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
In this paper, we study a boundary value problem of nonlinear fractional dif- ferential equations of order q （1 〈 q 〈 2） with non-separated integral boundary conditions. Some new existence and uniqueness results a...In this paper, we study a boundary value problem of nonlinear fractional dif- ferential equations of order q （1 〈 q 〈 2） with non-separated integral boundary conditions. Some new existence and uniqueness results are obtained by using some standard fixed point theorems and Leray-Schauder degree theory. Some illustrative examples are also presented. We extend previous results even in the integer case q = 2.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 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 dif...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).展开更多
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
Let T be the singular integral operator with variable kernel, T＊ be the adjoint of T and T# be the pseudo-adjoint of T. Let TIT2 be the product of T1 and T2, T1 o T2 be the pseudo product of T1 and T2. In this paper,...Let T be the singular integral operator with variable kernel, T＊ be the adjoint of T and T# be the pseudo-adjoint of T. Let TIT2 be the product of T1 and T2, T1 o T2 be the pseudo product of T1 and T2. In this paper, we establish the boundedness for commutators of these operators and the fractional differentiation operator D^γ on the weighted Morrey spaces.展开更多
We discuss the solution of Laplace’s differential equation by using operational calculus in the framework of distribution theory. We here study the solution of that differential Equation with an inhomogeneous term, a...We discuss the solution of Laplace’s differential equation by using operational calculus in the framework of distribution theory. We here study the solution of that differential Equation with an inhomogeneous term, and also a fractional differential equation of the type of Laplace’s differential equation.展开更多
Fractional calculus has been used in many fields, such as engineering, population, medicine, fluid mechanics and different fields of chemistry and physics. These fields were found to be best described using fractional...Fractional calculus has been used in many fields, such as engineering, population, medicine, fluid mechanics and different fields of chemistry and physics. These fields were found to be best described using fractional differential equations (FDEs) to model their processes and equations. One of the well-known methods for solving fractional differential equations is the Shifted Legendre operational matrix (LOM) method. In this article, I proposed a numerical method based on Shifted Legendre polynomials for solving a class of fractional differential equations. A fractional order operational matrix of Legendre polynomials is also derived where the fractional derivatives are described by the Caputo derivative sense. By using the operational matrix, the initial and boundary equations are transformed into the products of several matrixes and by scattering the coefficients and the products of matrixes. I got a system of linear equations. Results obtained by using the proposed method (LOM) presented here show that the numerical method is very effective and appropriate for solving initial and boundary value problems of fractional ordinary differential equations. Moreover, some numerical examples are provided and the comparison is presented between the obtained results and those analytical results achieved that have proved the method’s validity.展开更多
Making use of the fractional differential operator, we impose and study a new class of analytic functions in the unit disk （type fractional differential equation）. The main object of this paper is to investigate inc...Making use of the fractional differential operator, we impose and study a new class of analytic functions in the unit disk （type fractional differential equation）. The main object of this paper is to investigate inclusion relations, coefficient bound for this class. Moreover, we discuss some geometric properties of the fractional differential operator.展开更多
This paper studies the existence and uniqueness of solutions for a class of boundary value problems of nonlinear fractional order differential equations involving the Caputo fractional derivative by employing the Ban...This paper studies the existence and uniqueness of solutions for a class of boundary value problems of nonlinear fractional order differential equations involving the Caputo fractional derivative by employing the Banach’s contraction principle and the Schauder’s fixed point theorem. In addition, an example is given to demonstrate the application of our main results.展开更多
In this paper,we propose numerical methods for the Riesz space fractional advection-dispersion equations with delay(RFADED).We utilize the fractional backward differential formulas method of second order(FBDF2)and wei...In this paper,we propose numerical methods for the Riesz space fractional advection-dispersion equations with delay(RFADED).We utilize the fractional backward differential formulas method of second order(FBDF2)and weighted shifted Grünwald difference(WSGD)operators to approximate the Riesz fractional derivative and present the finite difference method for the RFADED.Firstly,the FBDF2 and the shifted Grünwald methods are introduced.Secondly,based on the FBDF2 method and the WSGD operators,the finite difference method is applied to the problem.We also show that our numerical schemes are conditionally stable and convergent with the accuracy of O(+h2)and O(2+h2)respectively.Thirdly we find the analytical solution for RFDED in terms Mittag-Leffler type functions.Finally,some numerical examples are given to show the efficacy of the numerical methods and the results are found to be in complete agreement with the analytical solution.展开更多
In this paper, we prove the boundedness from H1 （Rn） to Ln/n-α∞, （Rn） for the multilinear fractional integral operator with rough kernel related to the mi-th remainder of Taylor series of Ai at x about y for i =...In this paper, we prove the boundedness from H1 （Rn） to Ln/n-α∞, （Rn） for the multilinear fractional integral operator with rough kernel related to the mi-th remainder of Taylor series of Ai at x about y for i = 1,2,..., l.展开更多
The object of the present paper is to investigate various argument results of analytic and multivalent functions which are defined by using a certain fractional derivative operator. Some interesting applications are a...The object of the present paper is to investigate various argument results of analytic and multivalent functions which are defined by using a certain fractional derivative operator. Some interesting applications are also considered.展开更多
In this paper, a class of mixed monotone operators is studied. Some new fixed point theorems are presented by means of partial order theory, and the uniqueness and existence of fixed points are obtained without assumi...In this paper, a class of mixed monotone operators is studied. Some new fixed point theorems are presented by means of partial order theory, and the uniqueness and existence of fixed points are obtained without assuming the operator to be compact or continuous. Our conclusions extend the relevant results. Moreover,as an application of our result, the existence and uniqueness of positive solution for a class of fractional differential equation boundary value problem are proved.展开更多
By using a method based upon the Briot-Bouquet differential subordination, we investigate some subordination properties of the generalized fractional integral operator which was defined by Owa, Saigo and Srivastava [1...By using a method based upon the Briot-Bouquet differential subordination, we investigate some subordination properties of the generalized fractional integral operator which was defined by Owa, Saigo and Srivastava [1]. Some interesting further consequences are also considered.展开更多
We obtain subordination, superordination and sandwich-preserving new theorems for certain integral operators defined on the space of normalized analytic functions in the open unit disk. The sandwich-type theorem for t...We obtain subordination, superordination and sandwich-preserving new theorems for certain integral operators defined on the space of normalized analytic functions in the open unit disk. The sandwich-type theorem for these integral operators is also derived, and the results generalize some recently ones.展开更多
In a preceding paper, we discussed the solution of Laplace’s differential equation by using operational calculus in the framework of distribution theory. We there studied the solution of that differential equation wi...In a preceding paper, we discussed the solution of Laplace’s differential equation by using operational calculus in the framework of distribution theory. We there studied the solution of that differential equation with an inhomogeneous term, and also a fractional differential equation of the type of Laplace’s differential equation. We there considered derivatives of a function on , when is locally integrable on , and the integral converges. We now discard the last condition that should converge, and discuss the same problem. In Appendices, polynomial form of particular solutions are given for the differential equations studied and Hermite’s differential equation with special inhomogeneous terms.展开更多
The existence,uniqueness,and continuous dependence to the mild solutions of the nonlocal Cauchy problem were proved for a class of semilinear fractional neutral differential equations.The results are obtained by using...The existence,uniqueness,and continuous dependence to the mild solutions of the nonlocal Cauchy problem were proved for a class of semilinear fractional neutral differential equations.The results are obtained by using the Krasnoselskii's fixed point theorem and the theory of resolvent operators for integral equations.展开更多
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.
文摘In this paper, we study a boundary value problem of nonlinear fractional dif- ferential equations of order q （1 〈 q 〈 2） with non-separated integral boundary conditions. Some new existence and uniqueness results are obtained by using some standard fixed point theorems and Leray-Schauder degree theory. Some illustrative examples are also presented. We extend previous results even in the integer case q = 2.
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 by NSFC(11471033),NCET of China(NCET-11-0574)the Fundamental Research Funds for the Central Universities(FRF-BR-16-011A)
文摘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).
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 by NSF of China (Grant No. 11471033)NCET of China (Grant No. NCET-11-0574)the Fundamental Research Funds for the Central Universities (FRF-TP-12-006B)
文摘Let T be the singular integral operator with variable kernel, T＊ be the adjoint of T and T# be the pseudo-adjoint of T. Let TIT2 be the product of T1 and T2, T1 o T2 be the pseudo product of T1 and T2. In this paper, we establish the boundedness for commutators of these operators and the fractional differentiation operator D^γ on the weighted Morrey spaces.
文摘We discuss the solution of Laplace’s differential equation by using operational calculus in the framework of distribution theory. We here study the solution of that differential Equation with an inhomogeneous term, and also a fractional differential equation of the type of Laplace’s differential equation.
文摘Fractional calculus has been used in many fields, such as engineering, population, medicine, fluid mechanics and different fields of chemistry and physics. These fields were found to be best described using fractional differential equations (FDEs) to model their processes and equations. One of the well-known methods for solving fractional differential equations is the Shifted Legendre operational matrix (LOM) method. In this article, I proposed a numerical method based on Shifted Legendre polynomials for solving a class of fractional differential equations. A fractional order operational matrix of Legendre polynomials is also derived where the fractional derivatives are described by the Caputo derivative sense. By using the operational matrix, the initial and boundary equations are transformed into the products of several matrixes and by scattering the coefficients and the products of matrixes. I got a system of linear equations. Results obtained by using the proposed method (LOM) presented here show that the numerical method is very effective and appropriate for solving initial and boundary value problems of fractional ordinary differential equations. Moreover, some numerical examples are provided and the comparison is presented between the obtained results and those analytical results achieved that have proved the method’s validity.
文摘Making use of the fractional differential operator, we impose and study a new class of analytic functions in the unit disk （type fractional differential equation）. The main object of this paper is to investigate inclusion relations, coefficient bound for this class. Moreover, we discuss some geometric properties of the fractional differential operator.
文摘This paper studies the existence and uniqueness of solutions for a class of boundary value problems of nonlinear fractional order differential equations involving the Caputo fractional derivative by employing the Banach’s contraction principle and the Schauder’s fixed point theorem. In addition, an example is given to demonstrate the application of our main results.
文摘In this paper,we propose numerical methods for the Riesz space fractional advection-dispersion equations with delay(RFADED).We utilize the fractional backward differential formulas method of second order(FBDF2)and weighted shifted Grünwald difference(WSGD)operators to approximate the Riesz fractional derivative and present the finite difference method for the RFADED.Firstly,the FBDF2 and the shifted Grünwald methods are introduced.Secondly,based on the FBDF2 method and the WSGD operators,the finite difference method is applied to the problem.We also show that our numerical schemes are conditionally stable and convergent with the accuracy of O(+h2)and O(2+h2)respectively.Thirdly we find the analytical solution for RFDED in terms Mittag-Leffler type functions.Finally,some numerical examples are given to show the efficacy of the numerical methods and the results are found to be in complete agreement with the analytical solution.
基金Supported in part by the NNSF of China under grant #10771110 and #10471069sponsored by the NSF of Ningbo city under grant #2009A610090
文摘In this paper, we prove the boundedness from H1 （Rn） to Ln/n-α∞, （Rn） for the multilinear fractional integral operator with rough kernel related to the mi-th remainder of Taylor series of Ai at x about y for i = 1,2,..., l.
文摘The object of the present paper is to investigate various argument results of analytic and multivalent functions which are defined by using a certain fractional derivative operator. Some interesting applications are also considered.
基金The NSF(201701D0503-9)of Shanxi Provincethe Innovation Foundation for College Teaching Team of Shanxi University of Finance and Economics2015 Education and Teaching Reform Project(2015234)of Shanxi University of Finance and Economics
文摘In this paper, a class of mixed monotone operators is studied. Some new fixed point theorems are presented by means of partial order theory, and the uniqueness and existence of fixed points are obtained without assuming the operator to be compact or continuous. Our conclusions extend the relevant results. Moreover,as an application of our result, the existence and uniqueness of positive solution for a class of fractional differential equation boundary value problem are proved.
文摘By using a method based upon the Briot-Bouquet differential subordination, we investigate some subordination properties of the generalized fractional integral operator which was defined by Owa, Saigo and Srivastava [1]. Some interesting further consequences are also considered.
文摘We obtain subordination, superordination and sandwich-preserving new theorems for certain integral operators defined on the space of normalized analytic functions in the open unit disk. The sandwich-type theorem for these integral operators is also derived, and the results generalize some recently ones.
文摘In a preceding paper, we discussed the solution of Laplace’s differential equation by using operational calculus in the framework of distribution theory. We there studied the solution of that differential equation with an inhomogeneous term, and also a fractional differential equation of the type of Laplace’s differential equation. We there considered derivatives of a function on , when is locally integrable on , and the integral converges. We now discard the last condition that should converge, and discuss the same problem. In Appendices, polynomial form of particular solutions are given for the differential equations studied and Hermite’s differential equation with special inhomogeneous terms.
文摘The existence,uniqueness,and continuous dependence to the mild solutions of the nonlocal Cauchy problem were proved for a class of semilinear fractional neutral differential equations.The results are obtained by using the Krasnoselskii's fixed point theorem and the theory of resolvent operators for integral equations.
