Caputo-Fabrizio Derivative Research Articles

Comparative study of blood sugar–insulin model using fractional derivatives

ABSTRACT Many years have passed since the model that outlined the connection between sugar and insulin was developed, and a great deal of study and investigation has subsequently been done on it. In this work, we investigated the disparities in the Modified Bergman's glucose–insulin model between Yang–Abdel–Cattani derivative and Caputo–Fabrizio derivative. With our improvements, the prior model now includes the diet, which is a crucial component of the blood glucose model. Based on results, the new model outperforms the previous one in terms of accuracy. To highlight the significance of fractional derivatives, we also established the fractional models. In addition, we verified the existence and uniqueness of the results and offered a graphical and numerical depiction of the findings.

Series solution of time-fractional mhd viscoelastic model through non-local kernel approach

The study of ramped condition in the context of unsteady incompressible magnetohydrodynamic Casson fluid flow over a moving vertical plate is a complex and important topic in fluid dynamics and heat transfer. This scenario combines several physical phenomena and has practical applications in various engineering and scientific fields. In this study, Casson fluid is considered unsteady under the influence of magnetic field. The fractional mathematical model is proposed by considering the effect of chemical reaction parameter of the flowing fluid. The governing equations are transformed into the dimensionless form and developed fractional models like Caputo-Fabrizio and Atangana-Baleanu Derivative. We used the Laplace transform technique to find the solution of the dimensionless governing equation analytically. The transformed solutions for velocity, energy and momentum balances developed in terms of series. MATHCAD software is being used for numerical computations and the physical attributes of material and fractional parameters are discussed. To analyze their behavior clearly, two-dimensional graphical results are plotted for velocity profile and temperature as well. It has been concluded that the fluid’s velocity are reduced for larger values of the fractional parameter and Prandtl number and is maximum for small values of both parameters. Further, the velocity behavior becomes larger for isothermal condition as compared to ramped conditions.

Magnetohydrodynamic influence on fractional nanofluid convection in infinite vertical porous media with constant heat flux

This study investigates the complex dynamics governing free convection flow in nanofluids under Magnetohydrodynamics (MHD) within a porous medium along an infinite vertical surface. We specifically focus on scenarios involving constant and uniform heat flux as well as heat sources. The governing equations, incorporating the Boussinesq approximation, continuity, and momentum equations, are formulated and solved using Caputo-Fabrizio derivatives and Laplace transforms. This comprehensive approach integrates the Boussinesq approximation and Caputo-Fabrizio fractional derivatives, while the use of Laplace transforms enables precise analytical solutions. Additionally, our investigation highlights the significant impact of nanometer-sized copper particles on fluid properties, transforming the behavior from Newtonian (water-like) to non-Newtonian. This transformation profoundly affects thermal conductivity and velocity behavior within the system. Overall, this research establishes a robust foundation for future studies and provides a framework for exploring non-Newtonian nanofluid systems in the context of MHD-driven free convection flow.

A NUMERICAL SOLUTION OF THE FRACTIONAL NAVIER-STOKES EQUATION USING THE CAPUTO-FABRIZIO ABOODH TRANSFORM METHOD WITH THE REDUCED DIFFERENTIAL POLYNOMIALS

A combination of the Aboodh transform method and the reduced differential polynomial technique was employed in this work to solve the Navier-Stokes equations with the Caputo-Fabrizio derivative. Two illustrations are presented to show the efficacy of the used method. The results gotten are showcased with the aid of tables and graphs. It is discovered that the results derived are close to the actual solution of the problems illustrated. This work will thus make it simple to study nonlinear process that arise in various aspect of innovations and researches.

Using the Elzaki decomposition method to solve nonlinear fractional differential equations with the Caputo-Fabrizio fractional operator

يتم تطبيق تقنيات حساب التفاضل والتكامل الجزئي بنجاح في العديد من فروع العلوم والهندسة ، احدى هذه التقنيات طريقة تحليل الزكي وادوميان  (EADM) ، والتي لم يدرسها الباحثون باستخدام المشتق الكسري  لـ Caputo-Fabrizio. يهدف هذا العمل لدراسة طريقة تحليل الزكي وادوميان (EADM) لحل المعادلات التفاضلية الكسرية بمشتق Caputo-Fabrizio. قدمنا خوارزمية هذه الطريقة مع المؤثر CF وناقشنا تقاربها باستخدام طريقة سلسلة كوشي بعد ذلك ، تم تطبيق الطريقة لحل معادلات برجر، الشبية بالحرارة ، و برجر المزدوجة مع المؤثر Caputo -Fabrizio. في الختام أن الطريقة كانت متقاربة وفعالة لحل هذا النوع من المعادلات التفاضلية الكسرية.

New semi-analytical solution of fractional Newell–Whitehead–Segel equation arising in nonlinear optics with non-singular and non-local kernel derivative

In this paper, a combined form of Laplace transform is applied with the Adomian Decomposition technique for the first time to obtain new semi-analytical solutions of the fractional Newell–Whitehead–Segel equation which is a model arising in nonlinear optics with Caputo–Fabrizio derivative which involves non-singular and non-local kernels in its definition. The obtained results by the suggested method are compared with exact solutions, as a result of remarkable concurrence between the acquired results and the exact proposed method and the exacted solutions. Plotted graphs and given tables illustrate the efficiency and accuracy of the proposed technique. All the calculations are made by the computer software called MAPLE and Mathematica.

Analysis of the proportional Caputo-Fabrizio derivative

Analysis of the proportional Caputo-Fabrizio derivative

A study on fractional order financial model by using Caputo–Fabrizio derivative

This paper aims to provide an understanding of the dynamics and behavior of the financial model in the context of fractional differential equations. The model explores the interactions and connections among four key financial variables: interest rates, investment demand, price index, and savings amounts. We establish the existence of unique solution for the fractional financial model and stability of the model using iterative approach by employing Banach’s fixed point theory. A numerical method is carried out to perform the numerical simulation of the Caputo–Fabrizio financial model. The work capitalizes on the insights gained by scrutinizing the repercussions stemming from variations in model parameters on the dynamics of the solutions obtained through our numerical scheme. Our findings highlight that the interplay among diverse financial components can lead to chaotic behavior in specific scenarios.

Solution of the fractional diffusion equation by using Caputo-Fabrizio derivative: application to intrinsic arsenic diffusion in germanium

In this work, we focused on solving the space-time fractional diffusion equation with an application on the intrinsic arsenic diffusion in germanium. At first we have treated the differential equation in a semi-infinite medium by using Caputo-Fabrizio fractional derivative. We have introduced the Laplace transform to solve this type of equations. Secondly, Based on the obtained solution, we have simulated an profile of arsenic diffusion in germanium under intrinsic conditions. Accurate simulations have been achieved showing that the fractional derivative orders affect on the estimation of the diffusion coefficient, where increasing the time fractional derivative order α reduces the value of the diffusion coefficient, while increasing the space fractional derivative order β increases the value of the diffusion coefficient.

Efficient numerical method for multi-term time-fractional diffusion equations with Caputo-Fabrizio derivatives

<abstract><p>In this paper, we consider a numerical method for the multi-term Caputo-Fabrizio time-fractional diffusion equations (with orders $ \alpha_i\in(0, 1) $, $ i = 1, 2, \cdots, n $). The proposed method employs a fast finite difference scheme to approximate multi-term fractional derivatives in time, requiring only $ O(1) $ storage and $ O(N_T) $ computational complexity, where $ N_T $ denotes the total number of time steps. Then we use a Legendre spectral collocation method for spatial discretization. The stability and convergence of the scheme have been thoroughly discussed and rigorously established. We demonstrate that the proposed scheme is unconditionally stable and convergent with an order of $ O\left(\left(\Delta t\right)^{2}+N^{-m}\right) $, where $ \Delta t $, $ N $, and $ m $ represent the timestep size, polynomial degree, and regularity in the spatial variable of the exact solution, respectively. Numerical results are presented to validate the theoretical predictions.</p></abstract>

Mathematical analysis and numerical simulations of the piecewise dynamics model of Malaria transmission: A case study in Yemen

<abstract><p>This study presents a mathematical model capturing Malaria transmission dynamics in Yemen, incorporating a social hierarchy structure. Piecewise Caputo-Fabrizio derivatives are utilized to effectively capture intricate dynamics, discontinuities, and different behaviors. Statistical data from 2000 to 2021 is collected and analyzed, providing predictions for Malaria cases in Yemen from 2022 to 2024 using Eviews and Autoregressive Integrated Moving Average models. The model investigates the crossover effect by dividing the study interval into two subintervals, establishing existence, uniqueness, positivity, and boundedness of solutions through fixed-point techniques and fractional-order properties of the Laplace transformation. The basic reproduction number is computed using a next-generation technique, and numerical solutions are obtained using the Adams-Bashforth method. The results are comprehensively discussed through graphs. The obtained results can help us to better control and predict the spread of the disease.</p></abstract>

On chaos control of nonlinear fractional Newton-Leipnik system via fractional Caputo-Fabrizio derivatives

In this work, we present a design for a Newton-Leipnik system with a fractional Caputo-Fabrizio derivative to explain its chaotic characteristics. This time-varying fractional Caputo-Fabrizio derivative approach is applied to solve the model numerically, and to check the solution’s existence and uniqueness. The existence and uniqueness of results of a fractional-order model under the Caputo-Fabrizio fractional operator have been proved by fixed point theory. As well, we achieved a stable result by applying the Ulam-Hyers concept. Chaos is controlled by linear controllers. Furthermore, the Lyapunov exponent of the system indicates that the chaos control findings are accurate. Based on weighted covariant Lyapunov vectors we construct a background covariance matrix using the Kaplan-Yorke dimension. Using a numerical example, this suggested method is illustrated for its applicability and efficiency.

Fractional Nadeem trigonometric non-Newtonian (NTNN) fluid model based on Caputo-Fabrizio fractional derivative with heated boundaries

The fractional operator of Caputo-Fabrizio has significant advantages in various physical flow problems due to the implementations in manufacturing and engineering fields such as viscoelastic damping in polymer, image processing, wave propagation, and dielectric polymerization. The current study has the main objective of implementation of Caputo-Fabrizio fractional derivative on the flow phenomenon and heat transfer mechanism of trigonometric non-Newtonian fluid. The time-dependent flow mechanism is assumed to be developed through a vertical infinite plate. The thermal radiation’s effects are incorporated into the analysis of heat transfer. With the help of mathematical formulations, the physical flow system is expressed. The governing equations of the flow system acquire the dimensionless form through the involvement of the dimensionless variables. The application of Caputo-Fabrizio derivative is implemented to achieve the fractional model of the dimensionless system. An exact solution of the fractional-based dimensionless system of the equations is acquired through the technique of the Laplace transform. Physical interpretation of temperature and velocity distributions relative to the pertinent parameters is visualized via graphs. The current study concludes that the velocity distributions exhibit an accelerating nature corresponding to the increasing order of the fractional operator. Moreover, the graphical results are more significant corresponding to the greater time period.

Caputo-fabrizio fractional-order systems: periodic solution and stabilization of non-periodic solution with application to gunn diode oscillator

Fractional-order autonomous systems do not possess any non-constant periodic solutions, and to the best of our knowledge, there are no existing results regarding the existence of the periodic solution for fractional-order non-autonomous systems. The main objective of this work is to fill the above gap by studying the existence of a periodic solution of the Caputo-Fabrizio fractional-order system and also to find ways to stabilize a non-periodic solution. First, by using the concepts of an equilibrium point, it is proved that an autonomous Caputo-Fabrizio system cannot admit a non-constant periodic solution. Under a similar assumption as the one for an integer-order differential system, and by using the properties of the Caputo-Fabrizio derivative, the existence of a periodic solution of a non-autonomous Caputo-Fabrizio fractional-order differential system is established. The main result is utilized in constructing and finding the periodic solution of the linear non-homogeneous Caputo-Fabrizio system. By using the result on linear systems, we derive a periodic solution of a fractional-order Gunn diode oscillator under a periodic input voltage, and observe that the diameter of the periodic orbit keeps reducing as the fractional-order continuously increases. In the end, by using the result on a linear non-homogeneous system, and by constructing a suitable linear feedback control, the solution of the linear non-homogeneous fractional-order system is stabilized to a periodic solution. An example is presented to support the obtained result. The main advantage of the proposed method over others is the simple considerations like the concept of equilibrium point and the utilization of the property of the Caputo-Fabrizio derivatives instead of other types of fractional derivatives.

Application of Heat and Mass Transfer to Convective Flow of Casson Fluids in a Microchannel with Caputo–Fabrizio Derivative Approach

It has been demonstrated that fractional derivatives exhibit a range of solutions that are helpful in the engineering, medical, and manufacturing sciences. Particularly in analytical research, investigations on using fractional derivatives in fluid flow are still in their infancy. Therefore, it is still being determined whether fractional derivatives may be represented geometrically in the mechanics of the flow of fluids. However, theoretical research will be helpful in supporting upcoming experimental research. Therefore, the aim of this study is to showcase an application of Caputo–Fabrizio toward the Casson fluid flowing in an unsteady boundary layer. Mass diffusion and heat radiation are taken into account while analyzing the PDEs that governed the problem. Dimensionless governing equations are formed from the fractional PDEs by utilizing the necessary dimensionless variables. Once the equations have been transformed into linear ODEs, the solution may then be found by applying the Laplace transform technique. Inverting Laplace transforms by Stehfest’s and Tzou’s Algorithm is then used to retrieve the original variables and the solutions as concentration, temperature, and velocity fields. Graphical illustrations sketched using the Mathcad program are used to show how physical parameters affect temperature, velocity, and concentration profiles. Findings show that the velocity, temperature, and concentration profiles have been improved by thermal radiation, mass diffusion, and fractional parameters. The fractional derivative is a more general derivative due to its nonlocal and flexible nature the flow model that is formulated by applying the fractional derivative is suitable to address the memory effect. The present fractionalized results of velocity, concentration, and temperature are more general and applicable to the wide range of orders of fractional derivatives.

Two-dimensional Cattaneo-Hristov heat diffusion in the half-plane

In this paper, Cattaneo-Hristov heat diffusion is discussed in the half plane for the first time, and solved under two different boundary conditions. For the solution purpose, the Laplace, and the sine- and exponential- Fourier transforms with respect to time and space variables are applied, respectively. Since the fractional term in the problem is the Caputo-Fabrizio derivative with the exponential kernel, the solutions are in terms of time-dependent exponential and spatial-dependent Bessel functions. Behaviors of the temperature functions due to the change of different parameters of the problem are interpreted by giving 2D and 3D graphics.

Deterministic-stochastic analysis of fractional differential equations malnutrition model with random perturbations and crossover effects

To boost the handful of nutrient-dense individuals in the societal structure, adequate health care documentation and comprehension are permitted. This will strengthen and optimize the well-being of the community, particularly the girls and women of the community that are welcoming the new generation. In this article, we extensively explored a deterministic-stochastic malnutrition model involving nonlinear perturbation via piecewise fractional operators techniques. This novel concept leads us to analyze and predict the process from the beginning to the end of the well-being growth, as it offers the possibility to observe many behaviors from cross over to stochastic processes. Moreover, the piecewise differential operators, which can be constructed with operators such as classical, Caputo, Caputo-Fabrizio, Atangana-Baleanu and stochastic derivative. The threshold parameter is developed and the role of malnutrition in society is examined. Through a rigorous analysis, we first demonstrated that the stochastic model’s solution is positive and global. Then, using appropriate stochastic Lyapunov candidates, we examined whether the stochastic system acknowledges a unique ergodic stationary distribution. The objective of this investigation is to design a nutritional deficiency in pregnant women using a piecewise fractional differential equation scheme. We examined multiple options and outlined numerical methods of coping with problems. To exemplify the effectiveness of the suggested concept, graphical conclusions, including chaotic and random perturbation patterns, are supplied. Consequently, fractional calculus’ innovative aspects provide more powerful and flexible layouts, enabling us to more effectively adapt to the system dynamics tendencies of real-world representations. This has opened new doors to readers in different disciplines and enabled them to capture different behaviors at different time intervals.

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.

Hermite wavelets collocation method for solving a fredholm integro-differential equation with fractional Caputo-Fabrizio derivative

In this paper, we investigate the numerical study of nonlinear Fredholm integro-differential equation with the fractional Caputo-Fabrizio derivative. We use the Hermite wavelets and collocation technique to approximate the exact solution by reducing the Fredholm integro-differential equation to a nonlinear algebraic system. Furthermore, we apply this numerical method on certain examples to check its accuracy and validity.

Modeling and Analysis of a Fractional Visceral Leishmaniosis with Caputo and Caputo–Fabrizio derivatives

Visceral leishmaniosis is one recent example of a global illness that demands our best efforts at understanding. Thus, mathematical modeling may be utilized to learn more about and make better epidemic forecasts. By taking into account the Caputo and Caputo-Fabrizio derivatives, a frictional model of visceral leishmaniosis was mathematically examined based on real data from Gedaref State, Sudan. The stability analysis for Caputo and Caputo-Fabrizio derivatives is analyzed. The suggested ordinary and fractional differential mathematical models are then simulated numerically. Using the Adams-Bashforth method, numerical simulations are conducted. The results demonstrate that the Caputo-Fabrizio derivative yields more precise solutions for fractional differential equations.