Fractal-fractional Derivative Research Articles

Numerical approach for solving two dimensional fractal-fractional PDEs using peridynamic method

This paper develops a numerical approach for solving two-dimensional fractal-fractional parabolic partial differential equations. The fractal-fractional derivative is defined in the Atangana-Riemann-Liouville sense with Mittage-Leffler kernel. To solve this equation, we first eliminate the spatial derivatives using peridynamic differential operators. Then, we derive an operational matrix (OM) of fractal-fractional derivative in terms of the Legendre polynomials to simplify the time derivative. The aim of the formulated method is to transform the original problem into an uncomplicated system of linear algebraic equations which can be solved by mathematical software. The applicability of the approach is examined for several examples and numerical results show the computational efficiency of the method.

Fractal Fractional Operator Method on HER2+ Breast Cancer Dynamics

In this paper, an in vitro model of HER2+ breast cancer cells dynamics resulting from various dosages and timings of paclitaxel and trastuzumab combination regimens is considered. Since, the combined in vitro results and development of dynamics of drug synergy has a potential to evaluate and improve standard of care, then combination therapies in timings of paclitaxel and trastuzumab combination regimens, thus, HER2+ breast cancer cells dynamics are extended to a system of fractal fractional partial differential equations in order to enable one to capture the dynamics of the deadly breast cancer in terms of combination of the two therapies. Moreover, the well-posedness of solutions is presented and the extended dynamics are analysed to that effect. Since it is not that easy to obtain the analytic solution a novel numerical method based on fractal fractional derivatives is design, implemented and the results with respect to the stability conditions are presented.

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.

An efficient operation matrix method for solving fractal–fractional differential equations with generalized Caputo-type fractional–fractal derivative

In this study, we present the new generalized derivative and integral operators which are based on the newly constructed new generalized Caputo fractal–fractional derivatives (NGCFFDs). Based on these operators, a numerical method is developed to solve the fractal–fractional differential equations (FFDEs). We approximate the solution of the FFDEs as basis vectors of shifted Legendre polynomials (SLPs). We also extend the derivative operational matrix of SLPs to the generalized derivative operational matrix in the sense of NGCFFDs. The efficiency of the developed numerical method is tested by taking various test examples. We also compare the results of our proposed method with the methods existed in the literature In this paper, we specified the fractal–fractional differential operator of new generalized Caputo in three categories: (i) different values in ρ and fractal parameters, (ii) different values in fractional parameter while fractal and ρ parameters are fixed, and (iii) different values in fractal parameter controlling fractional and ρ parameters.

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.

Orthonormal Bernoulli polynomials for space–time fractal-fractional modified Benjamin–Bona–Mahony type equations

In this study, a new fractal-fractional (FF) derivative is defined by coupling the local conformable derivative and non-local Caputo fractional derivative. Using the defined derivative, a space–time FF version of the modified Benjamin–Bona–Mahony type equations is introduced. A collocation technique based on the orthonormal Bernoulli polynomials and their derivative matrices (including the ordinary and FF derivative matrices obtained in this study) is adopted for solving such equations. The presented method converts solving this equation to solve a simple system of algebraic equations. Some numerical problems are provided to show the accuracy of the expressed scheme.

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.

A meshless technique based on the moving least squares shape functions for nonlinear fractal-fractional advection-diffusion equation

This paper introduces a fractal-fractional version of the nonlinear 2D advection-diffusion equation and proposes a meshless method based on the moving least squares shape functions for its numerical solution. The fractal-fractional derivative in the Atangana-Riemann-Liouville is considered to define this equation. The proposed method includes the following steps: We first approximate the fractal-fractional derivative using the finite differences method and derive a recursive algorithm by applying the θ-weighted method. Next, using the moving least squares shape functions, we expand the solution of the problem and its corresponding partial derivatives and substitute them into the recurrence formula. Finally, in accordance with the previous step, we obtain a linear system of algebraic equations which must be solved at each time step. The validity and accuracy of the method are investigated by solving some numerical examples.

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.

MODELING AND ANALYSIS OF NOVEL COVID-19 UNDER FRACTAL-FRACTIONAL DERIVATIVE WITH CASE STUDY OF MALAYSIA

In this paper, new model on novel coronavirus disease (COVID-19) with four compartments including susceptible, exposed, infected, and recovered class with fractal-fractional derivative is proposed. Here, Banach and Leray–Schauder alternative type theorems are used to establish some appropriate conditions for the existence and uniqueness of the solution. Also, stability is needed in respect of the numerical solution. Therefore, Ulam–Hyers stability using nonlinear functional analysis is used for the proposed model. Moreover, the numerical simulation using the technique of fundamental theorem of fractional calculus and the two-step Lagrange polynomial known as fractional Adams-Bashforth (AB) method is proposed. The obtained results are tested on real data of COVID-19 outbreak in Malaysia from 25 January till 10 May 2020. The numerical simulation of the proposed model has performed in terms of graphs for different fractional-order [Formula: see text] and fractal dimensions [Formula: see text] via number of considered days of disease spread in Malaysia. Since COVID-19 transmits rapidly, perhaps, the clear understanding of transmission dynamics of COVID-19 is important for countries to implement suitable strategies and restrictions such as Movement Control Order (MCO) by the Malaysian government, against the disease spread. The simulated results of the presented model demonstrate that movement control order has a great impact on the transmission dynamics of disease outbreak in Malaysia. It can be concluded that by adopting precautionary measures as restrictions on individual movement the transmission of the disease in society is reduced. In addition, for such type of dynamical study, fractal-fractional calculus tools may be used as powerful tools to understand and predict the global dynamics of the mentioned disease in other countries as well.

Optimal control problem of advection-diffusion-reaction equation of kind fractal-fractional applying shifted Jacobi polynomials

Optimal control is always intended for optimization of different systems. In this study, new introduced differential operators called fractal-fractional derivatives have been used to investigate the behavior of one of the attractions of applied mathematics in physics and engineering. A novel version of the optimal control problem (OCP) generated using a dynamic system of type space-time fractal-fractional advection-diffusion-reaction equation is provided. An exact method based on the operational matrix derived from shifted Jacobi polynomials (SJPs) and collocation schemes is proposed. To define these new class of problems, the fractal-fractional derivative is implemented in to the Atangana-Riemann-Liouville sense whit Mittag-Leffler kernel applied for the time variable, and the fractional derivatives carried out for the space variable in the Caputo sense and the Atangana-Baleano-Caputo concept. In other words, the proposed method turns the principal problems solution into solving a system of algebraic equations. For this purpose, we substitute the approximate state and control values of the variables obtained by SJPs with unknown coefficients into the objective function, the dynamic system, and the initial and Dirichlet boundary conditions. Eventually, this problem becomes a quadratic optimization problem with linear constraint solved by the Lagrange multipliers method. In addition, the convergence analysis of SJPs expansion has been proved. To assess the effectiveness and precision of the process, some scenarios are illustrated.

Fractal fractional derivative on chemistry kinetics hires problem

<abstract> <p>In this work, we construct the fractional order model for chemical kinetics issues utilizing novel fractal operators such as fractal fractional by using generalized Mittag-Leffler Kernel. To overcome the constraints of the traditional Riemann-Liouville and Caputo fractional derivatives, a novel notion of fractional differentiation with non-local and non-singular kernels was recently presented. Many scientific conclusions are presented in the study, and these results are supported by effective numerical results. These findings are critical for solving the nonlinear models in chemical kinetics. These concepts are very important to use for real life problems like brine tank cascade, recycled brine tank cascade, pond pollution, home heating and biomass transfer problem. Many scientific results are presented in the paper also prove these results by effective numerical results. These results are very important for solving the nonlinear model in chemistry kinetics which will be helpful to understand the chemical reactions and its actual behavior; also the observation can be developed for future kinematic chemical reactions with the help of these results.</p> </abstract>

New Aspects of Bloch Model Associated with Fractal Fractional Derivatives

Abstract To model complex real world problems, the novel concept of non-local fractal-fractional differential and integral operators with two orders (fractional order and fractal dimension) have been used as mathematical tools in contrast to classical derivatives and integrals. In this paper, we consider Bloch equations with fractal-fractional derivatives. We find the general solutions for components of magnetization ℳ = (Mu , Mv , Mw ) by using descritization and Lagrange's two step polynomial interpolation. We analyze the model with three different kernels namely power function, exponential decay function and Mittag-Leffler type function. We provide graphical behaviour of magnetization components ℳ = (Mu , Mv , Mw ) on different orders. The examination of Bloch equations with fractal-fractional derivatives show new aspects of Bloch equations.

Fractal fractional derivative on chemistry kinetics hires problem

<abstract> <p>In this work, we construct the fractional order model for chemical kinetics issues utilizing novel fractal operators such as fractal fractional by using generalized Mittag-Leffler Kernel. To overcome the constraints of the traditional Riemann-Liouville and Caputo fractional derivatives, a novel notion of fractional differentiation with non-local and non-singular kernels was recently presented. Many scientific conclusions are presented in the study, and these results are supported by effective numerical results. These findings are critical for solving the nonlinear models in chemical kinetics. These concepts are very important to use for real life problems like brine tank cascade, recycled brine tank cascade, pond pollution, home heating and biomass transfer problem. Many scientific results are presented in the paper also prove these results by effective numerical results. These results are very important for solving the nonlinear model in chemistry kinetics which will be helpful to understand the chemical reactions and its actual behavior; also the observation can be developed for future kinematic chemical reactions with the help of these results.</p> </abstract>

On the Caputo-Fabrizio fractal fractional representation for the Lorenz chaotic system

<abstract><p>The widespread application of chaotic dynamical systems in different fields of science and engineering has attracted the attention of many researchers. Hence, understanding and capturing the complexities and the dynamical behavior of these chaotic systems is essential. The newly proposed fractal-fractional derivative and integral operators have been used in literature to predict the chaotic behavior of some of the attractors. It is argued that putting together the concept of fractional and fractal derivatives can help us understand the existing complexities better since fractional derivatives capture a limited number of problems and on the other side fractal derivatives also capture different kinds of complexities. In this study, we use the newly proposed Caputo-Fabrizio fractal-fractional derivatives and integral operators to capture and predict the behavior of the Lorenz chaotic system for different values of the fractional dimension $ q $ and the fractal dimension $ k $. We will look at the well-posedness of the solution. For the effect of the Caputo-Fabrizio fractal-fractional derivatives operator on the behavior, we present the numerical scheme to study the graphical numerical solution for different values of $ q $ and $ k $.</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.

New applications related to Covid-19

Analysis of mathematical models projected for COVID-19 presents in many valuable outputs. We analyze a model of differential equation related to Covid-19 in this paper. We use fractal-fractional derivatives in the proposed model. We analyze the equilibria of the model. We discuss the stability analysis in details. We apply very effective method to obtain the numerical results. We demonstrate our results by the numerical simulations.

Novel chaotic systems with fractional differential operators: Numerical approaches

Abstract The purpose of this paper is to study numerically the behavior of two novel different classes of fractional order chaotic systems. These systems are; the fractal-fractional hyperchaotic finance system and the fractal-fractional Bloch system with time delay. The fractal-fractional derivatives are defined in the Caputo and Riemann-Liouville senses. Two Grunwald-Letnikov nonstandard finite difference schemes are presented to study the proposed chaotic systems. Moreover the stability analysis of the used methods are proved. In order to show the simplicity and effectively of the proposed methods, numerical simulations and comparative studies are given.

Fixed-Point Theorems for Rational Interpolative-Type Operators with Applications

In the current manuscript, two fixed-point theorems for Dass-Gupta and Gupta-Saxena rational interpolative-type operators are studied in the setting of metric spaces. For the authenticity of the presented work, examples and applications to the existence of a solution to the Caputo-Fabrizio fractional derivative and Caputo-Fabrizio fractal-fractional derivative are also discussed.

Fractal-fractional study of the hepatitis C virus infection model

The main purpose of this article is proposed and investigated a new model of fractal-fractional-order model for hepatitis C virus (HCV) infection with immune response, in which the effect of dendritic cells (DC) and cytotoxic T lymphocytes (CTL) on HCV infection is considered. Fractal-Fractional new model is studied by replacing the classic derivative by fractal-fractional derivatives based on Caputo-Fabrizio (CF). By employing the basic theory of fractional calculus, and the Lagrange polynomial interpolation (LGPI) (the Newton polynomial interpolation (NPI)), we can construct an algorithm, through which we obtain the numerical solutions for the new model. We compare the numerical solutions obtained with help of LGPI and NPI with the numerical solutions using finite differences method (FDM). Also, the comparison between the numerical solutions based on LGPI and NPI is presented. We calculated the absolute error of these solutions, the results are very accurate, and the order of the error decreases as we increase the number of the iterations. The chaos of the numerical solutions for the fractal-fractional-order model are obtained for different values of parameters and represented graphically.