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Monkeypox is a highly infectious disease and spreads very easily, hence posing several health concerns or risks as it may lead to outbreak. This article proposes a new mathematical model to simulate the transmission rate of the monkeypox virus-infected fractional-order differential equations using the Caputo–Fabrizio derivative. The existence, uniqueness, and stability under contraction mapping of the fixed point of the model are discussed using Krasnoselskii’s and Banach’s fixed point theorems. To verify the model proposed, we employ data that record the actual dynamics, and based on these data, the model can capture the observed transmission patterns in Ghana. Also, the analytic algorithm is used to find the result applying the Laplace Adomian decomposition method (LADM). Performance analysis of LADM is made regarding Runge-Kutta fourth order, which is the most commonly employed method for solving second-order ordinary differential equations. This comparison therefore offers information on the truth and reliability of the two techniques toward modeling the transmission pattern of the monkey pox virus. The information obtained through this study provides a better understanding of the antibodies linked to monkeypox virus spreading and provides effective strategies to doctors and politicians. This article helps shape better strategies about combating the impact of monkeypox virus in public health since it makes it easy to predict and prevent the occurrence of the disease.

In this paper, we apply the fractional homotopy perturbation transform method (FHPTM) to deliver an effective semi-analytical technique for determining fractional-order Kuramoto–Sivashinsky equations. The project technique combines the Laplace transform with the Caputo–Fabrizio fractional derivative of order α where α ∈ (0 , 1 ] . Fractional-order Kuramoto–Sivashinsky equation is indeed important in the field of nonlinear physics and mathematics. It is a fractional partial differential equation that describes the behaviour of waves in certain dissipative media, such as flames and chemicals. The FHPTM is described to be fast and accurate. Illustrative examples are included to demonstrate the efficiency and reliability of the presented techniques. [ABSTRACT FROM AUTHOR]

Cancer develops through cells when mutations build up in different genes that control cell proliferation. To treat these abnormal cells and minimize their growth, various cancer tumor samples have been modeled and analyzed in literature. The current study is focused on the investigation of more generalized cancer tumor model in fractional environment, where net killing rate is taken into account in different domains. Three types of killing rates are considered in the current study including time and position dependent killing rates, and concentration of cells based killing rate. A hybrid mechanism is proposed in which different homotopies are used with perturbation technique and Laplace transform. This leads to a convenient algorithm to tackle all types of fractional derivatives efficiently. The convergence and error bounds of the proposed scheme are computed theoretically by proving related theorems. In the next phase, convergence and validity is analyzed numerically by calculating residual errors round the fractional domain. It is observed that computed errors are very less in the entire fractional domain. Moreover, comparative analysis of Caputo, Caputo-Fabrizio (CF), and Atangana–Baleanu (AB) fractional derivatives is also performed graphically to discern the effect of different fractional approaches on the solution profile. Analysis asserts the reliability of proposed methodology in the matter of intricate fractional tumor models, and hence can be used to other complex physical phenomena.

The main objective of this article on Wireless sensor network of the Internet of Things (IoT). The wireless network, B bluetooth network, infrared network, and other networks are the key components of the Internet of Things (IoT). The major emphasis of this work was on the impact of Caputo-Fabrizio fractional derivative on worm propagation in heterogeneous susceptible-exposed-infected-recovered wireless IoT devices. We first determined the equilibrium points and fundamental reproduction number for the Caputo-Fabrizio HSEIR system, and then we discussed the stability of the system at the worm propagation equilibrium point. Using the Picard-Lindeof method, we determine the existence and unique solution for the fractional CF system of the heterogeneous SEIR model. Next, we use fixed point theory to judge the stability of the iterative method. We investigate the impact of the derivative order on the behaviour of the resultant functions and acquired numerical values by computing the model's findings for various fractional orders. In addition, we compute the integer-order model's results and contrast them with the results of the fractional-order model. We develop a periodically intermittent controller driven by white noise with the amazing benefits of reduced cost and more adaptable control technique to restrict the spread of worms in wireless IoT networks. To clearly define the conditions for stability in probability one, we employ the stochastic analysis technique. Our results show that the nonlinear worm propagation system may be stabilised by intermittent stochastic perturbation under the parameters of intermittent time linked to stochastic perturbation strength. Our theoretical conclusions may be used to analyse the observable processes of the worm, design countermeasures to prevent its spread, and evaluate the consequences of various system parameters.

Abstract The primary goal of the current work is to use a novel technique known as the natural transform decomposition method to approximate an analytical solution for the fractional smoking epidemic model. In the proposed method, fractional derivatives are considered in the Caputo, Caputo–Fabrizio, and Atangana–Baleanu–Caputo senses. An epidemic model is proposed to explain the dynamics of drug use among adults. Smoking is a serious issue everywhere in the world. Notwithstanding the overwhelming evidence against smoking, it is nonetheless a harmful habit that is widespread and accepted in society. The considered nonlinear mathematical model has been successfully used to explain how smoking has changed among people and its effects on public health in a community. The two states of being endemic and disease-free, which are when the disease dies out or persists in a population, have been compared using sensitivity analysis. The proposed technique has been used to solve the model, which consists of five compartmental agents representing various smokers identified, such as potential smokers V, occasional smokers G, smokers T, temporarily quitters O, and permanently quitters W. The results of the suggested method are contrasted with those of existing numerical methods. Finally, some numerical findings that illustrate the tables and figures are shown. The outcomes show that the proposed method is efficient and effective.

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.

In this paper, an implicit compact finite difference (CFD) scheme was constructed to get the numerical solution for time fractional diffusion-wave equation (TFDWE), in which the time fractional derivative was denoted by Caputo-Fabrizio (C-F) sense. We proved that the full discrete scheme is unconditionally stable. We also proved that the rate of convergence in time is near to $ O(\tau^{2}) $ and the rate of convergence in space is near to $ O(h^{4}) $. Test problem was considered for regular domain with uniform points to validate the efficiency and accuracy of the method. The numerical results can support the theoretical claims.

This study demonstrates the use of fractional calculus in the field of epidemiology, specifically in relation to dengue illness. Using noninteger order integrals and derivatives, a novel model is created to examine the impact of temperature on the transmission of the vector–host disease, dengue. A comprehensive strategy is proposed and illustrated, drawing inspiration from the first dengue epidemic recorded in 2009 in Cape Verde. The model utilizes a fractional-order derivative, which has recently acquired popularity for its adaptability in addressing a wide variety of applicable problems and exponential kernel. A fixed point method of Krasnoselskii and Banach is used to determine the main findings. The semi-analytical results are then investigated using iterative techniques such as Laplace-Adomian decomposition method. Computational models are utilized to support analytical experiments and enhance the credibility of the results. These models are useful for simulating and validating the effect of temperature on the complex dynamics of the vector–host interaction during dengue outbreaks. It is essential to note that the research draws on dengue outbreak studies conducted in various geographic regions, thereby providing a broader perspective and validating the findings generally. This study not only demonstrates a novel application of fractional calculus in epidemiology but also casts light on the complex relationship between temperature and the dynamics of dengue transmission. The obtained results serve as a foundation for enhancing our understanding of the complex interaction between environmental factors and infectious diseases, leading the way for enhanced prevention and control strategies to combat global dengue outbreaks.

This study investigates solutions for the time-fractional coupled system of the shallow-water equations. The shallow-water equations are employed for the purpose of elucidating the dynamics of water motion in oceanic or sea environments. Also, the aforementioned system characterizes a thin fluid layer that maintains a hydrostatic equilibrium while exhibiting uniform density. Shallow water flows have a vertical dimension that is considerably smaller in magnitude than the typical horizontal dimension. In the current work, we employ an innovative and effective technique, known as the natural transform decomposition method, to obtain the solutions for these fractional systems. The present methodology entails the utilization of both singular and non-singular kernels for the purpose of handling fractional derivatives. The Banach fixed point theorem is employed to demonstrate the uniqueness and convergence of the obtained solution. The outcomes obtained from the application of the suggested methodology are compared to the exact solution and the results of other numerical methods found in the literature, including the modified homotopy analysis transform method, the residual power series method and the new iterative method. The results obtained from the proposed methodology are presented through the use of tabular and graphical simulations. The current framework effectively captures the behavior exhibited by different fractional orders. The findings illustrate the efficacy of the proposed method.

Modeling several physical events leads to the Bagley–Torvik equation (BTE). In this study, we have taken into account the BTE, including the Caputo–Fabrizio and Atangana–Baleanu derivatives. It becomes challenging to find the analytical solution to these kinds of problems using standard methods in many circumstances. Therefore, to arrive at the required outcome, numerical techniques are used. The Laplace transform is a promising method that has been utilized in the literature to address a variety of issues that come up when modeling real-world data. For complicated functions, the Laplace transform approach can make the analytical inversion of the Laplace transform excessively laborious. As a result, numerical techniques are utilized to invert the Laplace transform. The numerical inverse Laplace transform is generally an ill-posed problem. Numerous numerical techniques for inverting the Laplace transform have been developed as a result of this challenge. In this article, we use the Weeks method, which is one of the most efficient numerical methods for inverting the Laplace transform. In our proposed methodology, first the BTE is transformed into an algebraic equation using Laplace transform. Then the reduced equation solved the Laplace domain. Finally, the Weeks method is used to convert the obtained solution from the Laplace domain into the real domain. Three test problems with Caputo–Fabrizio and Atangana–Baleanu derivatives are considered to demonstrate the accuracy, effectiveness, and feasibility of the proposed numerical method.

ABSTRACTMany 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.

The main objective of this article on Wireless sensor network of the Internet of Things (IoT). The wireless network, B bluetooth network, infrared network, and other networks are the key components of the Internet of Things (IoT). The major emphasis of this work was on the impact of Caputo-Fabrizio fractional derivative on worm propagation in heterogeneous susceptible-exposed-infected-recovered wireless IoT devices. We first determined the equilibrium points and fundamental reproduction number for the Caputo-Fabrizio HSEIR system, and then we discussed the stability of the system at the worm propagation equilibrium point. Using the Picard-Lindeof method, we determine the existence and unique solution for the fractional CF system of the heterogeneous SEIR model. Next, we use fixed point theory to judge the stability of the iterative method. We investigate the impact of the derivative order on the behaviour of the resultant functions and acquired numerical values by computing the model's findings for various fractional orders. In addition, we compute the integer-order model's results and contrast them with the results of the fractional-order model. We develop a periodically intermittent controller driven by white noise with the amazing benefits of reduced cost and more adaptable control technique to restrict the spread of worms in wireless IoT networks. To clearly define the conditions for stability in probability one, we employ the stochastic analysis technique. Our results show that the nonlinear worm propagation system may be stabilised by intermittent stochastic perturbation under the parameters of intermittent time linked to stochastic perturbation strength. Our theoretical conclusions may be used to analyse the observable processes of the worm, design countermeasures to prevent its spread, and evaluate the consequences of various system parameters. [ABSTRACT FROM AUTHOR]

In this paper, we consider a numerical method for the multi-term Caputo-Fabrizio time-fractional diffusion equations (with orders α i ∈ (0 , 1) , i = 1 , 2 , ⋯ , 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 ( (Δ t) 2 + N − m ) , where Δ 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. In this paper, we consider a numerical method for the multi-term Caputo-Fabrizio time-fractional diffusion equations (with orders , ). The proposed method employs a fast finite difference scheme to approximate multi-term fractional derivatives in time, requiring only storage and computational complexity, where 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 , where , , and 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. [ABSTRACT FROM AUTHOR]

The primary goal of the current work is to use a novel technique known as the natural transform decomposition method to approximate an analytical solution for the fractional smoking epidemic model. In the proposed method, fractional derivatives are considered in the Caputo, Caputo–Fabrizio, and Atangana–Baleanu–Caputo senses. An epidemic model is proposed to explain the dynamics of drug use among adults. Smoking is a serious issue everywhere in the world. Notwithstanding the overwhelming evidence against smoking, it is nonetheless a harmful habit that is widespread and accepted in society. The considered nonlinear mathematical model has been successfully used to explain how smoking has changed among people and its effects on public health in a community. The two states of being endemic and disease-free, which are when the disease dies out or persists in a population, have been compared using sensitivity analysis. The proposed technique has been used to solve the model, which consists of five compartmental agents representing various smokers identified, such as potential smokers V, occasional smokers G, smokers T, temporarily quitters O, and permanently quitters W. The results of the suggested method are contrasted with those of existing numerical methods. Finally, some numerical findings that illustrate the tables and figures are shown. The outcomes show that the proposed method is efficient and effective. [ABSTRACT FROM AUTHOR]

In this paper, an implicit compact finite difference (CFD) scheme was constructed to get the numerical solution for time fractional diffusion-wave equation (TFDWE), in which the time fractional derivative was denoted by Caputo-Fabrizio (C-F) sense. We proved that the full discrete scheme is unconditionally stable. We also proved that the rate of convergence in time is near to O (τ 2) and the rate of convergence in space is near to O (h 4). Test problem was considered for regular domain with uniform points to validate the efficiency and accuracy of the method. The numerical results can support the theoretical claims. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A spectral collocation-based approximate algorithm is adapted for the numerical evaluation of a class of fractional-order delay differential equations. The involved fractional operators are defined as the Caputo-Fabrizio and Liouville-Caputo derivatives. A novel family of polynomials called Narayana polynomials and their generalization forms are utilized in our collocation procedure. We study the convergent analysis of the Narayana polynomials in a weighted L2 norm and obtain an upper bound for their series expansion form. The performance of the present matrix collocation is justified by solving three test examples using both fractional operators and various fractional orders. The outcomes are compared with two existing numerical approaches, i.e., the modified operational matrix method (MOMM) and the operational matrix of integration relied on Taylor bases (OMTB). A comparison study reveals that our results are more accurate than these two methods and thus the presented matrix algorithm is superior in terms of efficacy and applicability. [ABSTRACT FROM AUTHOR]

This study investigates solutions for the time-fractional coupled system of the shallowwater equations. The shallow-water equations are employed for the purpose of elucidating the dynamics of water motion in oceanic or sea environments. Also, the aforementioned system characterizes a thin fluid layer that maintains a hydrostatic equilibrium while exhibiting uniform density. Shallow water flows have a vertical dimension that is considerably smaller in magnitude than the typical horizontal dimension. In the current work, we employ an innovative and effective technique, known as the natural transform decomposition method, to obtain the solutions for these fractional systems. The present methodology entails the utilization of both singular and non-singular kernels for the purpose of handling fractional derivatives. The Banach fixed point theorem is employed to demonstrate the uniqueness and convergence of the obtained solution. The outcomes obtained from the application of the suggested methodology are compared to the exact solution and the results of other numerical methods found in the literature, including the modified homotopy analysis transform method, the residual power series method and the new iterative method. The results obtained from the proposed methodology are presented through the use of tabular and graphical simulations. The current framework effectively captures the behavior exhibited by different fractional orders. The findings illustrate the efficacy of the proposed method. [ABSTRACT FROM AUTHOR]

This research aims to provide a new identification of Lagrange multipliers using a novel hybrid method based on the Khalouta transform method and the variational iteration method and to study the solutions of nonlinear fractional problems. This method is called Khalouta variational iteration method (KHVIM). The fractional derivative is considered in Caputo–Fabrizio sense. The uniqueness and convergence results are investigated using Banach fixed point theorem. The results are obtained in the form of successive approximations corresponding to the proposed problem. To demonstrate the accuracy and effectiveness of the proposed method, three different nonlinear fractional partial differential equations are provided. Approximate solutions of the fractional equations were obtained. These solutions quickly converged to exact solutions with lower computational cost. Furthermore, the method used in this study is more generalized and allows our results to be more extensive and cover several new and existing fractional problems in the literature.

This article is related to the utilization of the Laplace Adomian Decomposition Method (LADM) to analyze the solution of a fractional order system under different operators and normalized functions. In this connection, the three new and updated operators i.e., Caputo, Caputo–Fabrizio, and Atangana–Baleano are used to define the fractional order derivative with the help of four different normalization functions. It is observed that, among the four normalized functions, the fourth kind of normalized function is the most suitable function for the above-mentioned operators. The LADM series solutions are obtained for both integer and fractional orders, implying that the proposed approach has a greater accuracy and convergence rate. Figures and Tables are constructed to verify the validity and efficiency of the suggested technique. LADM simulations confirmed, that the more iterations of LADM reduced the associated absolute error. Moreover, fractional order solutions are obtained which are convergent towards integer order solutions.

Ebola is a serious disease that affects people; in many cases, it results in death. Ebola outbreaks have also occurred in communities where residents keep pets, particularly dogs. Due to a lack of food, the dogs must hunt for food. Dogs eat the internal organs of wildlife that the locals have killed and eaten, as well as small dead animals that are found within the communities which may contain the Ebola virus. This study introduces a mathematical model based on the Caputo–Fabrizio derivative to describe the Ebola transmission dynamics between dogs and humans. The model’s existence and the uniqueness of its solution were investigated using fixed-point theory. Furthermore, through the Sumudu transform criterion, we established that the Caputo–Fabrizio Ebola model is Picard stable. Some qualitative analysis was also carried out to investigate the Ebola propagation trend in the dog-to-human model. The proposed model is fitted to the reported Ebola incidence in Uganda between October 15, 2022, and November 2, 2022. The Ebola reproduction number obtained using the cumulative data was 2.65. It is noticed that as the fractional order reduces, the Ebola reproduction number also reduces. We derived a numerical scheme for our model using the two-step Lagrange interpolation. It has been discovered that the fractional orders significantly influence the model, indicating that natural occurrences could affect the dynamics of Ebola. It is observed that when the recovery rate is enhanced, such as through the hospitalisation of Ebola-infected individuals, the disease will reduce. Finally, as we ensure a reduction in the contact rate among the dog’s compartments, the disease does not spread adiabatically. Therefore, we urge that quarantine measures be put in place to control interactions among the dogs during the outbreak.

The aim of this study is to characterize the micro-structural effects for the thermoelastic interactions in a moving finite medium due to a time-dependent laser heat source. The heat transport equation has been formulated in the context of dual-phase-lag (DPL) model of generalized thermoelasticity. The medium is subjected to a time-dependent laser heat source, in which the capacity of the source decays exponentially. It is assumed that the boundaries of the medium are thermally insulated. The heat transport equation is formulated using the Caputo–Fabrizio derivative. Laplace transform technique is used to solve the governing equations and a suitable numerical scheme is adopted to arrive at the solutions. The corresponding numerical inversion has been carried out using the Zakian method. According to the graphical representations, conclusions about the new theory are constructed. The graphical representation interprets a novel observation of how the numerical scheme of the Zakian method gives superior results compared to Tzou algorithm or Stehfest method. Moreover, a comparative study between the Cattaneo–Vernotte equation and the DPL equation has been analyzed and, due to the microstructural effects into the delayed response in time, the superiority of the DPL model compared to the CV model is analyzed too. [ABSTRACT FROM AUTHOR]

Dusty Casson fluids and tetra-hybrid nanofluids are complex phenomena that find their extensive uses in engineering and industrial applications. For instance, dusty fluids are used in gas-freezing systems and nuclear power reactors. The main objective of this article is to focus on the characterization of generalized two-phase free convection magnetohydrodynamic flow of dusty tetra-hybrid Casson nanofluid among parallel microplates: dusty Casson fluid and tetra-hybrid nanofluid exhibit free movement and electrical conductivity. The Caputo–Fabrizio fractional derivative recently discovered generalizes the partial differential equations governing the flow. Highly accurate temperature and velocity distributions can be obtained using finite sine Fourier and Laplace transform together. This study examines the relationships between temperature, dust particle velocity, and Casson fluid velocity, along with the effects of magnetic parameter, Grashof number, dusty fluid parameter, Peclet number, Reynold number, and particle mass parameter. The Mathcad-15 software provides Casson, dusty, and temperature profiles graphically. The Nusselt number and skin friction are also examined for the tetra-hybrid nanofluid. The fractional Casson fluid model is more accurate than the classical model in terms of velocity, temperature, heat transfer, and skin friction. Graphical results conclude that the fractional Casson fluid model describes a more realistic aspect of both (fluid and dust particle) velocities and temperature profiles, heat transfer rate, and skin friction than the classical Casson fluid model. Furthermore, the heat transfer rate enhanced from 0 to 39.3111% of the tetra-hybrid nanofluid.

This paper uses a fractional-order epidemic model to describe the transmission dynamics of the Ebola virus. The proposed model uses the fractional-order derivative in Caputo-Fabrizio’s sense. It calculates the time-independent solutions of the proposed model, and the next-generation matrix method is used to calculate the basic reproduction number. It provides the conditions for the existence and uniqueness of solutions to the model. Further, the conditions for generalized Ulam-Hyers-Rassias stability of the proposed model are obtained. Numerical simulations show how the proposed model’s approximate solution varies for integer and fractional orders. They also show the behavior of the Ebola in terms of infections, deceased, and susceptible counts, for various contact rates. To demonstrate efficiency while using less time, CPU times are given in tabular form.

In this work, a coupled system under coupled integral boundary conditions with Caputo-Fabrizio derivative (CFD) is considered. We intend to derive some necessary and sufficient results for the existence of at least one solution. In addition, we extend our analysis further to develop a monotone iterative scheme coupled with the upper and lower solution method to compute extremal solutions. Therefore, in this regard, Perov's fixed point theorem is applied to study the existing criteria for the solution. Also, results related to at least one solution are derived by using Schauder's fixed point theorem. Finally, we use a monotone iterative procedure together with upper and lower solution methods to study extremal solutions. Graphical presentations of upper and lower solutions are provided for some examples to illustrate our results.

This work aims at providing a concise review of various agri-food models that employ fractional differential operators. In this context, various mathematical models based on fractional differential equations have been used to describe a wide range of problems in agri-food. As a result of this review, we found out that this new area of research is finding increased acceptance in recent years and that some reports have employed fractional operators successfully in order to model real-world data. Our results also show that the most commonly used differential operators in these problems are the Caputo, the Caputo–Fabrizio, the Atangana–Baleanu, and the Riemann–Liouville derivatives. Most of the authors in this field are predominantly from China and India. [ABSTRACT FROM AUTHOR]

In this work, a coupled system under coupled integral boundary conditions with Caputo-Fabrizio derivative (CFD) is considered. We intend to derive some necessary and sufficient results for the existence of at least one solution. In addition, we extend our analysis further to develop a monotone iterative scheme coupled with the upper and lower solution method to compute extremal solutions. Therefore, in this regard, Perov's fixed point theorem is applied to study the existing criteria for the solution. Also, results related to at least one solution are derived by using Schauder's fixed point theorem. Finally, we use a monotone iterative procedure together with upper and lower solution methods to study extremal solutions. Graphical presentations of upper and lower solutions are provided for some examples to illustrate our results. [ABSTRACT FROM AUTHOR]

The approximate solution of the Kersten-Krasil'shchik coupled Korteweg-de Vries-modified Korteweg-de Vries system is obtained in this study by employing a natural decomposition method in association with the newly established Atangana-Baleanu derivative and Caputo-Fabrizio derivative of fractional order. The Korteweg-de Vries equation is considered a classical super-extension in this system. This nonlinear model scheme is commonly used to describe waves in traffic flow, electromagnetism, electrodynamics, elastic media, multi-component plasmas, shallow water waves and other phenomena. The acquired results are compared to exact solutions to demonstrate the suggested method's effectiveness and reliability. Graphs and tables are used to display the numerical results. The results show that the natural decomposition technique is a very user-friendly and reliable method for dealing with fractional order nonlinear problems.

A key challenge for drug dosing schedules is the ability to learn an optimal control policy even when there is a paucity of accurate information about the systems. Artificial intelligence has great potential for shaping a smart control policy for the dosage of drugs for any treatment. Motivated by this issue, in the present research paper a Caputo–Fabrizio fractional-order model of cancer chemotherapy treatment was elaborated and analyzed. A fix-point theorem and an iterative method were implemented to prove the existence and uniqueness of the solutions of the proposed model. Afterward, in order to control cancer through chemotherapy treatment, a fuzzy-reinforcement learning-based control method that uses the State-Action-Reward-State-Action (SARSA) algorithm was proposed. Finally, so as to assess the performance of the proposed control method, the simulations were conducted for young and elderly patients and for ten simulated patients with different parameters. Then, the results of the proposed control method were compared with Watkins's Q-learning control method for cancer chemotherapy drug dosing. The results of the simulations demonstrate the superiority of the proposed control method in terms of mean squared error, mean variance of the error, and the mean squared of the control action—in other words, in terms of the eradication of tumor cells, keeping normal cells, and the amount of usage of the drug during chemotherapy treatment. [ABSTRACT FROM AUTHOR]

This paper uses a fractional-order epidemic model to describe the transmission dynamics of the Ebola virus. The proposed model uses the fractional-order derivative in Caputo-Fabrizio's sense. It calculates the time-independent solutions of the proposed model, and the next-generation matrix method is used to calculate the basic reproduction number. It provides the conditions for the existence and uniqueness of solutions to the model. Further, the conditions for generalized Ulam-Hyers-Rassias stability of the proposed model are obtained. Numerical simulations show how the proposed model's approximate solution varies for integer and fractional orders. They also show the behavior of the Ebola in terms of infections, deceased, and susceptible counts, for various contact rates. To demonstrate efficiency while using less time, CPU times are given in tabular form. [ABSTRACT FROM AUTHOR]

In this paper, we study the analytical solutions of two-dimensional fractional-order linear system $ \mathcal{D}^{\alpha}_{t}X(t) = AX(t) $ described by fractional differential equations, where $ \mathcal{D} $ is the fractional derivative in the Caputo-Fabrizio sense and $ A = (a_{ij})_{2\times2} $ is nonsingular coefficient matrix with $ a_{ij}\in\mathbb{R} $. The analytical solutions of fractional-order linear system will be compared to the solution of classical linear system. Examples are provided to characterize the behavior of the solutions for fractional-order linear system.

In this paper, the solutions of some typical nonlinear fractional differential equations are discussed, and the implicit analytical solutions are obtained. The fractional derivative concerned here is the Caputo-Fabrizio form, which has a nonsingular kernel. The calculation results of different fractional orders are compared through images. In addition, by comparing the results obtained in this paper with those under Caputo fractional derivative, it is found that the solutions change relatively gently under Caputo-Fabrizio fractional derivative. It can be concluded that the selection of appropriate fractional derivatives and appropriate fractional order is very important in the modeling process.

The approximate solution of the Kersten-Krasil'shchik coupled Korteweg-de Vriesmodified Korteweg-de Vries system is obtained in this study by employing a natural decomposition method in association with the newly established Atangana-Baleanu derivative and Caputo-Fabrizio derivative of fractional order. The Korteweg-de Vries equation is considered a classical super-extension in this system. This nonlinear model scheme is commonly used to describe waves in traffic flow, electromagnetism, electrodynamics, elastic media, multi-component plasmas, shallow water waves and other phenomena. The acquired results are compared to exact solutions to demonstrate the suggested method's effectiveness and reliability. Graphs and tables are used to display the numerical results. The results show that the natural decomposition technique is a very user-friendly and reliable method for dealing with fractional order nonlinear problems. [ABSTRACT FROM AUTHOR]

HIV-1 infection is a dangerous diseases like Cancer, AIDS, etc. Many mathematical models have been introduced in the literature, which are investigated with different approaches. In this article, we generalize the HIV-1 model through nonsingular fractional operator. The non-integer mathematical model of HIV-1 infection under the Caputo-Fabrizio derivative is presented in this paper. The concept of Picard-Lindelof and fixed-point theory are used to address the existence of a unique solution to the HIV-1 model under the suggested operator. Also, the stability of the suggested model is proved through the Picard iteration and fixed point theory approach. The model's approximate solution is constructed through three steps Adams-Bashforth numerical method. Numerical simulations are provided for different values of fractional-order to study the complex dynamics of the model. Lastly, we provide the oscillatory and chaotic behavior of the proposed model for various fractional orders.

The virus that begins from Wuhan China, known as COVID-19 or coronavirus is still a huge panic for humans around the globe. The elimination of this virus from our society needs proper attentions to follows the rule suggested by World Health Organization (WHO). A vast literature on the modeling of this infection in various perspective is available. In the present work, we design a new mathematical model for COVID-19 pandemic by utilizing the real infected cases reported from Kingdom of Saudi Arabia. Initially, we formulate the model with the help of classical integer order nonlinear differential equations. The treatment class is considered the model to analyze the impact of treatment on the disease dynamics. The Caputo-Fabrizio derivative with the non-singular exponential kernel is applied in order to reformulate the proposed COVID-19 transmission model with a fractional order. The biologically important parameter called the basic reproductive number is investigated both theoretically and numerically. The estimated values of R0 for the selected period are approximated to be 1.63. Further, by making use of the Picard Lindelöf theorem we provide the existence and uniqueness of the COVID-19 fractional epidemic model. Moreover, the fractional model is solved numerically and a number of simulation results are depicted using the real estimated parameters. The impact of various model parameters and memory index are shown graphically. We conclude that the fractional order epidemic models are more appropriate and provide deep insights into the disease dynamics.

This study investigates the fractional-order Swift–Hohenberg equations using the natural decomposition method with non-singular kernel derivatives. The fractional derivative in the sense of Caputo–Fabrizio is considered. The Adomian decomposition technique (ADT) is a great deal to the overall natural transformation to create closed-form results of the given models. This technique provides a closed-form result for the suggested models. In addition, this technique is attractive, simple, and preferred over other techniques. The graphs of the solution in fractional and integer-order show that the achieved solutions are very close to the actual result of the examples. It is also investigated that the result of fractional-order models converges to the integer-order model's solution. Furthermore, the proposed method validity is examined using numerical examples. The obtained results for the given problems fully support the theory of the proposed method. The present method is a straightforward and accurate analytical method to analyze other fractional-order partial differential equations, such as many evolution equations that govern the dynamics of nonlinear waves in plasma physics. [ABSTRACT FROM AUTHOR]

In this paper, we study the analytical solutions of two-dimensional fractional-order linear system Dαt X(t) = AX(t) described by fractional differential equations, where D is the fractional derivative in the Caputo-Fabrizio sense and A = (ai j)2x2 is nonsingular coefficient matrix with aij ∈ R. The analytical solutions of fractional-order linear system will be compared to the solution of classical linear system. Examples are provided to characterize the behavior of the solutions for fractional-order linear system. [ABSTRACT FROM AUTHOR]

In this manuscript, we study the fractional-order SIRC epidemiological model for influenza A in the human population in the Caputo-Fabrizio sense. The existence and uniqueness of the solution of the proposed problem are established using fixed point theory. The local stability of both disease-free equilibrium and endemic equilibrium points is investigated. Using the three-step fractional Adams-Bashforth scheme, an iterative solution of our system is generated. In the numerical simulation, many plots are given for different values of the fractional-order to check the stability of equilibrium points. Also, the effect of varying some parameters of the model was presented. Furthermore, we compared our numerical solutions with those using Caputo fractional derivative model via graphical representations. The obtained results show the efficiency and accuracy of our approach. [ABSTRACT FROM AUTHOR]

The aim of the article is two-fold. We first analyze and investigate free convective, unsteady, MHD blood flow with single- and multiwalled carbon nanotubes (S&MWCNTs) as nanoparticles. The blood flow has been taken across an upright vertical plate, oscillating in its own plane, and engrafted in a porous medium with slip, radiation, and porosity effects. Nanofluids consist of human blood as the base fluid and SWCNTs and MWCNTs as nanoparticles. The second aim is to discuss the three different definitions of fractional derivatives, namely, Caputo (C), Caputo–Fabrizio (CF), and Atangana–Baleanu (ABC), to obtain the solutions of such proposed models by the Adomian decomposition method. The impact of fractional and physical parameters on the concentration, velocity, and temperature of human blood in the presence of the slip effect is studied and projected diagrammatically. The article ends by providing numerical results such as the reliableness, efficiency, and significant features that are simple in computation with eminent accuracy of the process for non-Newtonian Casson nanofluid fractional order models. It is observed that the velocity of the fluid decreases with SWCNTs’ and MWCNTs’ volume fraction, and an increase in the CNTs’ volume fraction increases blood temperature, which ultimately enhances heat transfer rates. The results acquired are in excellent correspondence with the reported results.

This study uses efficient techniques to evaluate a non-linear system of Korteweg–de Vries (KdV) equations with fractional Caputo Fabrizio derivative, including the modified decomposition approach and the novel iterative transform method. The system of KdV equations and the modify scheme of KdV equations used as a model in non-linear physical processes emerging in biology, chemistry, physics and sciences are the non-linear fractional coupled systems explored in this present analysis. Approximate analytical outcomes are represented as a series with simple components, and some features revealed a proper dependency on the fractional-order derivatives’ values. An examination of convergence and uniqueness is performed. Three test cases for the analytic findings of the fractional-order KdV equations are supplied to help understand the analytical technique of both methods. Furthermore, the efficiency of the aforementioned operations, as well as the decrease in computations, allow for a larger application. It is also demonstrated that the present methodology’s conclusions are in close agreement with the precise answers. With few computations, the series result obtained using this approach has been proven to be accurate and dependable. For various fractional-order values, numerical simulations for derived solutions are described.

This work aims at providing a concise review of various agri-food models that employ fractional differential operators. In this context, various mathematical models based on fractional differential equations have been used to describe a wide range of problems in agri-food. As a result of this review, we found out that this new area of research is finding increased acceptance in recent years and that some reports have employed fractional operators successfully in order to model real-world data. Our results also show that the most commonly used differential operators in these problems are the Caputo, the Caputo–Fabrizio, the Atangana–Baleanu, and the Riemann–Liouville derivatives. Most of the authors in this field are predominantly from China and India.

In this study, numerical results of a fractional-order multi-dimensional model of the Navier–Stokes equations will be achieved via adoption of two analytical methods, i.e., the Adomian decomposition transform method and the q-Homotopy analysis transform method. The Caputo–Fabrizio operator will be used to define the fractional derivative. The proposed methods will be implemented to provide the series form results of the given models. The series form results of proposed techniques will be validated with the exact results available in the literature. The proposed techniques will be investigated to be efficient, straightforward, and reliable for application to many other scientific and engineering problems. [ABSTRACT FROM AUTHOR]

The purpose of this investigation is to theoretically shed some light on the effect of the unsteady electroosmotic flow (EOF) of an incompressible fractional second-grade fluid with low-dense mixtures of two spherical nanoparticles, copper, and titanium. The flow of the hybrid nanofluid takes place through a vertical micro-channel. A fractional Cattaneo model with heat conduction is considered. For the DC-operated micropump, the Lorentz force is responsible for the pressure difference through the microchannel. The Debye-Hükel approximation is utilized to linearize the charge density. The semi-analytical solutions for the velocity and heat equations are obtained with the Laplace and finite Fourier sine transforms and their numerical inverses. In addition to the analytical procedures, a numerical algorithm based on the finite difference method is introduced for the given domain. A comparison between the two solutions is presented. The variations of the velocity heat transfer against the enhancements in the pertinent parameters are thoroughly investigated graphically. It is noticed that the fractional-order parameter provides a crucial memory effect on the fluid and temperature fields. The present work has theoretical implications for biofluid-based microfluidic transport systems. [ABSTRACT FROM AUTHOR]

Abstract This manuscript is devoted to a study of the existence and uniqueness of solutions to a mathematical model addressing the transmission dynamics of the coronavirus-19 infectious disease (COVID-19). The mentioned model is considered with a nonsingular kernel type derivative given by Caputo–Fabrizo with fractional order. For the required results of the existence and uniqueness of solution to the proposed model, Picard’s iterative method is applied. Furthermore, to investigate approximate solutions to the proposed model, we utilize the Laplace transform and Adomian’s decomposition (LADM). Some graphical presentations are given for different fractional orders for various compartments of the model under consideration.

A key challenge for drug dosing schedules is the ability to learn an optimal control policy even when there is a paucity of accurate information about the systems. Artificial intelligence has great potential for shaping a smart control policy for the dosage of drugs for any treatment. Motivated by this issue, in the present research paper a Caputo–Fabrizio fractional-order model of cancer chemotherapy treatment was elaborated and analyzed. A fix-point theorem and an iterative method were implemented to prove the existence and uniqueness of the solutions of the proposed model. Afterward, in order to control cancer through chemotherapy treatment, a fuzzy-reinforcement learning-based control method that uses the State-Action-Reward-State-Action (SARSA) algorithm was proposed. Finally, so as to assess the performance of the proposed control method, the simulations were conducted for young and elderly patients and for ten simulated patients with different parameters. Then, the results of the proposed control method were compared with Watkins’s Q-learning control method for cancer chemotherapy drug dosing. The results of the simulations demonstrate the superiority of the proposed control method in terms of mean squared error, mean variance of the error, and the mean squared of the control action—in other words, in terms of the eradication of tumor cells, keeping normal cells, and the amount of usage of the drug during chemotherapy treatment.

This paper presents the semi-analytical analysis of the fractional-order non-linear coupled system of Whitham-Broer-Kaup equations. An iterative process is designed to analyze analytical findings to the specified non-linear partial fractional derivatives scheme utilizing the Yang transformation coupled with the Adomian technique. The fractional derivative is considered in the sense of Caputo-Fabrizio. Two numerical problems show the suggested method. Moreover, the results of the suggested technique are compared with the solution of other well-known numerical techniques such as the Homotopy perturbation technique, Adomian decomposition technique, and the Variation iteration technique. Numerical simulation has been carried out to verify that the suggested methodologies are accurate and reliable, and the results are revealed using graphs and tables. Comparing the analytical and actual solutions demonstrates that the proposed approaches effectively solve complicated non-linear problems. Furthermore, the proposed methodologies control and manipulate the achieved numerical solutions in a vast acceptable region in an extreme manner. It will provide us with a simple process to control and adjust the convergence region of the series solution. [ABSTRACT FROM AUTHOR]

The capacitors and inductors in actual circuits often fail to exhibit the ideal integer-order characteristics, so as the circuits containing these types of electronic components. The errors can be compensated by introducing the concept of fractional calculus. The Caputo–Fabrizio​ fractional derivative, which fixes the defect of the singularity problem in Caputo derivative, has been proposed in recent years. This work uses Caputo–Fabrizio fractional derivative to model the fractional Buck converter working in continuous conduction mode and to deduce its analytical solutions by Laplace transformation and its inverse transformation. The analytical solution of the output voltage is obtained and the waveform of the output voltage is simulated. The correctness of the model is verified and the impact of the fractional order on the dynamic performance of the Buck converter is analyzed. The steady-state average output voltage of the open-loop fractional Buck converter is also calculated.

This study introduces a new and promising stability approach for Caputo-Fabrizio (CF)-fractional-order system. A new fractional comparison principle for this nonsingular kernel fractional derivative is proposed. Next, a key inequality is suggested to analysis the Lyapunov-based stability of assumed systems. Afterwards, class-K functions are established to analysis of fractional Lyapunov direct method. At last, an explanatory example is given to validate the proposed idea. This new and novel approach can be expanded to the other types of nonsingular kernel derivatives due to a simple and effective idea beyond the proposed procedure.

In this paper, we investigate the solution of the Riccati differential equations of fractional order (FRDEs) involving Caputo derivative (CD), Caputo-Fabrizio derivative (CFD) or Atangana-Baleanu derivative (ABD) using a semi-analytical iterative approach. Temimi and Ansari introduced this method and called it TAM. The linearization of the method involves splitting the problem into a linear and nonlinear part. While this approach is a semi-analytical iterative technique, there is a significant amount of analytical work where the computational times of integrals are needed to be carried out numerically. The novelty and the motivation behind this paper is the comparison of the time used in minutes which is given for three derivatives: CD, CFD and ABD. Meanwhile, the comparison of the approximate solutions with CD, CFD and ABD are presented. With the help of the software Mathematica, the computational results have been obtained. [ABSTRACT FROM AUTHOR]

This research focus on the determination of the numerical solution for the math- ematical model of Fokker-Planck equations utilizing a new method, in which Sumudu trans- formation and homotopy analysis method (SHAM) are used together. By SHAM analytical series solution of any mathematical model including fractional derivative can be obtained. By this method, we constructed the solution of fractional Fokker-Planck equations in Caputo and Caputo-Fabrizio senses. The results show that this method is advantageous and applicable to form the series resolution of the fractional mathematical models. [ABSTRACT FROM AUTHOR]