Radiation pattern is given for different parameters related to the geometry.

Propagation of transverse electric mode in a non-integer dimensional dielectric slab waveguide

In this paper, solution for the transverse electric (TEz) mode propagating in a non-integer dimensional (NID) dielectric slab waveguide has been derived. For region occupied by the slab, the medium is assumed to be NID along the direction normal to interfaces of the slab. Transcendental equations yield solution in form of non-integer order Hankel functions. The order of the Hankel functions depends on parameter describing the dimension of the NID space. Graphical results has been used to depict that the behavior of modes changes substantially with respect to NID parameter. It also affects the value of propagation constant, attenuation constant and cutoff frequency. The classical result is reproduced taking NID parameter equal to two. It is observed that the value of propagation constant for TM modes is higher than the value of propagation constant for TE modes. Moreover, the value of attenuation constant for TE modes is higher than the value of attenuation constant for TM modes.

Analysis of reflection and transmission from a planar NID-dispersive slab

Behavior of transmitted and reflected power from a planar non-integer dimensional lossless dispersive slab, placed in free space, is explored for uniform electromagnetic plane wave excitation. Dispersion has been incorporated through the Lorentz-Drude model. As the frequency of excitation increases, the dispersive slab respectively behaves as epsilon positive, double negative, mu negative, double positive meta-material. Effects of variation of angle of incidence, width of the slab and frequency of excitation on behavior of the powers have been noted taking different values of the parameter modeling non-integer dimensional space.