Fractal Order Research Articles

Comparative empirical evaluation of the aqueous adsorptive sequestration potential of low-cost feldspar-biochar composites for ivermectin

Several recent studies are shedding light on the potential of synergistically combined low-cost adsorbents for organic and inorganic contaminants removal in water treatment. However, there exists a dearth of data for their potential for removal of emerging contaminants. In this study, low-cost feldspar (FLC) was combined with blended Carica papaya or pine cone seeds (ratio of 1:1), and calcined to obtain FLC-Carica papaya seeds (FPA) or FLC-pine cone seeds (FPC) composites which were characterized. The composites were evaluated for ivermectin sorption by varying experimental pH, time, concentration and temperature, as well as their potential for reusability. The FLC structural lattice was not significantly affected by the modification process but the composites exhibited physicochemical features which were markedly different, having new FTIR spectra bands, well reduced surface area, and enhanced cation exchange capacity. The composites ivermectin sorption rates and equilibrium were faster (60 min) than the FLC (360 min). Ivermectin sorption was pH and concentration dependent. The sorption data fitted better to the homogeneous fractal pseudo-second order (FPSO) kinetics and Freundlich adsorption isotherm model; implying that the process involves complex sorption mechanism. The process was exothermic and increase in solution temperature up to 39.5 ºC did not enhance sorption. Ivermectin sorption capacity and rate trends were similar: FPA (110.9 µg/g) >FLC (84.6 µg/g) >FPC (62.7 µg/g). The FPA exhibited the best sorption capacity as well as the best reusability with over 64% of its initial sorption capacity retained on first reuse. The FPA also exhibited high potential to reduce ivermectin in low concentration solution (≈75 µg/L) to zero, with the trend of FPA ≤FLC (5.0 ± 1.5 µg/L) >FPC (≈27.5 µg/L). Though this kind of composite possesses economic potential in water treatment applications, not every biochar modification may result in enhanced sorption process.

NUMERICAL SOLUTION OF NONLINEAR FRACTAL–FRACTIONAL DYNAMICAL MODEL OF INTERPERSONAL RELATIONSHIPS WITH POWER, EXPONENTIAL AND MITTAG-LEFFLER LAWS

The term fractional differentiation has recently been merged with the term fractal differentiation to create a new fractional differentiation operator. Several kernels were used to explore these new operators, including the power-law, exponential decay, and Mittag-Leffler functions. In this study, we analyze three forms of interpersonal relationships model. The numerical solution of the fractal–fractional interpersonal relationships model based on different kernels has been investigated. The new operators contain two parameters: one is for fractional order [Formula: see text], and the other is for fractal dimension [Formula: see text]. We use Lagrangian polynomial interpolation along with numerical method and the concept of fractional theory to solve these three forms of the titled model. All three forms of the numerical computation are compared with the solutions of the other existing method when [Formula: see text] that leads to a good agreement. The existence and uniqueness of the models have been studied using the Picard–Lindelöf theorem. To understand how the effects of fractal dimension and fractional order influence the model, we have illustrated various plots taking different values of [Formula: see text] and [Formula: see text].

TWO-SCALE FRACTAL THEORY FOR THE POPULATION DYNAMICS

This paper aims to study a two-scale population growth model in a closed system by He–Laplace method together with the fractional complex transform (FCT). The two-scale derivative is described with the help of He’s fractional derivative. The FCT approach is used to convert differential equation of the two-scale fractal order in its traditional partner, which is then readily solved by He–Laplace iterative scheme. The results are computed as a series of easily computed components. The validation of the proposed methodology is illustrated by a quantitative comparison of numerical results with those obtained using other techniques. The results show that the proposed method is fast, accurate, straightforward, and computationally reasonable.

Multi-waves interaction and optical solitons for Heisenberg models of fractal order

Multi-waves interaction and optical solitons for Heisenberg models of fractal order

Complex dynamics of multi strain TB model under nonlocal and nonsingular fractal fractional operator

Researchers have recently begun to use fractal fractional operators in the Atangana–Baleanu sense to analyze complicated dynamics of various models in applied sciences, as the Atangana–Baleanu operator generalizes the integer and fractional order operators. To analyze the complex dynamics of the multi-strain TB model, we use the AB-fractal fractional operator. We use the Banach fixed point theorem to ensure that at most one solution exists to the model. Further, the Ulam–Hyers type stability of the model is investigated with the help of functional analysis. The Adams–Bashforth approach is used to get numerical results for the proposed model. The analysis of the chaotic behavior of the proposed TB model was missing in the literature. Therefore, for different values of fractional and fractal order, we study the nonlinear dynamics and chaotic behavior of the obtained results of the proposed model.

Harnessing fractal cuts to design robust lattice metamaterials for energy dissipation

Lattice metamaterials exhibit remarkable mechanical properties and novel functionalities, such as high specific stiffness, fracture toughness, tunable vibroacoustic properties, and energy absorption. This study explores a group of lattice metamaterials with fractal cuts that feature energy dissipation via structural sliding friction mechanisms and intrinsic material damping. Lattice metamaterials with three types of fractal cuts patterns were designed and fabricated using additive manufacturing. Three-point bending tests and numerical analysis have been combined to investigate their bending behavior. Experimental results show that the structural bending compliance of the metamaterials can be sharply enhanced by increasing the fractal orders, while at the same time keeping the shape recoverability during cyclic loading. Loss factors associated with the cyclic bending of these metamaterials are almost independent of the fractal order used, which is attributed to the synergistic effect between friction and viscoelasticity. It is further demonstrated via validated finite element models that the sliding friction plays a critical role by investigating the effects of the bending displacement and sample thickness on the flexural behavior. Results suggest that the magnitude of the maximum bending displacement has a negligible effect on the loss factors, and a power scaling law exists between bending stiffness and sample thickness. This study suggests that fractal lattice metamaterials have unique energy dissipation properties, with potential applications in many industrial sectors such as defense, energy, and transportation.

Effect of vaccination to control COVID-19 with fractal fractional operator

Currently, Atangana proposed new fractal-fractional operators that had extensively used to observe the unpredictable elements of an issue. COVID-19 is a pervasive infection today and is hard to fix. In this structure, the novel operators have been used to observing the effect of vaccination in the COVID-19 model with different values of η1,η2 which are used to show the effect of vaccination. The system will be converted into disease-free according to reproductive number. We used the Atangana-Baleanu fractal-fractional operator to investigate the COVID-19 model qualitatively and quantitatively. By using fixed point theorems we proved the existence and uniqueness of the model with the Atangana-Baleanue fractal- fractional operator. A non-linear assessment helped to find out the stability of the Ulam-Hyres. We simulate the mathematical outcomes, to understand the relationship of operators in several senses, for numerous arrangements of fractional and fractal orders.

Empirical aspects of an emerging agricultural pesticide contaminant retention on two sub-Saharan soils

Soils are highly heterogeneous dynamic mixtures with unique properties. The general properties of each type of soil affect its contaminants sorption; thus, obtaining sorption information about any regional soil may require actual sorption experiments. Hence, two soil types (alfisol–AFL and oxisol–OXL) were sampled and used for ivermectin sorption in order to obtain data on sub-Saharan soils for this new emerging contaminant from agriculture. Environmental parameters such as sorption time, ambient solution pH, ivermectin concentration, temperature and desorption were studied, while data were evaluated using two adsorption isotherm models, four kinetic models and the thermodynamics. The soils were neutral, typical of neutral West African soils and having slightly acidic pHpzc with similar particle size description and medium-to-low CECeff and SOM. Ivermectin sorption equilibrium was fast at 180 min for both soils. The rate constant k for sorption on AFL soil was slightly faster than for the OXL. Solution pH has some degree of influence on the sorption process which exhibited two sorption optimum peaks; one around pH 3 and the other around pH 9.5. Ivermectin sorption was concentration dependent; there was higher sorption as initial concentration increased, while sorption increased significantly with ambient temperature from 19.5 to 29.5 °C (≈55%) but there was slight reduction on further temperature increase to 39.5 °C compared to sorption at 29.5 °C (≤1%). The magnitude of the estimated energetics signaled a non-spontaneous and increasingly random process, with small size of the ΔH° values which were compatible with low energy interactive sorption forces and the overall process was exothermic. The ivermectin sorption process was controlled by external mass transfer (which was concentration dependent), with approximately 81.6% of sorption occurring on the soil surfaces, while 18.4% was within the pores or soil phases. The better fits of the sorption data to both the Freundlich adsorption isotherm model and the homogeneous fractal pseudo-second order model are complementary, and confirms that ivermectin sorption process on both soils involves complex interactions on heterogeneous sorption surfaces which include electrostatic interactions, multi–layer adsorption probably facilitated by π-π interactions between surface sorbed ivermectin molecules and those in solution, possibly trapping of ivermectin molecules within the soil pores, and various van der Waals attractive forces. Though ivermectin was rapidly dissipated on these soils, the hysteresis was high; irrespective of the soil, the amount of ivermectin leached per soil was small, in fact less than 6% of the initial amount sorbed. Our study has vital implications in predicting the fate of specific contaminants in the environment.

Crashworthiness analysis of bio-inspired fractal tree-like multi-cell circular tubes under axial crushing

This study proposes novel bio-inspired fractal multi-cell circular (BFMC) tubes for energy absorption. The inner structures of the proposed BFMC tubes were constructed based on the fractal tree-like forms found in many biological structures such as giant water lily and dragon blood tree. The crashworthiness performances of the proposed structures with different fractal orders and mass were numerically investigated. The numerical results indicated that the specific energy absorption (SEA) increased with the fractal order and the SEA of the 2nd-order BFMC tube was 35.43% higher than that of the conventional multi-cell circular tube. Furthermore, the complex proportional assessment (COPRAS) method was adopted to optimize the performance of the BFMC. The results demonstrated that the proposed structure with four number of tree-like branches and 2nd-order fractal provided the best performance. Finally, a theoretical derivation of the mean crushing force (MCF) was developed for the proposed tubes based on the simplified super folding element theory. The theoretical results of MCF agreed well with the numerical results. The findings of this study provide an effective guide for using the biomimetic approach with the fractal tree-like forms for the design of a multi-cell energy absorber with high energy absorption efficiency.

Numerical solution of a fractal-fractional order chaotic circuit system

The dynamical system has an important research area and due to its wide applications many researchers and scientists working to develop new model and techniques for their solution. We present in this work the dynamics of a chaotic model in the presence of newly introduced fractal-fractional operators. The model is formulated initially in ordinary differential equations and then we utilize the fractal-fractional (FF) in power law, exponential and Mittag-Leffler to generalize the model. For each fractal-fractional order model, we briefly study its numerical solution via the numerical algorithm. We present some graphical results with arbitrary order of fractal and fractional orders, and present that these operators provide different chaotic attractors for different fractal and fractional order values. The graphical results demonstrate the effectiveness of the fractal-fractional operators.

Investigation of complex behaviour of fractal fractional chaotic attractor with mittag-leffler Kernel

This article aims to study the behaviour of a chaotic attractor in the fractal-fractional Mittag-Leffler perspective. The different aspects of the chaotic attractor are observed with different fractal and fractional orders. The existence and uniqueness of the system are presented by using Schauder and Banach fixed point theorems. The stability analysis of the equilibrium points of the system is presented together with Ulam-Hyers stability for the system under consideration. The Lyapunov spectra and the bifurcation in the system with respect to control parameter ζ are studied. The numerical scheme based on the Adam-Bashforth method is established with Lagrangian piecewise interpolation. The complex behaviour of the considered system is numerically illustrated using various fractal and fractional orders. It is observed that, the chaotic attractor self-replicates its pattern in the fractal process when fractal dimension varies.

New optical soliton solutions via two distinctive schemes for the DNA Peyrard–Bishop equation in fractal order

The deoxyribonucleic acid (DNA) dynamical equation, which emerges from the oscillator chain known as the Peyrard–Bishop (PB) model for abundant optical soliton solutions, is presented, along with a novel fractional derivative operator. The Kudryashov expansion method and the extended hyperbolic function (HF) method are used to construct novel abundant exact soliton solutions, including light, dark, and other special solutions that can be directly evaluated. These newly formed soliton solutions acquired here lead one to ask whether the analytical approach could be extended to deal with other nonlinear evolution equations with fractional space–time derivatives arising in engineering physics and nonlinear sciences. It is noted that the newly proposed methods’ performance is most reliable and efficient, and they will be used to construct new generalized expressions of exact closed-form solutions for any other NPDEs of fractional order.

In-plane wave propagation analysis for waveguide design of hexagonal lattice with Koch snowflake

In this study, the band structure and directional characteristics of the hexagonal lattice with Koch snowflake are investigated. The finite element method and Bloch theorem are used to analyze the wave propagation in hexagonal lattice with Koch snowflake for different fractal orders. The effects of the geometric parameters on band structures are discussed in detail. The group velocities at given frequencies are calculated for analyzing the directional characteristics of the wave propagation. The dynamic responses of the periodic lattice are investigated via numerical simulation to confirm the directional features. The vibration experiments are conducted to verify the existence of band gap and vibration suppression performances of the structures, which are used to design the waveguide in the lattices. The results demonstrate that the hexagonal lattice with Koch snowflake for different fractal orders can tune the elastic wave behaviors in a specific way. As the fractal order increases, multiple band gaps appear and the band structure has self-similarity in the low-frequency range. The directional characteristics of wave propagation in the hexagonal lattice with Koch snowflake can be used to further expand the vibration attenuation range. The results also show the potential importance and feasibility of using the hexagonal lattice with Koch snowflake for vibration isolation and control using the hexagonal lattice with Koch snowflake.

Modeling and analysis of the dynamics of HIV/AIDS with non-singular fractional and fractal-fractional operators

In the present study, we analyze the dynamics of HIV/AIDS epidemic model with fractional and the new definition of a nonlocal fractal-fractional derivative in the Atangana-Baleanu-Caputo (ABC) sense. The proposed model is initially formulated with the help of classical integer-order differential equations and then the ABC fractional derivative is used to obtain the fractional HIV/AIDS model with arbitrary order p 1. Then we present some basic properties including equilibria and the basic reproduction number of the model. The local and global stability of the fractional model at disease free equilibrium is shown and it is found that the model is stable when the threshold quantity is less than 1. Further, an efficient numerical approach is applied to present an iterative scheme for the fractional model and present graphical results for various values of p 1. After this, we make use of the new fractal-fractional operator to formulate the HIV/AIDS model in the ABC sense. The uniqueness and existence of the solutions are shown through the Picard-Lindelof theorem. Furthermore, we derive an iterative scheme of the fractal-fractional ABC model using a novel numerical method based on Adams-Bashforth fractal-fractional approachand depicts detailed graphical results for many values of fractional and fractal orders p 1 and p 2 respectively. The simulation results show that the HIV/AIDS model with the novel fractal-fractional operator provides biologically more feasible analysis than that of the classical fractional and integer-order models.

Local Fractal Fourier Transform and Applications

In this manuscript, we review fractal calculus and the analogues of both local Fourier transform with its related properties and Fourier convolution theorem are proposed with proofs in fractal calculus. The fractal Dirac delta with its derivative and the fractal Fourier transform of the Dirac delta are also defined. In addition, some important applications of the local fractal Fourier transform are presented in this paper such as the fractal electric current in a simple circuit, the fractal second order ordinary differential equation, and the fractal Bernoulli-Euler beam equation. All discussed applications are closely related to the fact that, in fractal calculus, a useful local fractal derivative is a generalized local derivative in the standard calculus sense. In addition, a comparative analysis is also carried out to explain the benefits of this fractal calculus parameter on the basis of the additional alpha parameter, which is the dimension of the fractal set, such that when $alpha=1$, we obtain the same results in the standard calculus.

A NEW PERSPECTIVE ON THE STUDY OF THE FRACTAL COUPLED BOUSSINESQ–BURGER EQUATION IN SHALLOW WATER

The well-known coupled Boussinesq–Burger equation can be used to describe the flow of the shallow water of the harbor. But when the boundary is nonsmooth, it becomes powerless. So, the fractal calculus is needed to be applied to it and the fractal coupled Boussinesq–Burger equation (FCBBe) is presented for the first time in this paper. By using the semiinverse method, we have successfully established the fractal variational formulation of the FCBBe, which can not only provide the conservation laws in an energy form in the fractal space but also reveal the possible solution structures of the equation. Then a novel variational approach based on He’s variational method and the two-scale transform are used to seek its periodic wave solutions. The main advantage of variational approach is that it can reduce the order of differential equation and make the equation more simple. Finally, the numerical results have been shown through graphs to discuss the effect of different fractal orders on the wave motion. The obtained results in this work are expected to shed a bright light on the study of fractal nonlinear partial differential equations in the fractal space.

Fractal-fractional order mathematical vaccine model of COVID-19 under non-singular kernel.

In this paper, the severe acute respiratory syndrome coronavirus (SARS-CoV-2) or COVID-19 is researched by employing mathematical analysis under modern calculus. In this context, the dynamical behavior of an arbitrary order p and fractal dimensional q problem of COVID-19 under Atangana Bleanu Capute (ABC) operator for the three cities, namely, Santos, Campinas, and Sao Paulo of Brazil are investigated as a case-study. The considered problem is analyzed for at least one solution and unique solution by the applications of the theorems of fixed point and non-linear functional analysis. The Ulam-Hyres stability condition via nonlinear functional analysis for the given system is derived. In order to perform the numerical simulation, a two-step fractional type, Lagrange plynomial (Adams Bashforth technique) is utilized for the present system. MATLAB simulation tools have been used for testing different fractal fractional orders considering the data of aforementioned three regions. The analysis of the results finally infer that, for all these three regions, the smaller order values provide better constraints than the larger order values.

New optical soliton solutions via generalized Kudryashov’s scheme for Ginzburg–Landau equation in fractal order

Here, we use the modified Kudryashov’s algorithm and addendum to Kudryashov’s technique to obtain new optical solitons of the Ginsburg–Landau fractional complex of non-linearity (CGLE) laws in Kerr, which demonstrate various phenomena in physics such as non-linear waves, the transformation of the secondary phase, superconductivity, surfaces, liquid crystals, and field-theory strings. Dark, bright, singular solitons and combo bright-singular solitons are retrieved. The modified Kudryashov’s algorithm provides dark, singular solitons, and combo bright-singular solitons, although the addendum to Kudryashov’s technique ensures bright and singular solitons. These solutions are practiced for three optional definitions of derivative viz. conformable derivative, beta derivative, and M-truncated. Also shown are graphic representations of the solutions obtained.

Modeling and analysis of an epidemic model with fractal-fractional Atangana-Baleanu derivative

Mathematical modeling of infectious diseases with non-integer order getting attentions from scientists and researchers day by day. It is obvious that classical models in epidemiology can only be described through a fixed order while models in fractional order derivative are of not fixed order. Having non fixed order the fractional derivative becomes more powerful in modeling real life problems. In the recent era, different novel concepts regarding fractional operators such as the exponential decay and the Mittag–Leffler kernel have been introduced which overcome the limitations of the previous fractional order derivatives. These new operators have been found effective in modeling problems arising in science and engineering. A more recent operator in fractional calculus was introduced that is known as fractal-fractional operator. In this study, we consider this novel approach and apply it to an epidemic model of dengue fever and explore their dynamics. We show some important analysis for the dengue epidemic model in the presence of this new operator. The uniqueness and existence results will be shown. We show the simulation results for the considered model with a novel numerical approach which is not yet considered by anyone for such epidemic model. We obtain results for fractal model when fractional order is one and will have fractional solution when fractal order is one and have when both are present. We show that the fractal-fractional approach is much suitable for an epidemic model rather than fractional operator.

Mechanical performance of fractal-like cementitious lightweight cellular structures: Numerical investigations

Cellular structures with tunable mechanical properties are promising lightweight structures for prefabricated construction. In this study, a 3D fractal structure named Menger-Sponge (MS) cube is analysed. Besides, six other alternative fractal-like structures are designed with respect to different shapes of their associated hollow sections. Finite element method (FEM) is employed to predict the structure's mechanical behaviours under a uniaxial compression load. The numerical model is validated by experimental results obtained from author's previous publication. Comparisons are made between triply periodic minimal surface (TPMS-Primitive) and the second-order MS cementitious structure, and also among other fractal structures with different hierarchies. Results indicate that the MS and Primitive blocks have similar mechanical responses. The fractal-square (FS) structures exhibit better structural performances, especially for the first and second fractal orders. Overall, this numerical investigation shows the effect of the shaped holes and thinnest cross-section area on the hierarchically porous structure.