Fractal Fract., Volume 6, Issue 11 (November 2022) – 72 articles

In this work, the exact and approximate solution for generalized linear, nonlinear, and coupled systems of fractional singular M-dimensional pseudo-hyperbolic equations are examined by using the multi-dimensional Laplace Adomian decomposition method (M-DLADM). In particular, some two-dimensional illustrative examples are provided to confirm the efficiency and accuracy of the present method. Full article

Fractional q-calculus plays an extremely important role in mathematics and physics. In this paper, we aim to investigate the existence of triple-positive solutions for nonlinear singular fractional q-difference equation boundary value problems at resonance by means of the fixed-point index theorem and the q-Laplace transform, where the nonlinearity $f(t,u,v)$ permits singularities at $t=0,1$ and $u=v=0$. The obtained theorem is well illustrated with the aid of an example. Full article

Fractal dimension (FD) together with advances in imaging technologies has provided an increasing application of digital images to interpret biological phenomena. In ophthalmology, topography-based images are increasingly used in common practices of clinical settings. They provide detailed information about corneal surfaces. Few-micron alterations of the corneal geometry to the elevation and curvature cause a highly multifocal surface, change the corneal optical power up to several diopters, and therefore adversely affect the individual’s vision. Keratoconus (KCN) is a corneal disease characterized by a local alteration of the corneal anatomical and mechanical features. The formation of cone-shaped regions accompanied by thinning and weakening of the cornea are the major manifestations of KCN. The implantation of tiny arc-like polymeric sections, known as intracorneal implants, is considered to be effective in restoring the corneal curvature. This study investigated the FD nature of healthy corneas (n = 7) and compared it to the corresponding values before and after intracorneal implant surgery in KCN patients (n = 7). The generalized Hurst exponent, Higuchi, and Katz FDs were computed for topography-based parameters of corneal surfaces: front elevation (ELE-front), back elevation (ELE-back), and corneal curvature (CURV). The Katz FD showed better discriminating ability for the diseased group. It could reveal a significant difference between the healthy corneas and both pre- and post-implantation topographies (p < 0.001). Moreover, the Katz dimension varied between the topographic features of KCN patients before and after the treatment (p < 0.036). We propose to describe the curvature feature of corneal topography as a “strange attractor” with a self-similar (i.e., fractal) structure according to the Katz algorithm. Full article

We apply two mathematical techniques, specifically, the unified solver approach and the exp$(-\phi (\xi \left)\right)$-expansion method, for constructing many new solitary waves, such as bright, dark, and singular soliton solutions via the fractional Biswas–Milovic (FBM) model in the sense of conformable fractional derivative. These solutions are so important for the explanation of some practical physical problems. Additionally, we study the stochastic modeling for the fractional Biswas–Milovic, where the parameter and the fraction parameters are random variables. We consider these parameters via beta distribution, so the mathematical methods that were used in this paper may be called random methods, and the exact solutions derived using these methods may be called stochastic process solutions. We also determined some statistical properties of the stochastic solutions such as the first and second moments. The proposed techniques are robust and sturdy for solving wide classes of nonlinear fractional order equations. Finally, some selected solutions are illustrated for some special values of parameters. Full article

Image processing remains an area that has impact on the software industry and is a field that is permanently developing in both IT and industrial contexts. Nowadays, the demand for fast computing times is becoming increasingly difficult to fulfill in the case of massive computing systems. This article proposes a particular case of efficiency for a specifically developed model for fractal generations. From the point of view of graphic analysis, the application can generate a series of fractal images. This process is analyzed and compared in this study from a programming perspective in terms of both the results at the processor level and the graphical generation possibilities. This paper presents the structure of the software and its implementation for generating fractal images using the Mandelbrot set. Starting from the complex mathematical set, the component iterations of the Mandelbrot algorithm lead to optimization variants for the calculation. The article consists of a presentation of an optimization variant based on applying parallel calculations for fractal generation. The method used in the study assumes a high grade of accuracy regarding the selected mathematical model for fractal generation and does not characterize a method specially built for a certain kind of image. A series of scenarios are analyzed, and details related to differences in terms of calculation times, starting from the more efficient proposed variant, are presented. The developed software implementation is parallelization-based and is optimized for generating a wide variety of fractal images while also providing a test package for the generated environment. The influence of parallel programming is highlighted in terms of its difference to sequential programming to, in turn, highlight recent methods of speeding up computing times. The purpose of the article is to combine the complexity of the mathematical calculation behind the fractal sets with programming techniques to provides an analysis of the graphic results from the point of view of the use of computing resources and working time. Full article

Under the influence of internal and external factors, a fracture network is easily generated in concrete and rock, which seriously endangers project safety. Fractal theory can be used to describe the formation and development of the fracture network and characterize its structure. Based on the flow balance in the node balance field, Forchheimer’s law is introduced to derive the control equation of high-velocity non-Darcy flow in the fracture network. The fracture network is established according to the geological parameters of Sellafield, Cumbria, England. A total of 120 internal fracture networks are intercepted according to 10 dimensions (1 m, 2 m, …, 10 m) and 12 directions (0°, 30°, …, 330°). The fractal dimension, equivalent hydraulic conductivity (K), and equivalent non-Darcy coefficient ($\beta$) of the fracture network are calculated, and the influence of the fractal dimension on K and $\beta$ is studied. The results indicate that the fractal dimension of the fracture network has a size effect; with the increase in the size, the fractal dimension of the fracture network undergoes three stages: rapid increase, slow increase, and stabilization. In the rapid increase stage, K and $\beta$ do not exist. In the slow increase stage, K exists and is stable, and $\beta$ does not exist. In the stabilization stage, K and $\beta$ both exist and are stable. The principal axes of the fitted seepage ellipses of K and $\beta$ are orthogonal, and the main influencing factors are the direction and continuity of the fracture. Full article

Impedance control is an important method in robot–environment interaction. In traditional impedance control, the damping force is regarded as a linear viscoelastic model, which limits the description of the dynamic model of the impedance system to a certain extent. For the robot manipulator, the optimal impedance parameters of the impedance controller are the key to improve the performance. In this paper, the damping force is described more accurately by fractional calculus than the traditional viscoelastic model, and a fractional-order impedance controller for the robot manipulator is proposed. A practical and systematic tuning procedure based on the frequency design method is developed for the proposed fractional-order impedance controller. The fairness of comparison between the fractional-order impedance controller and the integer-order impedance controller is addressed under the same specifications. Fair comparisons of the two controllers via the simulation and experiment tests show that, in the step response, the fractional-order impedance controller has a better integral time square error (ITSE) result, smaller overshoot and less settling time than the integer-order impedance controller. In terms of anti-disturbance, the fractional-order impedance controller can achieve stability with less recovering time and better ITSE index than integer order impedance controller. Full article

Singular systems, which can be applied to gauge field theory, condensed matter theory, quantum field theory of anyons, and so on, are important dynamic systems to study. The fractional order model can describe the mechanical and physical behavior of a complex system more accurately than the integer order model. Fractional singular systems within mixed integer and combined fractional derivatives are established in this paper. The fractional Lagrange equations, fractional primary constraints, fractional constrained Hamilton equations, and consistency conditions are analyzed. Then Noether and Lie symmetry methods are studied for finding the integrals of the fractional constrained Hamiltonian systems. Finally, an example is given to illustrate the methods and results. Full article

In the literature, all investigations dealing with regulator design in the AVR loop observe the AVR system as a single input single output (SISO) system, where the input is the generator reference voltage, while the output is the generator voltage. Besides, the regulator parameters are determined by analyzing the terminal generator voltage response for a step change from zero to the rated value of the generator voltage reference. Unlike literature approaches, in this study, tuning of the AVR controllers is conducted while modeling the AVR system as a double input single output (DISO) system, where the inputs are the setpoint of the generator voltage and the step disturbance on the excitation voltage, while the output is the generator voltage. The transfer functions of the generator voltage dependence on the generator voltage reference value and the excitation voltage change were derived in the developed DISO-AVR model. A novel objective function for estimating DISO-AVR regulator parameters is proposed. Also, a novel metaheuristic algorithm named hybrid simulated annealing and gorilla troops optimization is employed to solve the optimization problem. Many literature approaches are compared using different regulator structures and practical limitations. Furthermore, the experimental results of 120 MVA synchronous generators in HPP Piva (Montenegro) are presented to show the drawbacks of the literature approaches that observe generator setpoint voltage change from zero to the rated value. Based on the presented results, the proposed procedure is efficient and strongly applicable in practice. Full article

Deformable image registration is a very important topic in the field of image processing. It is widely used in image fusion and shape analysis. Generally speaking, image registration models can be divided into two categories: smooth registration and non-smooth registration. During the last decades, many smooth registration models (i.e., diffeomorphic registration) were proposed. However, image with strong noise may lead to discontinuous deformation, which cannot be modelled by smooth registration. To simulate this kind of deformation, some non-smooth registration models were also proposed. However, numerical algorithms for these models are easily trapped into a local minimum because of the nonconvexity of the object functional. To overcome the local minimum of the object functional, we propose a multiscale approach for a non-smooth registration model: the bounded deformation (BD) model. The convergence of the approach is shown, and numerical tests are also performed to show the good performance of the proposed multiscale approach. Full article

One of the main tasks in the problems of machine learning and curve fitting is to develop suitable models for given data sets. It requires to generate a function to approximate the data arising from some unknown function. The class of kernel regression estimators is one of main types of nonparametric curve estimations. On the other hand, fractal theory provides new technologies for making complicated irregular curves in many practical problems. In this paper, we are going to investigate fractal curve-fitting problems with the help of kernel regression estimators. For a given data set that arises from an unknown function m, one of the well-known kernel regression estimators, the Nadaraya–Watson estimator $\widehat{m}$, is applied. We consider the case that m is Hölder-continuous of exponent $\beta$ with $0<\beta \le 1$, and the graph of m is irregular. An estimation for the expectation of $|\widehat{m}{-m|}^2$ is established. Then a fractal perturbation $f_{\left[\widehat{m}\right]}$ corresponding to $\widehat{m}$ is constructed to fit the given data. The expectations of $|f_{\left[\widehat{m}\right]}-\widehat{m}{|}^2$ and $|f_{\left[\widehat{m}\right]}{-m|}^2$ are also estimated. Full article

The fuzzy-number valued up and down $\lambda$-convex mapping is originally proposed as an intriguing generalization of the convex mappings. The newly suggested mappings are then used to create certain Hermite–Hadamard- and Pachpatte-type integral fuzzy inclusion relations in fuzzy fractional calculus. It is also suggested to revise the Hermite–Hadamard integral fuzzy inclusions with regard to the up and down $\lambda$-convex fuzzy-number valued mappings (U∙D$\lambda$-convex F-N∙V∙Ms). Moreover, Hermite–Hadamard–Fejér has been proven, and some examples are given to demonstrate the validation of our main results. The new and exceptional cases are presented in terms of the change of the parameters $“i”$ and $“\alpha ”$ in order to assess the accuracy of the obtained fuzzy inclusion relations in this study. Full article

In this paper, the sliding-mode control method was used to control a class of general nonlinear fractional-order systems which covers a wide class of chaotic systems. A novel sliding manifold with an additional nonlinear part which achieved better control performance was designed. Furthermore, a novel fixed-time reaching law with a fractional adaptive gain is proposed, where the reaching time to the sliding manifold is determined by the first positive zero of a Mittag–Leffler function and is independent of initial conditions. We have provided some instructions on tuning the parameters of the proposed reaching law to avoid exacerbating the chattering phenomenon. Finally, simulation examples are presented to validate all results. Full article

Self-similar fractals can be generated using subdivision and the subdivision curves/surfaces are actually attractors. Such a connection has been studied between fractals and an extended family of subdivision including stationary and non-stationary schemes. This paper aims to move one step further on such a connection and introduce multiple-function systems, which has a set of function systems and choose one for each step of iteration. These multiple-function systems can be obtained by deriving the iterated function systems based on the subdivision operators and applying some modifications, including deleting some transformations, to them. Such multiple-function systems can be arranged in a tree structure and can generate different attractors along different paths in the tree. Several examples are presented to illustrate the performance of these multiple-function systems. Full article

Compression and mercury intrusion porosimetry (MIP) tests were conducted to analyze the effect of municipal solid waste incineration fly ash (MSWIFA) content on the mechanical performance and pore structure of geopolymer mortar. The MSWIFA weight contents were 0%, 5%, 15%, 25%, and 35% and the pore diameter distribution, specific surface area, and pore volume were considered to assess the pore structure of the geopolymer mortars. The popular fractal model was used to investigate the fractal features of the geopolymer mortars. Additionally, mathematical models of fractal dimension with pore structural parameters and compressive strength were established. The results showed that the compressive strength of geopolymer mortars decreased while the total pore volume and total specific surface area of mortars increased with the increase in MSWIFA content. As the MSWIFA content increased, the harmless pores (pore diameter < 20 nm) were refined. Specifically, the pores with a diameter of 5–10 nm increased in number but the pores with a diameter of 10–20 nm decreased in number with the increase in MSWIFA content. The pore structure in the mortars showed scale-dependent fractal characteristics. All fractal curves were divided into four segments according to the pore diameter, namely, Region I (<20 nm), Region II (20–50 nm), Region III (50–200 nm), and Region IV (>200 nm). The surface fractal dimension ($D_S$) in Region I and Region IV was between 2 and 3. However, the $D_S$ in Region II and Region III was greater than 3, indicating the pores in Region II and Region III were non-physical according to the surface geometry because of the presence of ink bottle pores which distorted the result of the MIP. The complexity of pores in Region I and Region IV was reduced by the addition of MSWIFA. The $D_S$ is a comprehensive parameter that well describes the spatial and morphological distribution of pores in geopolymer mortars and exhibited a good correlation with the specific surface area, pore volume, and compressive strength. A mathematical model based on the $D_S$ was established to predict the compressive strength of the geopolymer mortar containing MSWIFA. Full article

Pore structure is one of the important parameters for evaluating reservoirs, critical in controlling the storage capacity and transportation properties of hydrocarbons. The conventional pore characterization method cannot fully reflect the pore network morphology. The edge-threshold automatic processing method is applied to extract and quantify pore structures in shale scanning electron microscope (SEM) images. In this manuscript, a natural lacustrine oil-prone shale in the Qingshankou Formation of Songliao Basin is used as the research object. Based on FE-SEM, a high-resolution cross-section of shale was obtained to analyze the microstructure of pores and characterize the heterogeneity of pores by multifractal theory. The stringent representative elementary area (REA) of the SEM cross-section was determined to be 35 × 35. Four pore types were found and analyzed in the stringent REA: organic pores, organic cracks, inorganic pores, inorganic cracks. The results showed that inorganic pores and cracks were the main pore types and accounted for 87.8% of the total pore area, and organic cracks were of the least importance in the Qingshankou shale. Inorganic pores were characterized as the simplest pore morphologies, with the largest average MinFeret diameter, and the least heterogeneity. Moreover, the inorganic cracks had a long extension distance and stronger homogeneity, which could effectively connect the inorganic pores. Organic pores were found to be the most complex for pore structure, with the least average MinFeret diameter, but the largest heterogeneity. In addition, the extension distance of the organic cracks was short and could not effectively connect the organic pore. We concluded that inorganic pores and cracks are a key factor in the storage and seepage capacity of the Qingshankou shale. Organic pores and cracks provide limited storage space. Full article

Abrasion resistance and cracking resistance are two important properties determining the normal operation and reliability of hydropower projects that are subjected to erosion and abrasive action. In this study, polyvinyl alcohol (abbreviated as PVA) fiber and magnesium oxide expansive agents (abbreviated as MgO) were used together to solve the problems of cracking and abrasive damage. The effects of PVA fiber and MgO on the mechanical property, abrasion and cracking resistance, pore structures and fractal features of high-strength hydraulic concrete were investigated. The main results are: (1) The incorporation of 4–8% Type I MgO reduced the compressive strength, splitting tensile strength and the abrasion resistance by about 5–12% at 3, 28 and 180 days. Adding 1.2–2.4 kg/m³ PVA fibers raised the splitting tensile strength of concrete by about 8.5–15.7% and slightly enhanced the compressive strength and abrasion resistance of concrete. (2) The incorporation of 4–8% Type I MgO prolongs the initial cracking time of concrete rings under drying by about 6.5–11.4 h, increased the cracking tensile stress by about 6–11% and lowered the cracking temperature by 2.3–4.5 °C during the cooling down stage. Adding 1.2–2.4 kg/m³ PVA fibers was more efficient than adding 4–8% MgO in enhancing the cracking resistance to drying and temperature decline. (3) Although adding 4% MgO and 1.2–2.4 kg/m³ PVA fibers together could not enhance the compressive strength and abrasion resistance, it could clearly prolong the cracking time, noticeably increase the tensile stress and greatly lower the racking temperature; that is, it efficiently improved the cracking resistance to drying and thermal shrinkage compared with the addition of MgO or PVA fiber alone. The utilization of a high dosage of Type I MgO of less than 8% and PVA fiber of no more than 2.4 kg/m³ together is a practical technique to enhance the cracking resistance of hydraulic mass concretes, which are easy to crack. (4) The inclusion of MgO refined the pores, whereas the PVA fiber incorporation marginally coarsened the pores. The compressive strength and the abrasion resistance of hydraulic concretes incorporated with MgO and/or PVA fiber are not correlated with the pore structure parameters and the pore surface fractal dimensions. Full article

In this paper, we investigate the exact and approximate controllability, finite time stability, and $\beta$–Hyers–Ulam–Rassias stability of a fractional order neutral impulsive differential system. The controllability criteria is incorporated with the help of a fixed point approach. The famous generalized Grönwall inequality is used to study the finite time stability and $\beta$–Hyers–Ulam–Rassias stability. Finally, the main results are verified with the help of an example. Full article

We continue the development of the basic theory of generalized derivatives as introduced and give some of their applications. In particular, we formulate necessary conditions for extrema, Rolle’s theorem, the mean value theorem, the fundamental theorem of calculus, integration by parts, along with an existence and uniqueness theorem for a generalized Riccati equation, each of which provides simple proofs of the corresponding version for the so-called conformable fractional derivatives considered by many. Finally, we show that for each $\alpha >1$ there is a fractional derivative and a corresponding function whose fractional derivative fails to exist everywhere on the real line. Full article

In this paper, a chaotic circuit based on a memcapacitor and meminductor is constructed, and its dynamic equation is obtained. Then, the mathematical model is obtained by normalization, and the system is decomposed and summed by an Adomian decomposition method (ADM) algorithm. So as to study the dynamic behavior in detail, not only the equilibrium stability of the system is analyzed, but also the dynamic characteristics are analyzed by means of a Bifurcation diagram and Lyapunov exponents (Les). By analyzing the dynamic behavior of the system, some special phenomena, such as the coexistence of attractor and state transition, are found in the system. In the end, the circuit implementation of the system is implemented on a Digital Signal Processing (DSP) platform. According to the numerical simulation results of the system, it is found that the system has abundant dynamical characteristics. Full article

We begin by introducing some function spaces $L_c^p\left({\mathbb{R}}^{+}\right),X_c^p\left(J\right)$ made up of integrable functions with exponent or power weights defined on infinite intervals, and then we investigate the properties of Mellin convolution operators mapping on these spaces, next, we derive some new boundedness and continuity properties of Hadamard integral operators mapping on $X_c^p\left(J\right)$ and $X^p\left(J\right)$. Based on this, we investigate a class of boundary value problems for Hadamard fractional differential equations with the integral boundary conditions and the disturbance parameters, and obtain uniqueness results for positive solutions to the boundary value problem under some weaker conditions. Full article

Several identification approaches have recently been employed in human identification systems for forensic purposes to decrease human efforts and to boost the accuracy of identification. Dental identification systems provide automated matching by searching photographic dental features to retrieve similar models. In this study, the problem of dental image identification was investigated by developing a novel dental identification scheme (DIS) utilizing a fractional wavelet feature extraction technique and rule mining with an Apriori procedure. The proposed approach extracts the most discriminating image features during the mining process to obtain strong association rules (ARs). The proposed approach is divided into two steps. The first stage is feature extraction using a wavelet transform based on a k-symbol fractional Haar filter (k-symbol FHF), while the second stage is the Apriori algorithm of AR mining, which is applied to find the frequent patterns in dental images. Each dental image’s created ARs are saved alongside the image in the rules database for use in the dental identification system’s recognition. The DIS method suggested in this study primarily enhances the Apriori-based dental identification system, which aims to address the drawbacks of dental rule mining. Full article

An option is the right to buy or sell a good at a predetermined price in the future. For customers or financial companies, knowing an option’s pricing is crucial. It is well recognized that the Black–Scholes model is an effective tool for estimating the cost of an option. The Black–Scholes equation has an explicit analytical solution known as the Black–Scholes formula. In some cases, such as the fractional-order Black–Scholes equation, there is no closed form expression for the modified Black–Scholes equation. This article shows how to find the approximate analytic solutions for the two-dimensional fractional-order Black–Scholes equation based on the generalized Riemann–Liouville fractional derivative. The generalized Laplace variational iteration method, which incorporates the generalized Laplace transform with the variational iteration method, is the methodology used to discover the approximate analytic solutions to such an equation. The expression of the two-parameter Mittag–Leffler function represents the problem’s approximate analytical solution. Numerical investigations demonstrate that the proposed scheme is accurate and extremely effective for the two-dimensional fractional-order Black–Scholes Equation in the perspective of the generalized Riemann–Liouville fractional derivative. This guarantees that the generalized Laplace variational iteration method is one of the effective approaches for discovering approximate analytic solutions to fractional-order differential equations. Full article

An oscillating second-grade fluid through a rectangular cross duct is studied. A traditional integer time derivative in the kinematic tensors is substituted by a fractional operator that considers the memory characteristics. To treat the fractional governing equation, an analytical method was obtained. To analyze the impact of the parameters more intuitively, the difference method was applied to determine the numerical expression and draw with the help of computer simulation. To reduce the cost of the amount of computation and storage, a fast scheme was proposed, one which can greatly improve the calculation speed. To verify the correctness of the difference scheme, the contrast between the numerical expression and the exact expression—constructed by introducing a source term—was given and the superiority of the fast scheme is discussed. Furthermore, the influences of the involved parameters, including the parameter of retardation time, fractional parameter, magnetic parameter, and oscillatory frequency parameter, on the distributions of velocity and shear force at the wall surface with oscillatory flow are analyzed in detail. Full article

In this work, we consider linear and nonlinear fractional stochastic delay systems driven by the Rosenblatt process. With the aid of the delayed Mittag-Leffler matrix functions and the representation of solutions of these systems, we derive the controllability results as an application. By introducing a fractional delayed Gramian matrix, we provide sufficient and necessary criteria for the controllability of linear fractional stochastic delay systems. Furthermore, by employing Krasnoselskii’s fixed point theorem, we establish sufficient conditions for the controllability of nonlinear fractional stochastic delay systems. Finally, an example is given to illustrate the main results. Full article

Zeolite Y and Ti-containing zeolite Y (1%, 2% and 5% TiO₂) were synthesized by a hydrothermal seed-assisted method. In order to evidence the evolution of morphology, structure, and fractal dimensions during the zeolitization process at certain time intervals, a small volume from the reaction medium was isolated and frozen by lyophilization. The obtained samples were characterized by scanning electron microscopy (SEM), wide-angle X-ray diffraction (XRD), and small-angle X-ray scattering (SAXS). The fractal dimension values of the isolated samples, calculated from SAXS data, evidenced a transition from small particles with a smooth surface (2.021) to compact structures represented by zeolite crystallites with rough surfaces (2.498) and specific organization for zeolite Y. The formation of new structures during hydrothermal treatment, the increase in crystallite size and roughness due to the continuous growth were suggested by variation of fractal dimensions values, SEM microscopy images and X-ray diffractograms. The incorporation of titanium in low concentration into the zeolite Y framework led to the obtaining of low fractal dimensions of 2.034–2.275 (smooth surfaces and compact structures). On the other hand, higher titanium concentration (2%) led to an increase in fractal dimensions indicating structures with rougher surfaces and well-defined self-similarity properties. A mechanism for zeolite synthesis was proposed by correlation of the results obtained through morphological, structural, and fractal analysis. Full article

Fractals play a vital role in modeling the natural environment. The present aim is to investigate the escape criterion to generate specific fractals such as Julia sets, Mandelbrot sets and Multi-corns via F-iteration using complex functions $h\left(z\right)=z^n+c$, $h\left(z\right)=sin\left(z^n\right)+c$ and $h\left(z\right)=e^{z^n}+c$, $n\ge 2,c\in \mathbb{C}$. We observed some beautiful Julia sets, Mandelbrot sets and Multi-corns for n = 2, 3 and 4. We generalize the algorithms of the Julia set and Mandelbrot set to visualize some Julia sets, Mandelbrot sets and Multi-corns. Moreover, we calculate image generation time in seconds at different values of input parameters. Full article

The outbreak of coronavirus (COVID-19) began in Wuhan, China, and spread all around the globe. For analysis of the said outbreak, mathematical formulations are important techniques that are used for the stability and predictions of infectious diseases. In the given article, a novel mathematical system of differential equations is considered under the piecewise fractional operator of Caputo and Atangana–Baleanu. The system is composed of six ordinary differential equations (ODEs) for different agents. The given model investigated the transferring chain by taking non-constant rates of transmission to satisfy the feasibility assumption of the biological environment. There are many mathematical models proposed by many scientists. The existence of a solution along with the uniqueness of a solution in the format of a piecewise Caputo operator is also developed. The numerical technique of the Newton interpolation method is developed for the piecewise subinterval approximate solution for each quantity in the sense of Caputo and Atangana-Baleanu-Caputo (ABC) fractional derivatives. The numerical simulation is drawn against the available data of Pakistan on three different time intervals, and fractional orders converge to the classical integer orders, which again converge to their equilibrium points. The piecewise fractional format in the form of a mathematical model is investigated for the novel COVID-19 model, showing the crossover dynamics. Stability and convergence are achieved on small fractional orders in less time as compared to classical orders. Full article