In this paper,we establish the new forms of Riemann-type fractional integral and derivative operators.The novel fractional integral operator is proved to be bounded in Lebesgue space and some classical fractional inte...In this paper,we establish the new forms of Riemann-type fractional integral and derivative operators.The novel fractional integral operator is proved to be bounded in Lebesgue space and some classical fractional integral and differential operators are obtained as special cases.The properties of new operators like semi-group,inverse and certain others are discussed and its weighted Laplace transform is evaluated.Fractional integro-differential freeelectron laser(FEL)and kinetic equations are established.The solutions to these new equations are obtained by using the modified weighted Laplace transform.The Cauchy problem and a growth model are designed as applications along with graphical representation.Finally,the conclusion section indicates future directions to the readers.展开更多
Compared with the Hamiltonian mechanics and the Lagrangian mechanics,the Birkhoffian mechanics is more general.The Birkhoffian mechanics is discussed on the basis of the generalized fractional operators,which are prop...Compared with the Hamiltonian mechanics and the Lagrangian mechanics,the Birkhoffian mechanics is more general.The Birkhoffian mechanics is discussed on the basis of the generalized fractional operators,which are proposed recently.Therefore,differential equations of motion within generalized fractional operators are established.Then,in order to find the solutions to the differential equations,Noether symmetry,conserved quantity,perturbation to Noether symmetry and adiabatic invariant are investigated.In the end,two applications are given to illustrate the methods and results.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
In this paper,we provide an alternative proof of the weak type(1,n/n-a)inequality for the fractional maximal operators.By using the discretization technique,we can get the main result,which shows that the weak type(1,...In this paper,we provide an alternative proof of the weak type(1,n/n-a)inequality for the fractional maximal operators.By using the discretization technique,we can get the main result,which shows that the weak type(1,n/n-a)bound of M_(α)is at worst 2^(n-a).The weak type(1,n/n-a)bound of M_(α)can be estimated more directly and easily in this method,which is different from the usual ways.展开更多
Lassa Fever(LF)is a viral hemorrhagic illness transmitted via rodents and is endemic in West Africa,causing thousands of deaths annually.This study develops a dynamic model of Lassa virus transmission,capturing the pr...Lassa Fever(LF)is a viral hemorrhagic illness transmitted via rodents and is endemic in West Africa,causing thousands of deaths annually.This study develops a dynamic model of Lassa virus transmission,capturing the progression of the disease through susceptible,exposed,infected,and recovered populations.The focus is on simulating this model using the fractional Caputo derivative,allowing both qualitative and quantitative analyses of boundedness,positivity,and solution uniqueness.Fixed-point theory and Lipschitz conditions are employed to confirm the existence and uniqueness of solutions,while Lyapunov functions establish the global stability of both disease-free and endemic equilibria.The study further examines the role of the Caputo operator by solving the generalized power-law kernel via a two-step Lagrange polynomial method.This approach offers practical advantages in handling additional data points in integral forms,though Newton polynomial-based schemes are generally more accurate and can outperform Lagrange-based Adams-Bashforth methods.Graphical simulations validate the proposed numerical approach for different fractional orders(ν)and illustrate the influence of model parameters on disease dynamics.Results indicate that increasing the fractional order accelerates the decline of Lassa fever in both human and rodent populations.Moreover,fractional-order modeling provides more nuanced insights than traditional integer-order models,suggesting potential improvements for medical intervention strategies.The study demonstrates that carefully chosen fractional orders can optimize convergence and enhance the predictive capacity of Lassa fever models,offering a promising direction for future research in epidemiological modeling.展开更多
From point of view of physics,especially of mechanics,we briefly introduce fractional operators(with emphasis on fractional calculus and fractional differential equa-tions)used for describing intermediate processes an...From point of view of physics,especially of mechanics,we briefly introduce fractional operators(with emphasis on fractional calculus and fractional differential equa-tions)used for describing intermediate processes and critical phenomena in physics and mechanics,their progress in theory and methods and their applications to modern me-chanics.Some authors’researches in this area in recent years are included.Finally,prospects and evaluation for this subject are made.展开更多
Fractional differential equations have garnered significant attention within the mathematical and physical sciences due to the diverse range of fractional operators available.Fractional calculus has demonstrated its u...Fractional differential equations have garnered significant attention within the mathematical and physical sciences due to the diverse range of fractional operators available.Fractional calculus has demonstrated its utility across various disciplines,including biological modeling[1–5],applications in physics[6,7],most notably in the formulation of fractional diffusion equations,in robotics,and emerging areas such as intelligent artificial systems,among others.Numerous types of fractional operators exist,including those characterized by singular kernels,such as the Caputo and Riemann-Liouville derivatives[8,9].It is important to highlight that the Riemann-Liouville derivative exhibits certain limitations;most notably,the derivative of a constant is not zero,which poses a significant inconvenience.To circumvent this issue,the Caputo derivative was introduced.Additionally,there are fractional derivatives with non-singular kernels,such as the Caputo-Fabrizio derivative[10]and the Atangana-Baleanu fractional derivative[11],each providing unique advantages for modeling purposes.Given the growing interest in utilizing fractional operators for various modeling scenarios,it is imperative to propose robust methodologies for obtaining both approximate and exact solutions.Consequently,this special issue emphasizes the exploration of diverse numerical schemes aimed at deriving approximate solutions for the models under consideration.Furthermore,analytical methods have also been discussed,providing additional avenues for obtaining exact solutions.展开更多
In this paper, it is proved that the commutator Hβ,b which is generated by the n-dimensional fractional Hardy operator Hβ and b ∈λα （R^n） is bounded from L^p（R^n） to L^q（R^n）, where 0 〈 α 〈 1, 1 〈 p, q ...In this paper, it is proved that the commutator Hβ,b which is generated by the n-dimensional fractional Hardy operator Hβ and b ∈λα （R^n） is bounded from L^p（R^n） to L^q（R^n）, where 0 〈 α 〈 1, 1 〈 p, q 〈 ∞ and 1/P - 1/q = （α＋β）/n. Furthermore, the boundedness of Hβ,b on the homogenous Herz space Kq^α,p（R^n） is obtained.展开更多
In this paper, we obtain the boundedness of the fractional integral operators, the bilineax fractional integral operators and the bilinear Hilbert transform on α-modulation spaces.
In this paper, we prove the boundedness of the fractional maximal operator, Hardy-Littlewood maximal operator and marcinkiewicz integrals associated with Schrodinger operator on Morrey spaces with variable exponent.
Let Tμ,b,m be the higher order commutator generated by a generalized fractional integral operator Tμ and a BMO function b. In this paper, we will study the boundedness of Tμ,b,m on classical Hardy spaces and Herz-t...Let Tμ,b,m be the higher order commutator generated by a generalized fractional integral operator Tμ and a BMO function b. In this paper, we will study the boundedness of Tμ,b,m on classical Hardy spaces and Herz-type Hardy spaces.展开更多
In this paper, we establish two weighted integral inequalities for commutators of fractional Hardy operators with Besov-Lipschitz functions. The main result is that this kind of commutator, denoted by H^ab, is bounded...In this paper, we establish two weighted integral inequalities for commutators of fractional Hardy operators with Besov-Lipschitz functions. The main result is that this kind of commutator, denoted by H^ab, is bounded from L^Pxy （R＋） to L^qxδ （R＋） with the bound explicitly worked out.展开更多
We obtain weak type （1, q） inequalities for fractional integral operators on generalized non-homogeneous Morrey spaces. The proofs use some properties of maximal operators. Our results are closely related to the str...We obtain weak type （1, q） inequalities for fractional integral operators on generalized non-homogeneous Morrey spaces. The proofs use some properties of maximal operators. Our results are closely related to the strong type inequalities in [13, 14, 15].展开更多
In this paper,we obtain that b∈ BMO（Rn） if and only if the commutator[b,Iα]is bounded from the Morrey spaces Lp1,λ1Rn×Lp2,λ2Rnto Lq,λ（Rn）,for some appropriate indices p,q,λ,μ.Also we show that b ∈ Lip...In this paper,we obtain that b∈ BMO（Rn） if and only if the commutator[b,Iα]is bounded from the Morrey spaces Lp1,λ1Rn×Lp2,λ2Rnto Lq,λ（Rn）,for some appropriate indices p,q,λ,μ.Also we show that b ∈ Lip_β（R^n） if and only if the commutator[b,I_α]is bounded from the Morrey spaces Lp1,λ1）Rn×Lp2,λ（Rnto Lq,λRn,for some appropriate indices p,q,λ,μ.展开更多
Let M α be the fractional maximal operators (0<α≤1) and (u,v) a pair of weight functions, u∈D ∞, σ=v~~~~^(-1/(p-1))∈A ∞. The boundedness of M α on some homogenous groups (G, ‖·‖, dx) and the cov...Let M α be the fractional maximal operators (0<α≤1) and (u,v) a pair of weight functions, u∈D ∞, σ=v~~~~^(-1/(p-1))∈A ∞. The boundedness of M α on some homogenous groups (G, ‖·‖, dx) and the covering Lemma of Calderon-Zygmund type are studied. Not only an adequate covering Lemma of Calderon-Zygmund type is shown, but also the boundedness of fractional maximal operators M α(0<α≤1) on some of homogeneous groups with respect to a given pair of weight functions (u,v) as above is proved. Moreover, a sufficient and necessary condition for M α∈B(u^qdx, v~~pdx), 0<α<1, 1<p<1α, and 1q=1p-α is also given. Finally, an application of the results is also obtained.展开更多
In this paper, we prove the M^(k)-type sharp maximal function estimates for the Toeplitz type operators associated to the fractional integral and singular integral operator with non-smooth kernel. As an application, w...In this paper, we prove the M^(k)-type sharp maximal function estimates for the Toeplitz type operators associated to the fractional integral and singular integral operator with non-smooth kernel. As an application, we obtain the boundedness of the operators on the Morrey space.展开更多
In the present paper, we discuss the solution of Euler-Darboux equation in terms of Dirichlet averages of boundary conditions on H?lder space and weighted H?lder spaces of continuous functions using Riemann-Liouville ...In the present paper, we discuss the solution of Euler-Darboux equation in terms of Dirichlet averages of boundary conditions on H?lder space and weighted H?lder spaces of continuous functions using Riemann-Liouville fractional integral operators. Moreover, the results are interpreted in alternative form.展开更多
Fractional discrete systems can enable the modeling and control of the complicated processes more adaptable through the concept of versatility by providing systemdynamics’descriptions withmore degrees of freedom.Nume...Fractional discrete systems can enable the modeling and control of the complicated processes more adaptable through the concept of versatility by providing systemdynamics’descriptions withmore degrees of freedom.Numerical approaches have become necessary and sufficient to be addressed and employed for benefiting from the adaptability of such systems for varied applications.A variety of fractional Layla and Majnun model(LMM)system kinds has been proposed in the current work where some of these systems’key behaviors are addressed.In addition,the necessary and sufficient conditions for the stability and asymptotic stability of the fractional dynamic systems are investigated,as a result of which,the necessary requirements of the LMM to achieve constant and asymptotically steady zero resolutions are provided.As a special case,when Layla and Majnun have equal feelings,we propose an analysis of the system in view of its equilibrium and fixed point sets.Considering that the system has marginal stability if its eigenvalues have both negative and zero real portions,it is demonstrated that the system neither converges nor diverges to a steady trajectory or equilibrium point.It,rather,continues to hover along the line separating stability and instability based on the fractional LMM system.展开更多
文摘In this paper,we establish the new forms of Riemann-type fractional integral and derivative operators.The novel fractional integral operator is proved to be bounded in Lebesgue space and some classical fractional integral and differential operators are obtained as special cases.The properties of new operators like semi-group,inverse and certain others are discussed and its weighted Laplace transform is evaluated.Fractional integro-differential freeelectron laser(FEL)and kinetic equations are established.The solutions to these new equations are obtained by using the modified weighted Laplace transform.The Cauchy problem and a growth model are designed as applications along with graphical representation.Finally,the conclusion section indicates future directions to the readers.
基金This work was supported by the National Natural Science Foun-dation of China(Grants 11802193 and 11972241)the Natural Sci-ence Foundation of Jiangsu Province(Grant BK20191454)the Young Scientific and Technological Talents Promotion Project of Suzhou Association for Science and Technology.
文摘Compared with the Hamiltonian mechanics and the Lagrangian mechanics,the Birkhoffian mechanics is more general.The Birkhoffian mechanics is discussed on the basis of the generalized fractional operators,which are proposed recently.Therefore,differential equations of motion within generalized fractional operators are established.Then,in order to find the solutions to the differential equations,Noether symmetry,conserved quantity,perturbation to Noether symmetry and adiabatic invariant are investigated.In the end,two applications are given to illustrate the methods and results.
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.
基金Supported by by Natural Science Foundation of Henan(202300410184 and242300421387)。
文摘In this paper,we provide an alternative proof of the weak type(1,n/n-a)inequality for the fractional maximal operators.By using the discretization technique,we can get the main result,which shows that the weak type(1,n/n-a)bound of M_(α)is at worst 2^(n-a).The weak type(1,n/n-a)bound of M_(α)can be estimated more directly and easily in this method,which is different from the usual ways.
文摘Lassa Fever(LF)is a viral hemorrhagic illness transmitted via rodents and is endemic in West Africa,causing thousands of deaths annually.This study develops a dynamic model of Lassa virus transmission,capturing the progression of the disease through susceptible,exposed,infected,and recovered populations.The focus is on simulating this model using the fractional Caputo derivative,allowing both qualitative and quantitative analyses of boundedness,positivity,and solution uniqueness.Fixed-point theory and Lipschitz conditions are employed to confirm the existence and uniqueness of solutions,while Lyapunov functions establish the global stability of both disease-free and endemic equilibria.The study further examines the role of the Caputo operator by solving the generalized power-law kernel via a two-step Lagrange polynomial method.This approach offers practical advantages in handling additional data points in integral forms,though Newton polynomial-based schemes are generally more accurate and can outperform Lagrange-based Adams-Bashforth methods.Graphical simulations validate the proposed numerical approach for different fractional orders(ν)and illustrate the influence of model parameters on disease dynamics.Results indicate that increasing the fractional order accelerates the decline of Lassa fever in both human and rodent populations.Moreover,fractional-order modeling provides more nuanced insights than traditional integer-order models,suggesting potential improvements for medical intervention strategies.The study demonstrates that carefully chosen fractional orders can optimize convergence and enhance the predictive capacity of Lassa fever models,offering a promising direction for future research in epidemiological modeling.
基金the National Natural Science Foundation of China(Grant Nos.10272067,10372007,10572006)the Doctoral Program Foundation of Ministry of Education of China(Grant No.20030422046)New Century Training Programme Foundation for the Talents by Ministry of Education of China.
文摘From point of view of physics,especially of mechanics,we briefly introduce fractional operators(with emphasis on fractional calculus and fractional differential equa-tions)used for describing intermediate processes and critical phenomena in physics and mechanics,their progress in theory and methods and their applications to modern me-chanics.Some authors’researches in this area in recent years are included.Finally,prospects and evaluation for this subject are made.
基金Supported by National Natural Science Foundation of China (Grant Nos. 10871024 and 10931001)the Key Laboratory of Mathematics and Complex System (at Beijing Normal University), Ministry of Education, China
文摘The author establishes weighted strong type estimates for iterated commutators of multi- linear fractional operators.
文摘Fractional differential equations have garnered significant attention within the mathematical and physical sciences due to the diverse range of fractional operators available.Fractional calculus has demonstrated its utility across various disciplines,including biological modeling[1–5],applications in physics[6,7],most notably in the formulation of fractional diffusion equations,in robotics,and emerging areas such as intelligent artificial systems,among others.Numerous types of fractional operators exist,including those characterized by singular kernels,such as the Caputo and Riemann-Liouville derivatives[8,9].It is important to highlight that the Riemann-Liouville derivative exhibits certain limitations;most notably,the derivative of a constant is not zero,which poses a significant inconvenience.To circumvent this issue,the Caputo derivative was introduced.Additionally,there are fractional derivatives with non-singular kernels,such as the Caputo-Fabrizio derivative[10]and the Atangana-Baleanu fractional derivative[11],each providing unique advantages for modeling purposes.Given the growing interest in utilizing fractional operators for various modeling scenarios,it is imperative to propose robust methodologies for obtaining both approximate and exact solutions.Consequently,this special issue emphasizes the exploration of diverse numerical schemes aimed at deriving approximate solutions for the models under consideration.Furthermore,analytical methods have also been discussed,providing additional avenues for obtaining exact solutions.
基金The NSF (Q2008A01) of Shandong,Chinathe NSF (10871024) of China
文摘In this paper, it is proved that the commutator Hβ,b which is generated by the n-dimensional fractional Hardy operator Hβ and b ∈λα （R^n） is bounded from L^p（R^n） to L^q（R^n）, where 0 〈 α 〈 1, 1 〈 p, q 〈 ∞ and 1/P - 1/q = （α＋β）/n. Furthermore, the boundedness of Hβ,b on the homogenous Herz space Kq^α,p（R^n） is obtained.
基金Supported by the National Natural Science Foundation of China(11271330)
文摘In this paper, we obtain the boundedness of the fractional integral operators, the bilineax fractional integral operators and the bilinear Hilbert transform on α-modulation spaces.
基金supported by NSFC (No. 11201003)University NSR Project of Anhui Province (No. KJ2014A087)
文摘In this paper, we prove the boundedness of the fractional maximal operator, Hardy-Littlewood maximal operator and marcinkiewicz integrals associated with Schrodinger operator on Morrey spaces with variable exponent.
基金Supported Partially by NSF of China (10371087) Education Committee of Anhui Province (2003kj034zd).
文摘Let Tμ,b,m be the higher order commutator generated by a generalized fractional integral operator Tμ and a BMO function b. In this paper, we will study the boundedness of Tμ,b,m on classical Hardy spaces and Herz-type Hardy spaces.
基金Supported in part by the Natural Science Foundation of China under the Grant 10771221Natural Science Foundation of Beijing under the Grant 1092004
文摘In this paper, we establish two weighted integral inequalities for commutators of fractional Hardy operators with Besov-Lipschitz functions. The main result is that this kind of commutator, denoted by H^ab, is bounded from L^Pxy （R＋） to L^qxδ （R＋） with the bound explicitly worked out.
基金Supported by Fundamental Research Program 2011-2012
文摘We obtain weak type （1, q） inequalities for fractional integral operators on generalized non-homogeneous Morrey spaces. The proofs use some properties of maximal operators. Our results are closely related to the strong type inequalities in [13, 14, 15].
基金Supported by the National Natural Science Foundation of China(Grant Nos.1126105511661075)
文摘In this paper,we obtain that b∈ BMO（Rn） if and only if the commutator[b,Iα]is bounded from the Morrey spaces Lp1,λ1Rn×Lp2,λ2Rnto Lq,λ（Rn）,for some appropriate indices p,q,λ,μ.Also we show that b ∈ Lip_β（R^n） if and only if the commutator[b,I_α]is bounded from the Morrey spaces Lp1,λ1）Rn×Lp2,λ（Rnto Lq,λRn,for some appropriate indices p,q,λ,μ.
文摘Let M α be the fractional maximal operators (0<α≤1) and (u,v) a pair of weight functions, u∈D ∞, σ=v~~~~^(-1/(p-1))∈A ∞. The boundedness of M α on some homogenous groups (G, ‖·‖, dx) and the covering Lemma of Calderon-Zygmund type are studied. Not only an adequate covering Lemma of Calderon-Zygmund type is shown, but also the boundedness of fractional maximal operators M α(0<α≤1) on some of homogeneous groups with respect to a given pair of weight functions (u,v) as above is proved. Moreover, a sufficient and necessary condition for M α∈B(u^qdx, v~~pdx), 0<α<1, 1<p<1α, and 1q=1p-α is also given. Finally, an application of the results is also obtained.
基金Supported by the National Natural Science Foundation of China (Grant No. 11901126)the Scientific Research Funds of Hunan Provincial Education Department (Grant No. 19B509)。
文摘In this paper, we prove the M^(k)-type sharp maximal function estimates for the Toeplitz type operators associated to the fractional integral and singular integral operator with non-smooth kernel. As an application, we obtain the boundedness of the operators on the Morrey space.
文摘In the present paper, we discuss the solution of Euler-Darboux equation in terms of Dirichlet averages of boundary conditions on H?lder space and weighted H?lder spaces of continuous functions using Riemann-Liouville fractional integral operators. Moreover, the results are interpreted in alternative form.
基金supported by Ajman University Internal Research Grant No.(DRGS Ref.2024-IRGHBS-3).
文摘Fractional discrete systems can enable the modeling and control of the complicated processes more adaptable through the concept of versatility by providing systemdynamics’descriptions withmore degrees of freedom.Numerical approaches have become necessary and sufficient to be addressed and employed for benefiting from the adaptability of such systems for varied applications.A variety of fractional Layla and Majnun model(LMM)system kinds has been proposed in the current work where some of these systems’key behaviors are addressed.In addition,the necessary and sufficient conditions for the stability and asymptotic stability of the fractional dynamic systems are investigated,as a result of which,the necessary requirements of the LMM to achieve constant and asymptotically steady zero resolutions are provided.As a special case,when Layla and Majnun have equal feelings,we propose an analysis of the system in view of its equilibrium and fixed point sets.Considering that the system has marginal stability if its eigenvalues have both negative and zero real portions,it is demonstrated that the system neither converges nor diverges to a steady trajectory or equilibrium point.It,rather,continues to hover along the line separating stability and instability based on the fractional LMM system.
