Fractal Fractional Operators Research Articles

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.

A robust study on the listeriosis disease by adopting fractal-fractional operators

Listeriosis is one of the zoonotic diseases affecting most parts of the Sub-Saharan countries. The infection is often transmitted by eating and it can also pass by respiratory and direct contact. In this paper, a listeriosis mathematical model is formulated involving fractal-fractional orders in both Caputo and Atangana–Baleanu derivatives. Moreover, future behaviors of the disease are investigated by considering the fractal–fractional operators that are very effective in modeling the real-life phenomena by virtue of their memory effect. The basic properties and steady states are also obtained. The threshold parameter for determining the spread of the disease is computed. Numerical results are presented for each fractal-fractional-order operator. The results obtained in the paper show that the numerical schemes are effective for predicting and analyzing complex phenomena.

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.

On the fractal dynamics for higher order traveling waves

Auto replication processes remain fascinating in sciences, engineering and technology as their applications in machining/biological systems have been widely used to solve number of outstanding problems in communities’ every day lives. Finding innovative techniques capable of generating auto replication processes in various fields has then become the priority for number of scientists. One of those fields includes wave motion. In this paper, we use the 7th order Korteweg–de Vries (KdV) model, combined with the fractal-fractional operator to artificially (numerically) generate auto replication processes characterizing the evolution of higher order traveling waves. The well-posedness for the combined model is first studied with the establishment of its existence and uniqueness results. Numerical simulations then follow and prove that the higher order traveling wave can be involved in a self replication process. There is generation of the exact or approximately exact copies of the initial traveling wave in different scales and where the fractal process produces other multiple traveling waves that look like the preceding ones. Furthermore, the fractal dynamics expand as the model’s parameter γ changes. We are on the right track for the artificially-formed fractal process of higher order traveling waves applicable in wave motion’s domain.

A numerical and analytical study of SE(Is)(Ih)AR epidemic fractional order COVID-19 model

This article describes the corona virus spread in a population under certain assumptions with the help of a fractional order mathematical model. The fractional order derivative is the well-known fractal fractional operator. We have given the existence results and numerical simulations with the help of the given data in the literature. Our results show similar behavior as the classical order ones. This characteristic shows the applicability and usefulness of the derivative and our numerical scheme.

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.

On solutions of an obesity model in the light of new type fractional derivatives

Obesity has become a serious problem in both developed and developing countries. In this work, we analyse the obesity model that involves three operators Caputo, Atangana–Baleanu in Caputo sense and fractal-fractional derivatives. The existence and uniqueness of the solution have been examined. The various numerical schemes have been presented for each operator. Many numerical analysis are carried out based on associated scheme to make meanings for the analytical analysis. Different fractional order values and fractal order values have been simulated. The fractal-fractional operator presents a unique dynamics of the Obesity model.

Tangent nonlinear equation in context of fractal fractional operators with nonsingular kernel.

In this manuscript, we investigate the approximate solutions to the tangent nonlinear packaging equation in the context of fractional calculus. It is an important equation because shock and vibrations are unavoidable circumstances for the packaged goods during transport from production plants to the consumer. We consider the fractal fractional Caputo operator and Atangana–Baleanu fractal fractional operator with nonsingular kernel to obtain the numerical consequences. Both fractal fractional techniques are equally good, but the Atangana–Baleanu Caputo method has an edge over Caputo method. For illustrations and clarity of our main results, we provided the numerical simulations of the approximate solutions and their physical interpretations. This paper contributes to the new applications of fractional calculus in packaging systems.

Analysis of fractal-fractional model of tumor-immune interaction

Recently, Atangana proposed new operators by combining the fractional and fractal calculus. These recently proposed operators, referred to as fractal-fractional operators, have been widely used to study the complex dynamics of a problem. Cancer is a prevalent disease today and is difficult to cure. The immune system tends to fight it as cancer sets up in the body. In this manuscript, the novel operators have been used to analyze the relationship between the immune system and cancer cells. The tumor-immune model has been studied qualitatively and quantitatively via Atangana-Baleanu fractal-fractional operator. The existence and uniqueness results of the model under Atangana-Baleanu fractal-fractional operator have proved through fixed point theorems. The Ulam-Hyres stability for the model has derived through non-linear analysis. Numerical results have developed through Lagrangian-piece wise interpolation for the different fractal-fractional operators. To visualize the relationship between immune cells and cancers cells under novel operators in a various sense, we simulate the numerical results for the different sets of fractional and fractal orders.

A fractal-fractional order Atangana-Baleanu model for Hepatitis B virus with asymptomatic class

Hepatitis B is still a major issue in most countries of the world. Due to many death and infection cases, the disease becoming a life-threatening issue and needs proper attention for its eradication. The main aim of this study is to design a new mathematical model with an asymptomatic class based on clinical investigations to study its dynamics. The asymptomatic carriers class do not possess symptoms but infect other healthy people. This new idea has been utilized for the first time in the present analysis with fractal-fractional operators. We formulate the model basically in integer-order and then apply the fractal-fractional derivative in Atangana-Baleanu type. For the fractional model, we study the related results and their numerical solution. Further, we apply the fractal operator together with fractional derivative which is known as fractal-fractional derivative in the Atangana-Baleanu case, and present the model. For the numerical solution, we provide a scheme based on the Adams-Bashforth method and obtained the results graphically. With various choices of the fractal and fractional orders, we present various graphical solutions. The model parameters that can reduce the infection of Hepatitis B are shown graphically. The disease in the population can be minimized well by taking into consideration the model important parameters. The important parameters and their effect have been shown graphically.

Analysis of MHD Couette flow by fractal-fractional differential operators

Abstract In this paper, an analysis is carried out to study the MHD Couette flow (flow between two parallel plates such that the upper plate is moving with constant velocity while the lower plate is at rest) for an incompressible viscous fluid under isothermal conditions. The governing equations are developed from the problem, formulated with the recently presented fractal-fractional operators in Riemann–Liouville sense with power law, exponential decay and the Mittag–Leffler law kernels. For each operator, we present a comprehensive analysis including, the numerical solutions, stability analysis and error analysis. We apply very accurate method to get the desired results. We demonstrate the numerical simulations to prove the efficiency of the proposed method.

Investigating the complex behaviour of multi-scroll chaotic system with Caputo fractal-fractional operator

Abstract A new set of differential and integral operators has recently been proposed by Abdon et al., merging the fractional derivative and the fractal derivative, taking into account non-locality, memory and fractal effects. These operators have demonstrated the complex behaviour of many physical phenomena, which generally does not predict in ordinary operators or sometimes also in fractional operators. In this article, we investigate a three dimensional quadratic multi scroll chaotic dynamical system under Caputo fractal-fractional operator. We study the proposed model by replacing the fractional derivative by fractal-fractional derivatives based on Caputo. Through Schauder’s fixed point theorem, we establish existence theory to ensure that the model possesses at least one solution. Also, Banach fixed theorem guarantees the uniqueness of solution of the proposed model. By mean of non-linear functional analysis, we derive that the proposed model is Ulam-Hyres stable under the new fractal-fractional derivative. We establish the numerical scheme of the considered model through Lagrangian piece-wise interpolation. For the different values of fractional order and fractal dimension, we present the complex behaviour of the proposed model.

Analytical study of transmission dynamics of 2019-nCoV pandemic via fractal fractional operator

Through this paper, we aim to study the dynamics of 2019-nCoV transmission using fractal-ABC type fractional differential equations by incorporating population self-protection behavior changes. The basic parameters of disease dynamics spread differently from country to country due to the different sensitive parameters. The proposed model in this study links the infection rate, the marginal value of the infection force for the population, the recovery rate, the rate of decomposition of the 2019-nCoV in the environment, and what methods are needed to stop the spread of the virus. We give a detailed analysis of the proposed model in this study by analyzing disease-free equilibrium point, the number of reproduction and the positivity of the model solutions, in addition to verifying the existence, uniqueness and stability of this disease using fixed point theories. Further on exploiting Adam Bash’s numerical scheme, we compute some numerical results for the required model. The concerned results have been simulated against some real initial data of three different counties including China, Brazil, and Italy.

A novel method for analysing the fractal fractional integrator circuit

In this article, we propose the integrator circuit model by the fractal-fractional operator in which fractional-order has taken in the Atangana-Baleanu sense. Through Schauder’s fixed point theorem, we establish existence theory to ensure that the model posses at least one solution and via Banach fixed theorem, we guarantee that the proposed model has a unique solution. We derive the results for Ulam-Hyres stability by mean of non-linear functional analysis which shows that the proposed model is Ulam-Hyres stable under the new fractal-fractional derivative. We establish the numerical results of the model under consideration through Atanaga-Toufik method. We simulate the numerical results for different sets of fractional order and fractal dimension.

A new auto-replication in systems of attractors with two and three merged basins of attraction via control

Largely recognized as leading concepts in network traffic prediction, machining or image processing, the processes of auto-duplication, self-organization and auto-replication are highly useful and fascinating for chaos and fractal theorists. Those processes appear naturally around us as observed on trees, river deltas, lightning, growth spirals, flowers, romanesco broccoli, frost, etc. Using mathematical notions, concepts and processes to reproduce and control auto-duplication and auto-replication dynamics have attracted the attention of scientists and engineers given their wide range of applications. We use the control technique combined to two different concepts, Julia’s process and fractal-fractional operator, to generate auto-replication in systems of chaotic attractors with two and three merged basins of attraction. The systems used here comprise a controller part, namely the switching-manifold control. Both systems are solved numerically using Julia’s scheme and Legendre wavelets methods. Numerical simulations reveal a fascinating capability for the systems to auto-replicate while conserving the initial properties of the systems, such as symmetry and number of basins of attraction. The merged basins of attraction are shown to be moving away as the result of the fractional dynamics’ impact. Great results that may interest scientists and engineers dealing with fractals and chaos.

Epidemiological analysis of fractional order COVID-19 model with Mittag-Leffler kernel

<abstract> <p>This paper derived fractional derivatives with Atangana-Baleanu, Atangana-Toufik scheme and fractal fractional Atangana-Baleanu sense for the COVID-19 model. These are advanced techniques that provide effective results to analyze the COVID-19 outbreak. Fixed point theory is used to derive the existence and uniqueness of the fractional-order model COVID-19 model. We also proved the property of boundedness and positivity for the fractional-order model. The Atangana-Baleanu technique and Fractal fractional operator are used with the Sumudu transform to find reliable results for fractional order COVID-19 Model. The generalized Mittag-Leffler law is also used to construct the solution with the different fractional operators. Numerical simulations are performed for the developed scheme in the range of fractional order values to explain the effects of COVID-19 at different fractional values and justify the theoretical outcomes, which will be helpful to understand the outbreak of COVID-19 and for control strategies.</p> </abstract>

New class of volterra integro-differential equations with fractal-fractional operators: Existence, uniqueness and numerical scheme

<p style='text-indent:20px;'>In this paper, we introduce a new fractional integro-differential equation involving newly introduced differential and integral operators so-called fractal-fractional derivatives and integrals. We present a numerical scheme that is convenient for obtaining solution of such equations. We give the general conditions for the existence and uniqueness of the solution of the considered equation using Banach fixed-point theorem. Both the suggested new equation and new numerical scheme will considerably contribute for our readers in theory and applications.

A Fractal-Fractional Model for the MHD Flow of Casson Fluid in a Channel

An emerging definition of the fractal-fractional operator has been used in this study for the modeling of Casson fluid flow. The magnetohydrodynamics flow of Casson fluid has cogent in a channel where the motion of the upper plate generates the flow while the lower plate is at a static position. The proposed model is non-dimensionalized using the Pi-Buckingham theorem to reduce the complexity in solving the model and computation time. The non-dimensional fractal-fractional model with the power-law kernel has been solved through the Laplace transform technique. The Mathcad software has been used for illustration of the influence of various parameters, i.e., Hartman number, fractal, fractional, and Casson fluid parameters on the velocity of fluid flow. Through graphs and tables, the results have been implemented and it is shown that the boundary conditions are fully satisfied. The results reveal that the flow velocity is decreasing with the increasing values of the Hartman number and is increasing with the increasing values of the Casson fluid parameter. The findings of the fractal-fractional model have elucidated that the memory effect of the flow model has higher quality than the simple fractional and classical models. Furthermore, to show the validity of the obtained closed-form solutions, special cases have been obtained which are in agreement with the already published solutions.

Role of fractal-fractional operators in modeling of rubella epidemic with optimized orders

Abstract Fractal-fractional (FF) differential and integral operators having the capability to subsume features of retaining memory and self-similarities are used in the present research analysis to design a mathematical model for the rubella epidemic while taking care of dimensional consistency among the model equations. Infectious diseases have history in their transmission dynamics and thus non-local operators such as FF play a vital role in modeling dynamics of such epidemics. Monthly actual rubella incidence cases in Pakistan for the years 2017 and 2018 have been used to validate the FF rubella model and such a data set also helps for parameter estimation. Using nonlinear least-squares estimation with MATLAB function lsqcurvefit, some parameters for the classical and the FF model are obtained. Upon comparison of error norms for both models (classical and FF), it is found that the FF produces the smaller error. Locally asymptotically stable points (rubella-free and rubella-present) of the model are computed when the basic reproduction number { {\mathcal R} }_{0} is less and greater than unity and the sensitivity is investigated. Moreover, solution of the FF rubella system is shown to exist. A new iterative method is proposed to carry out numerical simulations which resulted in getting insights for the transmission dynamics of the rubella epidemic.

Fractal-fractional mathematical modeling and forecasting of new cases and deaths of COVID-19 epidemic outbreaks in India

Fractional-order derivative-based modeling is very significant to describe real-world problems with forecasting and analyze the realistic situation of the proposed model. The aim of this work is to predict future trends in the behavior of the COVID-19 epidemic of confirmed cases and deaths in India for October 2020, using the expert modeler model and statistical analysis programs (SPSS version 23 & Eviews version 9). We also generalize a mathematical model based on a fractal fractional operator to investigate the existing outbreak of this disease. Our model describes the diverse transmission passages in the infection dynamics and affirms the role of the environmental reservoir in the transmission and outbreak of this disease. We give an itemized analysis of the proposed model including, the equilibrium points analysis, reproductive number R0, and the positiveness of the model solutions. Besides, the existence, uniqueness, and Ulam-Hyers stability results are investigated of the suggested model via some fixed point technique. The fractional Adams Bashforth method is applied to solve the fractal fractional model. Finally, a brief discussion of the graphical results using the numerical simulation (Matlab version 16) is shown.