AN EXPLANATION ON FOUR NEW DEFINITIONS OF FRACTIONAL OPERATORS

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.

On Riemann-Type Weighted Fractional Operators and Solutions to Cauchy Problems

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.

Carlson iterating rational approximation and performance analysis of fractional operator with arbitrary order

The performance analysis of the generalized Carlson iterating process,which can realize the rational approximation of fractional operator with arbitrary order,is presented in this paper.The reasons why the generalized Carlson iterating function possesses more excellent properties such as self-similarity and exponential symmetry are also explained.K-index,P-index,O-index,and complexity index are introduced to contribute to performance analysis.Considering nine different operational orders and choosing an appropriate rational initial impedance for a certain operational order,these rational approximation impedance functions calculated by the iterating function meet computational rationality,positive reality,and operational validity.Then they are capable of having the operational performance of fractional operators and being physical realization.The approximation performance of the impedance function to the ideal fractional operator and the circuit network complexity are also exhibited.

Lipschitz Estimates for Commutators of N-dimensional Fractional Hardy Operators

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.

BOUNDEDNESS OF FRACTIONAL MAXIMAL OPERATOR AND THEIR HIGHER ORDER COMMUTATORS IN GENERALIZED MORREY SPACES ON CARNOT GROUPS

In the article we consider the fractional maximal operator Mα, 0 ≤α 〈 Q on any Carnot group G （i.e., nilpotent stratified Lie group） in the generalized Morrey spaces Mp,φ（G）, where Q is the homogeneous dimension of G. We find the conditions on the pair （φ1, φ2） which ensures the boundedness of the operator Ms from one generalized Morrey space Mp,φ1 （G） to another Mq,φ2 （G）, 1. 〈 p ≤q 〈 ∞. 1/p - 1/q = α/Q, and from the space M1,φ1 （G） to the weak space Wq,φ2 （G）, 1 〈 q 〈 ∞, 1 - 1/q = α/Q. Also find conditions on the φ which ensure the Adams type boundedness of the Ms from M α （G） from Mp,φ^1/p（G）to Mq,φ^1/q（G） for 1 〈p〈q〈∞ and fromM1,φ（G） toWMq,φ^1/q（G）for 1〈q〈∞. In the case b ∈ BMO（G） and 1 〈 p 〈 q 〈 ∞, find the sufficient conditions on the pair （φ1, φ2） which ensures the boundedness of the kth-order commutator operator Mb,α,k from Mp,φ1 （G） to Mq,φ2（G） with 1/p - 1/q = α/Q. Also find the sufficient conditions on the φ which ensures the boundedness of the operator Mb,α,k from Mp,φ^1/p（G） tom Mp,φ^1/p （G） for 1 〈p〈q〈∞. In all the cases the conditions for the boundedness of Mα are given it terms of supremaltype inequalities on （φ1, φ2） and φ , which do not assume any assumption on monotonicity of （φ1, φ2） and φ in r. As applications we consider the SchrSdinger operator -△G ＋ V on G, where the nonnegative potential V belongs to the reverse Holder class B∞（G）. The MB,φ1 - Mq,φ2 estimates for the operators V^γ（-△G ＋ V）^-β and V^γ△↓G（-△G ＋ V）^-β are obtained.

Boundedness of fractional integral operators on α-modulation spaces

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.

ENDPOINT ESTIMATES FOR FRACTIONAL INTEGRAL ASSOCIATED TO SCHRDINGER OPERATORS ON THE HEISENBERG GROUPS

Let ∠= -△Hn＋ V be the Schrdinger operator on the Heisenberg groups Hn,where V is a nonnegative function satisfying the reverse Hlder inequality. In this article, the author obtains the BMO_∠ and BLO_∠ estimates of the fractional integrals associated to ∠.

The Fractional Maximal Operator and Marcinkiewicz Integrals Associated with Schr?dinger Operators on Morrey Spaces with Variable Exponent

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.

GRADIENT ESTIMATES FOR THE COMMUTATOR WITH FRACTIONAL DIFFERENTIATION FOR SECOND ORDER ELLIPTIC OPERATORS

Let L=-div(A▽) be a second order divergence form elliptic operator, where A is an accretive, n×n matrix with bounded measurable complex coefficients on R^n. Let L^α/2 (0 <α< 1) denotes the fractional differential operator associated with L and (-△)^α/2b ∈ L^n/α(R^n). In this article, we prove that the commutator[b, L^α/2] is bounded from the homogenous Sobolev space Lα%2 (R^n) to L^2(R^n).

BOUNDEDNESS OF HIGHER ORDER COMMUTATORS OF GENERALIZED FRACTIONAL INTEGRAL OPERATORS ON HARDY SPACES

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.

ON SOME BOUNDARY VALUE PROBLEMS FOR NONHOMOGENOUS POLYHARMONIC EQUATION WITH BOUNDARY OPERATORS OF FRACTIONAL ORDER

In the paper we study questions about solvability of some boundary value prob- lems for a non-homogenous poly-harmonic equation. As a boundary operator we consider differentiation operator of fractional order in Miller-Ross sense. The considered problem is a generalization of well-known Dirichlet and Neumann problems.

WEIGHTED BOUNDEDNESS OF COMMUTATORS OF FRACTIONAL HARDY OPERATORS WITH BESOV-LIPSCHITZ FUNCTIONS

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.

Composition Operators with Linear Fractional Symbols on Vector-Valued Bergman Spaces

Let φ and ψ be linear fractional self\|maps of the unit disk D and X a separable Hilbert space. In this paper we completely characterize the weak compactness of the product operators of a composition operation C φ with another one's adjoint C * ψ on the vector\|valued Bergman space B 1(X) for forms C φC * ψ and C * ψC φ.

Noether symmetry method for Birkhoffian systems in terms of generalized fractional operators

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.

Using the Fractional Order Method to Generalize Strengthening Buffer Operator and Weakening Buffer Operator

To reveal the relationship between a weakening buffer operator and strengthening buffer operator, the traditional integer order buffer operator is extended to one that is fractional order. Fractional order buffer operator not only can generalize the weakening buffer operator and the strengthening buffer operator, but also results in small adjustments of the buffer effect.The effectiveness of the grey model(GM(1,1)) with the fractional order buffer operator is validated by six cases.

Multilinear Fractional Integral Operators on Morrey Spaces with Variable Exponent on Bounded Domain

We prove the boundedness of multilinear fractional integral operators onproducts of the variable exponent Morrey spaces on bounded domain.

Analysis and Dynamics of Fractional Order Mathematical Model of COVID-19in Nigeria Using Atangana-Baleanu Operator

We propose a mathematical model of the coronavirus disease 2019(COVID-19)to investigate the transmission and control mechanism of the disease in the community of Nigeria.Using stability theory of differential equations,the qualitative behavior of model is studied.The pandemic indicator represented by basic reproductive number R0 is obtained from the largest eigenvalue of the next-generation matrix.Local as well as global asymptotic stability conditions for the disease-free and pandemic equilibrium are obtained which determines the conditions to stabilize the exponential spread of the disease.Further,we examined this model by using Atangana–Baleanu fractional derivative operator and existence criteria of solution for the operator is established.We consider the data of reported infection cases from April 1,2020,till April 30,2020,and parameterized the model.We have used one of the reliable and efficient method known as iterative Laplace transform to obtain numerical simulations.The impacts of various biological parameters on transmission dynamics of COVID-19 is examined.These results are based on different values of the fractional parameter and serve as a control parameter to identify the significant strategies for the control of the disease.In the end,the obtained results are demonstrated graphically to justify our theoretical findings.

ON SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR A NONHOMOGENEOUS BIHARMONIC EQUATION WITH A BOUNDARY OPERATOR OF A FRACTIONAL ORDER

This paper is concerned with the solvability of a boundary value problem for a nonhomogeneous biharmonic equation. The boundary data is determined by a differential operator of fractional order in the Riemann-Liouville sense. The considered problem is a generalization of the known Dirichlet and Neumann problems.

Fractional Radon Transform and Transform of Wigner Operator

Based on the Radon transform and fractional Fourier transform we introduce the fractional Radon trans-formation (FRT). We identify the transform kernel for FRT. The FRT of Wigner operator is derived, which naturallyreduces to the projector of eigenvector of the rotated quadrature in the usual Radon transform case.

WEAK TYPE INEQUALITIES FOR FRACTIONAL INTEGRAL OPERATORS ON GENERALIZED NON-HOMOGENEOUS MORREY SPACES

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].