在Banach空间中研究一类具有Caputo-Fabrizio分数阶导数且在非局部条件下的脉冲分数阶微分方程解的存在性。利用Schaefer不动点定理,压缩映射原理,Arzela-Ascoli定理,得到了该脉冲分数阶问题至少一个解和唯一解,并用一个例子验证其中一...在Banach空间中研究一类具有Caputo-Fabrizio分数阶导数且在非局部条件下的脉冲分数阶微分方程解的存在性。利用Schaefer不动点定理,压缩映射原理,Arzela-Ascoli定理,得到了该脉冲分数阶问题至少一个解和唯一解,并用一个例子验证其中一个结论。The existence of a class of impulsive fractional differential equations with Caputo-Fabrizio fractional derivatives under non-local conditions is studied in Banach space. Based on Schaefer’s fixed point theorem, compression mapping principle and Arzela-Ascoli theorem, at least one solution and the only solution of the pulse fractional order problem are obtained, and one of the conclusions is verified by an example.展开更多
Asymptotic stability of linear and interval linear fractional-order neutral delay differential systems described by the Caputo-Fabrizio (CF) fractional derivatives is investigated. Using Laplace transform, a novel cha...Asymptotic stability of linear and interval linear fractional-order neutral delay differential systems described by the Caputo-Fabrizio (CF) fractional derivatives is investigated. Using Laplace transform, a novel characteristic equation is derived. Stability criteria are established based on an algebraic approach and norm-based criteria are also presented. It is shown that asymptotic stability is ensured for linear fractional-order neutral delay differential systems provided that the underlying stability criterion holds for any delay parameter. In addition, sufficient conditions are derived to ensure the asymptotic stability of interval linear fractional order neutral delay differential systems. Examples are provided to illustrate the effectiveness and applicability of the theoretical results.展开更多
In this paper, we consider numerical solutions of fractional ordinary diferential equations with the Caputo-Fabrizio derivative, and construct and analyze a high-order time-stepping scheme for this equation. The propo...In this paper, we consider numerical solutions of fractional ordinary diferential equations with the Caputo-Fabrizio derivative, and construct and analyze a high-order time-stepping scheme for this equation. The proposed method makes use of quadratic interpolation function in sub-intervals, which allows to produce fourth-order convergence. A rigorous stability and convergence analysis of the proposed scheme is given. A series of numerical examples are presented to validate the theoretical claims. Traditionally a scheme having fourth-order convergence could only be obtained by using block-by-block technique. The advantage of our scheme is that the solution can be obtained step by step, which is cheaper than a block-by-block-based approach.展开更多
This paper studies the time-fractional Korteweg-de Vries (KdV) equations with Caputo-Fabrizio fractional derivatives. The scheme is presented by using a finite difference method in temporal variable and a local discon...This paper studies the time-fractional Korteweg-de Vries (KdV) equations with Caputo-Fabrizio fractional derivatives. The scheme is presented by using a finite difference method in temporal variable and a local discontinuous Galerkin method (LDG) in space. Stability and convergence are demonstrated by a specific choice of numerical fluxes. Finally, the efficiency and accuracy of the scheme are verified by numerical experiments.展开更多
This study focuses on the dynamics of drug concentration in the blood.In general,the concentration level of a drug in the blood is evaluated by themean of an ordinary and first-order differential equation.More precise...This study focuses on the dynamics of drug concentration in the blood.In general,the concentration level of a drug in the blood is evaluated by themean of an ordinary and first-order differential equation.More precisely,it is solved through an initial value problem.We proposed a newmodeling technique for studying drug concentration in blood dynamics.This technique is based on two fractional derivatives,namely,Caputo and Caputo-Fabrizio derivatives.We first provided comprehensive and detailed proof of the existence of at least one solution to the problem;we later proved the uniqueness of the existing solution.The proof was written using the Caputo-Fabrizio fractional derivative and some fixed-point techniques.Stability via theUlam-Hyers(UH)technique was also investigated.The application of the proposedmodel on two real data sets revealed that the Caputo derivative wasmore suitable in this study.Indeed,for the first data set,the model based on the Caputo derivative yielded a Mean Squared Error(MSE)of 0.03095 with a corresponding best value of fractional order of derivative of 1.00360.Caputo-Fabrizio-basedderivative appeared to be the second-best method for the problem,with an MSE of 0.04324 for a corresponding best fractional derivative order of 0.43532.For the second experiment,Caputo derivative-based model still performed the best as it yielded an MSE of 0.04066,whereas the classical and the Caputo-Fabrizio methods were tied with the same MSE of 0.07299.Another interesting finding was that the MSE yielded by the Caputo-Fabrizio fractional derivative coincided with the MSE obtained from the classical approach.展开更多
Diabetes is a burning issue in the whole world.It is the imbalance between body glucose and insulin.The study of this imbalance is very much needed from a research point of view.For this reason,Bergman gave an importa...Diabetes is a burning issue in the whole world.It is the imbalance between body glucose and insulin.The study of this imbalance is very much needed from a research point of view.For this reason,Bergman gave an important model named-Bergman minimalmodel.In the present work,using Caputo-Fabrizio(CF)fractional derivative,we generalize Bergman’s minimal blood glucose-insulin model.Further,we modify the old model by including one more component known as diet D(t),which is also essential for the blood glucose model.We solve the modified modelwith the help of Sumudu transformand fixed-point iteration procedures.Also,using the fixed point theorem,we examine the existence and uniqueness of the results along with their numerical and graphical representation.Furthermore,the comparison between the values of parameters obtained by calculating different values of t with experimental data is also studied.Finally,we draw the graphs of G(t),X(t),I(t),andD(t)for different values ofτ.It is also clear from the obtained results and their graphical representation that the obtained results of modified Bergman’s minimal model are better than Bergman’s model.展开更多
The aim of this paper is to investigate the dynamic behaviors of fractional-order logistic model with Allee effects in Caputo-Fabrizio sense.First of all,we apply the two-step Adams-Bashforth scheme to discretize the ...The aim of this paper is to investigate the dynamic behaviors of fractional-order logistic model with Allee effects in Caputo-Fabrizio sense.First of all,we apply the two-step Adams-Bashforth scheme to discretize the fractional-order logistic differential equation and obtain the two-dimensional discrete system.The parametric conditions for local asymptotic stability of equilibrium points are obtained by Schur-Chon criterion.Moreover,we discuss the existence and direction for Neimark-Sacker bifurcations with the help of center manifold theorem and bifurcation theory.Numerical simulations are provided to illustrate theoretical discussion.It is observed that Allee effect plays an important role in stability analysis.Strong Allee effect in population enhances the stability of the coexisting steady state.In additional,the effect of fractional-order derivative on dynamic behavior of the system is also investigated.展开更多
In this paper,we study two fractional models in the Caputo–Fabrizio sense and Atangana–Baleanu sense,in which the effects of malaria infection on mosquito biting behavior and attractiveness of humans are considered....In this paper,we study two fractional models in the Caputo–Fabrizio sense and Atangana–Baleanu sense,in which the effects of malaria infection on mosquito biting behavior and attractiveness of humans are considered.Using Lyapunov theory,we prove the global asymptotic stability of the unique endemic equilibrium of the integer-order model,and the fractional models,whenever the basic reproduction number R0 is greater than one.By using fixed point theory,we prove existence,and conditions of the uniqueness of solutions,as well as the stability and convergence of numerical schemes.Numerical simulations for both models,using fractional Euler method and Adams–Bashforth method,respectively,are provided to confirm the effectiveness of used approximation methods for different values of the fractional-orderγ.展开更多
Because of the features involved with their varied kernels,differential operators relying on convolution formulations have been acknowledged as effective mathematical resources for modeling real-world issues.In this p...Because of the features involved with their varied kernels,differential operators relying on convolution formulations have been acknowledged as effective mathematical resources for modeling real-world issues.In this paper,we constructed a stochastic fractional framework of measles spreading mechanisms with dual medication immunization considering the exponential decay and Mittag-Leffler kernels.In this approach,the overall population was separated into five cohorts.Furthermore,the descriptive behavior of the system was investigated,including prerequisites for the positivity of solutions,invariant domain of the solution,presence and stability of equilibrium points,and sensitivity analysis.We included a stochastic element in every cohort and employed linear growth and Lipschitz criteria to show the existence and uniqueness of solutions.Several numerical simulations for various fractional orders and randomization intensities are illustrated.展开更多
Fractional calculus has drawn more attentions of mathematicians and engineers in recent years.A lot of new fractional operators were used to handle various practical problems.In this article,we mainly study four new f...Fractional calculus has drawn more attentions of mathematicians and engineers in recent years.A lot of new fractional operators were used to handle various practical problems.In this article,we mainly study four new fractional operators,namely the CaputoFabrizio operator,the Atangana-Baleanu operator,the Sun-Hao-Zhang-Baleanu operator and the generalized Caputo type operator under the frame of the k-Prabhakar fractional integral operator.Usually,the theory of the k-Prabhakar fractional integral is regarded as a much broader than classical fractional operator.Here,we firstly give a series expansion of the k-Prabhakar fractional integral by means of the k-Riemann-Liouville integral.Then,a connection between the k-Prabhakar fractional integral and the four new fractional operators of the above mentioned was shown,respectively.In terms of the above analysis,we can obtain this a basic fact that it only needs to consider the k-Prabhakar fractional integral to cover these results from the four new fractional operators.展开更多
文摘在Banach空间中研究一类具有Caputo-Fabrizio分数阶导数且在非局部条件下的脉冲分数阶微分方程解的存在性。利用Schaefer不动点定理,压缩映射原理,Arzela-Ascoli定理,得到了该脉冲分数阶问题至少一个解和唯一解,并用一个例子验证其中一个结论。The existence of a class of impulsive fractional differential equations with Caputo-Fabrizio fractional derivatives under non-local conditions is studied in Banach space. Based on Schaefer’s fixed point theorem, compression mapping principle and Arzela-Ascoli theorem, at least one solution and the only solution of the pulse fractional order problem are obtained, and one of the conclusions is verified by an example.
文摘Asymptotic stability of linear and interval linear fractional-order neutral delay differential systems described by the Caputo-Fabrizio (CF) fractional derivatives is investigated. Using Laplace transform, a novel characteristic equation is derived. Stability criteria are established based on an algebraic approach and norm-based criteria are also presented. It is shown that asymptotic stability is ensured for linear fractional-order neutral delay differential systems provided that the underlying stability criterion holds for any delay parameter. In addition, sufficient conditions are derived to ensure the asymptotic stability of interval linear fractional order neutral delay differential systems. Examples are provided to illustrate the effectiveness and applicability of the theoretical results.
基金This research was supported by the National Natural Science Foundation of China(Grant numbers 11501140,51661135011,11421110001,and 91630204)the Foundation of Guizhou Science and Technology Department(No.[2017]1086)The first author would like to acknowledge the financial support by the China Scholarship Council(201708525037).
文摘In this paper, we consider numerical solutions of fractional ordinary diferential equations with the Caputo-Fabrizio derivative, and construct and analyze a high-order time-stepping scheme for this equation. The proposed method makes use of quadratic interpolation function in sub-intervals, which allows to produce fourth-order convergence. A rigorous stability and convergence analysis of the proposed scheme is given. A series of numerical examples are presented to validate the theoretical claims. Traditionally a scheme having fourth-order convergence could only be obtained by using block-by-block technique. The advantage of our scheme is that the solution can be obtained step by step, which is cheaper than a block-by-block-based approach.
文摘This paper studies the time-fractional Korteweg-de Vries (KdV) equations with Caputo-Fabrizio fractional derivatives. The scheme is presented by using a finite difference method in temporal variable and a local discontinuous Galerkin method (LDG) in space. Stability and convergence are demonstrated by a specific choice of numerical fluxes. Finally, the efficiency and accuracy of the scheme are verified by numerical experiments.
基金supported through the Annual Funding Track by the Deanship of Scientific Research,Vice Presidency for Graduate Studies and Scientific Research,King Faisal University,Saudi Arabia[Project No.AN000273],granted after a successful application by M.A.
文摘This study focuses on the dynamics of drug concentration in the blood.In general,the concentration level of a drug in the blood is evaluated by themean of an ordinary and first-order differential equation.More precisely,it is solved through an initial value problem.We proposed a newmodeling technique for studying drug concentration in blood dynamics.This technique is based on two fractional derivatives,namely,Caputo and Caputo-Fabrizio derivatives.We first provided comprehensive and detailed proof of the existence of at least one solution to the problem;we later proved the uniqueness of the existing solution.The proof was written using the Caputo-Fabrizio fractional derivative and some fixed-point techniques.Stability via theUlam-Hyers(UH)technique was also investigated.The application of the proposedmodel on two real data sets revealed that the Caputo derivative wasmore suitable in this study.Indeed,for the first data set,the model based on the Caputo derivative yielded a Mean Squared Error(MSE)of 0.03095 with a corresponding best value of fractional order of derivative of 1.00360.Caputo-Fabrizio-basedderivative appeared to be the second-best method for the problem,with an MSE of 0.04324 for a corresponding best fractional derivative order of 0.43532.For the second experiment,Caputo derivative-based model still performed the best as it yielded an MSE of 0.04066,whereas the classical and the Caputo-Fabrizio methods were tied with the same MSE of 0.07299.Another interesting finding was that the MSE yielded by the Caputo-Fabrizio fractional derivative coincided with the MSE obtained from the classical approach.
文摘Diabetes is a burning issue in the whole world.It is the imbalance between body glucose and insulin.The study of this imbalance is very much needed from a research point of view.For this reason,Bergman gave an important model named-Bergman minimalmodel.In the present work,using Caputo-Fabrizio(CF)fractional derivative,we generalize Bergman’s minimal blood glucose-insulin model.Further,we modify the old model by including one more component known as diet D(t),which is also essential for the blood glucose model.We solve the modified modelwith the help of Sumudu transformand fixed-point iteration procedures.Also,using the fixed point theorem,we examine the existence and uniqueness of the results along with their numerical and graphical representation.Furthermore,the comparison between the values of parameters obtained by calculating different values of t with experimental data is also studied.Finally,we draw the graphs of G(t),X(t),I(t),andD(t)for different values ofτ.It is also clear from the obtained results and their graphical representation that the obtained results of modified Bergman’s minimal model are better than Bergman’s model.
文摘The aim of this paper is to investigate the dynamic behaviors of fractional-order logistic model with Allee effects in Caputo-Fabrizio sense.First of all,we apply the two-step Adams-Bashforth scheme to discretize the fractional-order logistic differential equation and obtain the two-dimensional discrete system.The parametric conditions for local asymptotic stability of equilibrium points are obtained by Schur-Chon criterion.Moreover,we discuss the existence and direction for Neimark-Sacker bifurcations with the help of center manifold theorem and bifurcation theory.Numerical simulations are provided to illustrate theoretical discussion.It is observed that Allee effect plays an important role in stability analysis.Strong Allee effect in population enhances the stability of the coexisting steady state.In additional,the effect of fractional-order derivative on dynamic behavior of the system is also investigated.
文摘In this paper,we study two fractional models in the Caputo–Fabrizio sense and Atangana–Baleanu sense,in which the effects of malaria infection on mosquito biting behavior and attractiveness of humans are considered.Using Lyapunov theory,we prove the global asymptotic stability of the unique endemic equilibrium of the integer-order model,and the fractional models,whenever the basic reproduction number R0 is greater than one.By using fixed point theory,we prove existence,and conditions of the uniqueness of solutions,as well as the stability and convergence of numerical schemes.Numerical simulations for both models,using fractional Euler method and Adams–Bashforth method,respectively,are provided to confirm the effectiveness of used approximation methods for different values of the fractional-orderγ.
文摘Because of the features involved with their varied kernels,differential operators relying on convolution formulations have been acknowledged as effective mathematical resources for modeling real-world issues.In this paper,we constructed a stochastic fractional framework of measles spreading mechanisms with dual medication immunization considering the exponential decay and Mittag-Leffler kernels.In this approach,the overall population was separated into five cohorts.Furthermore,the descriptive behavior of the system was investigated,including prerequisites for the positivity of solutions,invariant domain of the solution,presence and stability of equilibrium points,and sensitivity analysis.We included a stochastic element in every cohort and employed linear growth and Lipschitz criteria to show the existence and uniqueness of solutions.Several numerical simulations for various fractional orders and randomization intensities are illustrated.
基金supported by the NSFC(11971475)the Natural Science Foundation of Jiangsu Province(BK20230708)+2 种基金the Natural Science Foundation for the Universities in Jiangsu Province(23KJB110003)Geng's research was supported by the NSFC(11201041)the China Postdoctoral Science Foundation(2019M651765)。
文摘Fractional calculus has drawn more attentions of mathematicians and engineers in recent years.A lot of new fractional operators were used to handle various practical problems.In this article,we mainly study four new fractional operators,namely the CaputoFabrizio operator,the Atangana-Baleanu operator,the Sun-Hao-Zhang-Baleanu operator and the generalized Caputo type operator under the frame of the k-Prabhakar fractional integral operator.Usually,the theory of the k-Prabhakar fractional integral is regarded as a much broader than classical fractional operator.Here,we firstly give a series expansion of the k-Prabhakar fractional integral by means of the k-Riemann-Liouville integral.Then,a connection between the k-Prabhakar fractional integral and the four new fractional operators of the above mentioned was shown,respectively.In terms of the above analysis,we can obtain this a basic fact that it only needs to consider the k-Prabhakar fractional integral to cover these results from the four new fractional operators.
