Fractal-fractional Derivative Research Articles

Dynamics analysis of dengue fever model with harmonic mean type under fractal-fractional derivative

<abstract><p>Dengue is a viral illness transmitted by Aedes mosquitoes and is a significant global threat. In this study, we developed a model of the dengue epidemic that incorporates larvicide and adulticide, as well as the harmonic mean incidence rate under fractal-fractional derivatives. We examined various theoretical aspects of the model, including nonnegativity, boundedness, existence, uniqueness, and stability. We computed the basic reproduction number $ \Re _{0} $ using the next-generation matrix. The model has two disease-free equilibriums, a trivial equilibrium, and a biologically realistic, along with one endemic equilibrium point. These findings enhanced our understanding of dengue transmission, providing valuable insights for awareness campaigns, control strategies, intervention approaches, decision support, guiding public health planning, and resource allocation to manage dengue effectively.</p></abstract>

A Computational Numerical Study of Burger Equation with Fractal Fractional Derivative

A Computational Numerical Study of Burger Equation with Fractal Fractional Derivative

Extension of Chaplygin's existence and uniqueness method for fractal-fractional nonlinear differential equations

<abstract><p>The existence and uniqueness of solutions to nonlinear ordinary differential equations with fractal-fractional derivatives, with Dirac-delta, exponential decay, power law, and generalized Mittag-Leffler kernels, have been the focus of this work. To do this, we used the Chaplygin approach, which entails creating two lower and upper sequences that converge to the solution of the equations under consideration. We have for each case provided the conditions under which these sequences are obtained and converge.</p></abstract>

Existence, uniqueness and numerical solution of stochastic fractional differential equations with integer and non-integer orders

<abstract><p>The parametrized approach is extended in this study to find solutions to differential equations with fractal, fractional, fractal-fractional, and piecewise derivatives with the inclusion of a stochastic component. The existence and uniqueness of the solution to the stochastic Atangana-Baleanu fractional differential equation are established using Caratheodory's existence theorem. For the solution of differential equations using piecewise differential operators, which take into account combining deterministic and stochastic processes utilizing certain significant mathematical tools such as fractal and fractal-fractional derivatives, the applicability of the parametrized technique is being examined. We discuss the crossover behaviors of the model obtained by including these operators and we present some illustrative examples for some problems with piecewise differential operators.</p></abstract>

Numerical investigation and deep learning approach for fractal–fractional order dynamics of Hopfield neural network model

This paper investigates the dynamics of Hopfield neural networks involving fractal–fractional derivatives. The incorporation of fractal–fractional derivatives in the neural network framework brings forth novel modeling capabilities, capturing nonlocal dependencies, complex scaling behaviors, and memory effects. The aim of this study is to provide a comprehensive analysis of the dynamics of Hopfield neural networks with fractal–fractional derivatives, including the existence and uniqueness of solutions, stability properties, and numerical analysis techniques. Numerical analysis techniques, including the Adams–Bashforth method, are employed to accurately simulate the fractal–fractional Hopfield neural network system. Moreover, the obtained numerical data serves as validation for developing predictions using Multilayer Perceptron (MLP) and Long Short-Term Memory (LSTM) neural network methods. The findings contribute to the advancement of both fractional calculus and neural network theory, providing valuable insights for theoretical investigations and practical applications in complex systems analysis.

A new modern scheme for solving fractal–fractional differential equations based on deep feedforward neural network with multiple hidden layer

The recent development of knowledge in fractional calculus introduced an advanced superior operator known as fractal–fractional derivative (FFD). This operator combines memory effect and self-similar property that give better accurate representation of real world problems through fractal–fractional differential equations (FFDEs). However, the existence of fresh and modern numerical technique on solving FFDEs is still scarce. Originally invented for machine learning technique, artificial neural network (ANN) is cutting-edge scheme that have shown promising result in solving the fractional differential equations (FDEs). Thus, this research aims to extend the application of ANN to solve FFDE with power law kernel in Caputo sense (FFDEPC) by develop a vectorized algorithm based on deep feedforward neural network that consists of multiple hidden layer (DFNN-2H) with Adam optimization. During the initial stage of the method development, the basic framework on solving FFDEs is designed. To minimize the burden of computational time, the vectorized algorithm is constructed at the next stage for method to be performed efficiently. Several example have been tested to demonstrate the applicability and efficiency of the method. Comparison on exact solutions and some previous published method indicate that the proposed scheme have give good accuracy and low computational time.

Analysis of Volterra Integrodifferential Equations with the Fractal-Fractional Differential Operator

In this paper, a class of integrodifferential equations with the Caputo fractal-fractional derivative is considered. We study the exact and numerical solutions of the said problem with a fractal-fractional differential operator. The abovementioned operator is arising widely in the mathematical modeling of various physical and biological problems. In our scheme, first, the integrodifferential equation with the fractal-fractional differential operator is converted to an integrodifferential equation with the Riemann–Liouville differential operator. Additionally, the obtained integrodifferential equation is then converted to the equivalent integrodifferential equation involving the Caputo differential operator. Then, we convert the integrodifferential equation under the Caputo differential operator using the Laplace transform to an algebraic equation in the Laplace space. Finally, we convert the obtained solution from the Laplace space into the real domain. Moreover, we utilize two techniques which include analytic inversion and numerical inversion. For numerical inversion of the Laplace transforms, we have to evaluate five methods. Extensive results are presented. Furthermore, for numerical illustration of the abovementioned methods, we consider three problems to demonstrate our results.

Mathematical Modeling of Breast Cancer Based on the Caputo–Fabrizio Fractal-Fractional Derivative

Breast cancer ranks among the most prevalent malignancies affecting the female population and is a prominent contributor to cancer-related mortality. Mathematical modeling is a significant tool that can be employed to comprehend the dynamics of breast cancer progression and dissemination and to formulate novel therapeutic approaches. This paper introduces a mathematical model of breast cancer that utilizes the Caputo–Fabrizio fractal-fractional derivative. The aim is to elucidate and comprehend the intricate dynamics governing breast cancer cells and cytotoxic T lymphocytes in the context of the fractional derivative. The derivative presented herein offers a broader perspective than the conventional derivative, as it incorporates the intricate fractal characteristics inherent in the process of tumor proliferation. The significance of this study lies in its contribution to a novel mathematical model for breast cancer, which incorporates the fractal characteristics of tumor development. The present model possesses the capability to investigate the impacts of diverse treatment strategies on the proliferation of breast cancer, as well as to formulate novel treatment strategies that exhibit enhanced efficacy.

Theoretical and numerical analysis of nonlinear Boussinesq equation under fractal fractional derivative

Abstract A nonlinear Boussinesq equation under fractal fractional Caputo’s derivative is studied. The general series solution is calculated using the double Laplace transform with decomposition. The convergence and stability analyses of the model are investigated under Caputo’s fractal fractional derivative. For the numerical illustrations of the obtained solution, specific examples along with suitable initial conditions are considered. The single solitary wave solutions under fractal fractional derivative are attained by considering small values of time ( t ) \left(t) . The wave propagation has a symmetrical form. The solitary wave’s amplitude diminishes over time, and its extended tail expands over a long distance. It is observed that the fractal fractional derivatives are an extremely constructive tool for studying nonlinear systems. An error analysis is also carried out for compactness.

Examining Dynamics of Emerging Nipah Viral Infection with Direct and Indirect Transmission Patterns: A Simulation-Based Analysis via Fractional and Fractal-Fractional Derivatives

In this study, we examine the transmission dynamics of global Nipah virus infection under direct (human-to-human) and indirect (contaminated foods-to-human) transmission routes via the Caputo fractional and fractional-fractal modeling approaches. The model is vigorously analyzed both theoretically and numerically. The possible equilibrium points of the system and their existence are investigated based on the reproduction number. The model exhibits three equilibrium points, namely, infection-free, infected flying foxes free, and endemic. Furthermore, novel numerical schemes are derived for the models in fractional and fractal-fractional cases. Finally, an extensive simulation is conducted to validate the theoretical results and provide an impact of the model on the disease incidence. We believe that this study will help to incorporate such mathematical techniques to examine the complex dynamics and control the spread of such infectious diseases.

Fractal‐fractional analysis and numerical simulation for the heat transfer of ZnO + Al2O3 + TiO2/DW based ternary hybrid nanofluid

AbstractNanofluids are used to achieve maximum thermal performance with the smallest concentration of nanoparticles and stable suspension in conventional fluids. The effectiveness of nanofluids in convection processes is significantly influenced by their increased thermophysical characteristics. However, this technology is not ended here; binary and ternary nanofluids are now used to improve the efficiency of regular fluids. Therefore, this paper aims to analyze the natural convection Newtonian ternary nanofluid flow in a vertical channel. The tri‐hybridized nanoparticles of zinc oxide ZnO, Aluminum oxide Al2O3, and titanium oxide TiO2 is dissolved in base fluid distilled water (DW) to form a homogenous suspension. The impact of thermal radiation, joule heating, and viscous dissipation are also assumed. The classical Newtonian ternary nanofluid model has been generalized by using fractal‐fractional derivative (FFD) operator. The generalized model has been discretized by using the Crank–Nicolson scheme and then solved by using computational software. To analyze the behavior of fluid flow and heat distribution in fluid, the obtained solution was computed numerically and then plotted in response to different physical parameters. It is noted from the figure that when the volume fraction ϕ reaches to 0.04 (4% of the base fluid), the ternary nanofluid flow shows a significant amount of enhancement in heat transfer rate as compared to binary and unary nanofluid flows. This enhancement in the rate of heat transfer leads to improve the thermophysical characteristics such as viscosity, thermal expansion, and heat capacity etc. of the base fluid. It is also worth mentioning here that the thermal field is also enhance with the higher values of Eckert number , radiation parameter , and joule heating parameter .

Fractional shifted Morgan–Voyce neural networks for solving fractal-fractional pantograph differential equations

We provide an effective numerical strategy for fractal-fractional pantograph differential equations (FFPDEs). The fractal-fractional derivative is considered in the Atangana–Riemann–Liouville sense. The scheme is based on fractional shifted Morgan-Voyce neural network (FShM-VNN). We introduce a new class of functions called fractional-order shifted Morgan-Voyce and some useful properties of these functions for the first time. The FShM-VNN method is utilized the fractional-order shifted Morgan-Voyce functions (FShM-VFs) and Sinh function as activation functions of the hidden layer and output layer of the neural network (NN), respectively. The approximate function contains the FShM-VFs with unknown weights. Using the classical optimization method and Newton’s iterative scheme, the weights are adjusted such that the approximate function satisfies the under study problem. Convergence analysis of the mentioned strategy is discussed. The scheme yields very accurate outcomes. The obtained numerical examples support this assertion.

Different strategies for diabetes by mathematical modeling: Applications of fractal–fractional derivatives in the sense of Atangana–Baleanu

For diabetes mellitus therapies, Bergman’s minimal model is proposed. The present study is devoted to mathematical modeling and analysis of the diabetes mellitus model without genetic factors in the fractal–fractional derivatives in the sense of Atangana–Baleanu. Diabetes mellitus disease dynamics are modeled using this modified derivative. The operator can be applied to formulate memory effects models. We proved the existence and uniqueness of the model solution using fractal–fractional derivatives in the sense of Atangana–Baleanu. The model’s solution and uniqueness, as well as the Hyers-Ulam stability analysis, are established. Moreover, we compare ordinary, Atangana–Baleanu, fractal–fractional derivatives in the sense of Atangana–Baleanu, and the real measured data of 10 subjects. Diabetes mellitus rates increase when both fractal dimensions θ1 and θ2 decrease in numerical experiments. Results from fractal–fractional derivatives in the sense of Atangana–Baleanu are closer to reality than integer-order models.

A numerical investigation of marriage divorce model: Fractal fractional perspective

Even if a marriage began as “till forever”, divorce is a legal way for married couples to end their relationship. Divorce, indeed, is a complex process that may be seen from several perspectives. Sociologists believe that the family gives more than simply individuals. It is the structure that allows individuals to develop mentally while also providing the necessary educational and financial assistance. Divorced spouses face economic, social, and financial exploitation. This, in turn, disintegrates families and diminishes the concept of the family as the core unit of society, causing society to become stalled. The breakup of families that will produce healthy and happy children in the future is a significant societal issue. According to Eurostat statistics, the number of crude marriages in European nations was 8 in 1964 and 4.3 in 2014. At the same period, crude divorce rates grew by more than twice as much, from 0.8 in 1964 to 1.8 in 2019. In 2022, Latvia, Lithuania, and Denmark had the most number of divorces. Considering the datas, the investigation of this phenomenon, which deeply injures and shakes society, by various branches of science and the study of its dynamics has become an important objective and mathematical analysis of divorce illuminates the motivation for this paper. Therefore, the mathematical model of divorce is considered with by the fractal fractional Caputo–Fabrizio derivative. Firstly, using Equilibrium points, the model’s linear stability is obtained. Then, the existence and uniqueness the solution of the model was proven by the Banach Fixed point theorem. Lastly, the behavior of the model is evaluated using graphics for different values of the fractal dimension and fractional derivative by developing a numerical method for the model that contains the fractal fractional derivative. Experimental results are analyzed for different instances of the key parameters that played major roles for each of the sub-population classes.

A mathematical model of corruption dynamics endowed with fractal–fractional derivative

Numerous organisations across the globe have significant challenges about corruption, characterised by a systematic, endemic, and pervasive nature that permeates various societal establishments. Hence, we propose the fractional order model of corruption, which encompasses the involvement of corrupt individuals across various stages of education and employment. Specifically, we examine the presence of corruption among children in elementary schools, youths in tertiary institutions, adults in civil services, adults in government and public offices, and individuals who have renounced their involvement in corrupt practices. The basic reproduction number of the system was determined by utilising the next-generation matrix. The strength number was obtained by calculating the second derivative of the corruption-related compartments. The examined model solution’s existence, uniqueness, and stability were established using the Krasnoselski fixed point theorem, the Banach contraction principle, and the Ulam–Hyers theorem, respectively. Based on the numerous figures presented, our simulations indicate a positive correlation between the decline in fractal–fractional order and the increase in the number of individuals susceptible to corruption. This phenomenon results in an increase in the prevalence of corruption among designated sectors of the general population. The persistence of corruption in society is a significant challenge to its eradication, as individuals who see personal gains from engaging in corrupt practices tend to exhibit a recurring inclination towards such behaviour. Nevertheless, it is recommended that to mitigate corruption within various corruption-prone subcategories, there is a need to enhance the level of consciousness and promotion of anti-corruption measures throughout all societal establishments.

A study on a monkeypox transmission model within the scope of fractal–fractional derivative with power-law kernel

A study on a monkeypox transmission model within the scope of fractal–fractional derivative with power-law kernel

Investigating the impact of memory effects on computer virus population dynamics: A fractal–fractional approach with numerical analysis

In this study, we investigate the impact of memory effects through the application of the fractal–fractional derivative operator on population dynamics in computer viruses. By utilizing numerical analysis techniques, we explore the influence of varying fractal–fractional operator orders on virus propagation and conduct a parameter study to examine the effects of fractal dimension and fractional order on computer virus spread. Furthermore, in order to comprehensively compare and analyze the outcomes from various perspectives, we introduce a novel application of the operational matrix technique to the proposed dynamical system, transforming the fractal–fractional model into the Caputo derivative form. Two different numerical techniques, Adams–Bashforth and Taylor operational matrix, have been employed to solve the two distinct forms of the model. The obtained results yielded highly consistent outcomes for the two numerical methods employed, despite their utilization of distinct derivative operators. The reported results highlight the memory effect associated with fractal–fractional derivatives, where past states and behaviors continue to influence the system. The ability to capture these memory effects through fractal–fractional derivatives can be considered a benefit, enabling a more comprehensive understanding, analysis and modeling of computer virus dynamics. Additionally, we investigate four variants to ensure numerical stability, employing both the Ulam–Hyers (UH) and Ulam–Hyers–Rassias (UHR) criteria.

New fractional approach for CMC and water based hybrid nanofluid with slip boundary layer: Applications of fractal fractional derivative

The physical significance of the Fractal fractional derivative, one of the most fascinating and modern analytical techniques, to the thermal flow is described in this paper. Free convection, which is the flow of an erratic, incompressible viscous fluid on a vertically inclined plate, has been considered in this work. Two types of base fluids, namely water and carboxymethyl cellulose, are used in the presence of graphene oxide (GO) and molybdenum disulfide (MoS2) nanoparticles. The flow is subject to slip possessions due to the inclined surface. The non-dimensional leading equations for thermal and momentum, under Boussinseq's approximation, can be solved by employing a Laplace transformation scheme, the fractal fractional derivative, and a more recent and improved definition of a fractional mathematical term. The thermal expressions are modeled using the Laplace transformation. The effect and comparison of various parameters are then visually and quantitatively analyzed after this activity. We, therefore, proposed that a water-based nanofluid's thermal profile and velocity are marginally higher than those of a hybrid nanofluid based on carboxymethyl cellulose. The momentum profile additionally illustrates the dual behavior of the flowing hybrid nanofluid as the fluid parameter of the Brinkman type increases.

Qualitative and quantitative analysis of a fractal fractional HIV/AIDS model

The fractal-fractional derivative is a type of fractional derivative that is more broadly applicable. This method is used to investigate a variety of real-world issues. This work focuses on investigating a fractional derivative-based sustainable strategy for the dynamics of HIV/AIDS. For the representation of a time-fractional order HIV/AIDS model, we presented a system of FDEs. The suggested model’s uniqueness and existence are revealed through equilibrium analysis. Analysis of the fractional order is accomplished using both sensitivity analysis and qualitative analysis. The first and second derivative tests are used to validate the analysis of the Lyapunov function for global stability. Solutions are produced using a two-step Lagrange polynomial in the generalized form of the Mlittag-Lefler kernel to analyze the influence of the fractional operator, which illustrates the impact of HIV/AIDS on society. Such an inquiry will help in the comprehension of HIV/AIDS behavior and the creation of preventative measures for the infected. Graphical representations of sensitive criteria that illustrate how to reduce and eradicate HIV/AIDS in society are provided. Finally, it is demonstrated that the smaller values of fractal-fractional order perform better than bigger values for all compartments of the proposed model of HIV/AIDS. We continue to believe that our work will be helpful to researchers working in diverse fields of applied science and engineering.

A New Class of Generalized Fractal and Fractal-Fractional Derivatives with Non-Singular Kernels

The present paper introduces a new class of generalized differential and integral operators. This class includes and generalizes a large number of definitions of fractal-fractional derivatives and integral operators used to model the complex dynamics of many natural and physical phenomena found in diverse fields of science and engineering. Some properties of the newly introduced class are rigorously established. As applications of this new class, two illustrative examples are presented, one for a simple problem and the other for a nonlinear problem modeling the dynamical behavior of a chaotic system.