In this paper, we consider the fractional q-integral with variable lower limit of integration. We prove the semigroup property of these integrals, and a formula of Leibniz type. Finally, we evaluate fractional q-integ...In this paper, we consider the fractional q-integral with variable lower limit of integration. We prove the semigroup property of these integrals, and a formula of Leibniz type. Finally, we evaluate fractional q-integrals of some functions. The consideration of q-exponential function in that sense leads to q-analogs of Mittag-Leffier function.展开更多
The article considers the controllability of a diffusion equation with fractional integro-differential expressions.We prove that the resulting equation is nullcontrollable in arbitrary small time.Our method reduces es...The article considers the controllability of a diffusion equation with fractional integro-differential expressions.We prove that the resulting equation is nullcontrollable in arbitrary small time.Our method reduces essentially to the study of classical moment problems.展开更多
The dynamics of a fractionalized semi-linear scalar differential equation is considered with a Caputo fractional derivative. By using a symbolic operational method, a fractional order initial value problem is converte...The dynamics of a fractionalized semi-linear scalar differential equation is considered with a Caputo fractional derivative. By using a symbolic operational method, a fractional order initial value problem is converted into an equivalent Volterra integral equation of second kind. A brief discussion is included to show that the fractional order derivatives and integrals incorporate a fading memory (also known as long memory) and that the order of the fractional derivative can be considered to be an index of memory. A variation of constants formula is established for the fractionalized version and it is shown by using the Fourier integral theorem that this formula reduces to that of the integer order differential equation as the fractional order approaches an integer. The global existence of a unique solution and the global Mittag-Leffler stability of an equilibrium are established by exploiting the complete monotonicity of one and two parameter Mittag-Leffler functions. The method and the analysis employed in this article can be used for the study of more general systems of fractional order differential equations.展开更多
This paper extends the homotopy perturbation Sumudu transform method (HPSTM) to solve linear and nonlinear fractional Klein-Gordon equations. To illustrate the reliability of the method, some examples are presented. T...This paper extends the homotopy perturbation Sumudu transform method (HPSTM) to solve linear and nonlinear fractional Klein-Gordon equations. To illustrate the reliability of the method, some examples are presented. The convergence of the HPSTM solutions to the exact solutions is shown. As a novel application of homotopy perturbation sumudu transform method, the presented work showed some essential difference with existing similar application four classical examples also highlighted the significance of this work.展开更多
Practical stabilities for linear fractional impulsive hybrid systems are investigated in detail.The transformation from a linear fractional differential system to a fractional impulsive hybrid system is interpreted.Wi...Practical stabilities for linear fractional impulsive hybrid systems are investigated in detail.The transformation from a linear fractional differential system to a fractional impulsive hybrid system is interpreted.With the help of the Mittag-Leffler functions for matrix-type,several practical stability criteria for fractional impulsive hybrid systems are derived.Finally,a numerical example is provided to illustrate the effectiveness of the results.展开更多
In this paper, we establish a Lyapunov-type inequality for fractional differential periodic boundary-value problems. As applications, a necessary condition is obtained to ensure the existence and uniqueness of nontriv...In this paper, we establish a Lyapunov-type inequality for fractional differential periodic boundary-value problems. As applications, a necessary condition is obtained to ensure the existence and uniqueness of nontrivial solutions to this problem.展开更多
In this paper, we discuss the Laplace transform of the Caputo fractional dierence and the fractional discrete Mittag-Leer functions. On these bases, linear and nonlinear fractional initial value problems are solved by...In this paper, we discuss the Laplace transform of the Caputo fractional dierence and the fractional discrete Mittag-Leer functions. On these bases, linear and nonlinear fractional initial value problems are solved by the Laplace transform method.展开更多
In this paper,we introduce mon-symbolic method to obtain the generating functions of the hybrid class of Hermite-associated Laguerre and its associated polynomials.We obtain the series definitions of these hybrid spec...In this paper,we introduce mon-symbolic method to obtain the generating functions of the hybrid class of Hermite-associated Laguerre and its associated polynomials.We obtain the series definitions of these hybrid special polynomials.Also,we derive the double lacunary generating functions of the Hermite-Laguerre polynomials and the Hermite-Laguerre-Wright polynomials.Further,we find multiplicative and derivative operators for the Hermite-Laguerre-Wright polynomials which helps to find the symbolic differential equation of the Hermite-Laguerre-Wright polynomials.Some concluding remarks are also given.展开更多
Fractional calculus and special functions have contributed a lot to mathematical physics and its various branches. The great use of mathematical physics in distinguished astrophysical problems has attracted astronomer...Fractional calculus and special functions have contributed a lot to mathematical physics and its various branches. The great use of mathematical physics in distinguished astrophysical problems has attracted astronomers and physicists to pay more attention to available mathematical tools that can be widely used in solving several problems of astrophysics/physics. In view of the great importance and usefulness of kinetic equations in certain astrophysical problems, the authors derive a generalized fractional kinetic equation involving the Lorenzo-Hartley function, a generalized function for fractional calculus. The fractional kinetic equation discussed here can be used to investigate a wide class of known （and possibly also new） fractional kinetic equations, hitherto scattered in the literature. A compact and easily computable solution is established in terms of the Lorenzo-Hartley function. Special cases, involving the generalized Mittag-Leffler function and the R-function, are considered. The obtained results imply the known results more precisely.展开更多
基金Supported by Ministry of Science,Technology and Development of Republic Serbia (Grant Nos.144023 and 144013)
文摘In this paper, we consider the fractional q-integral with variable lower limit of integration. We prove the semigroup property of these integrals, and a formula of Leibniz type. Finally, we evaluate fractional q-integrals of some functions. The consideration of q-exponential function in that sense leads to q-analogs of Mittag-Leffier function.
基金Supported by National Natural Science Foundation of China(No.11261024).
文摘The article considers the controllability of a diffusion equation with fractional integro-differential expressions.We prove that the resulting equation is nullcontrollable in arbitrary small time.Our method reduces essentially to the study of classical moment problems.
文摘The dynamics of a fractionalized semi-linear scalar differential equation is considered with a Caputo fractional derivative. By using a symbolic operational method, a fractional order initial value problem is converted into an equivalent Volterra integral equation of second kind. A brief discussion is included to show that the fractional order derivatives and integrals incorporate a fading memory (also known as long memory) and that the order of the fractional derivative can be considered to be an index of memory. A variation of constants formula is established for the fractionalized version and it is shown by using the Fourier integral theorem that this formula reduces to that of the integer order differential equation as the fractional order approaches an integer. The global existence of a unique solution and the global Mittag-Leffler stability of an equilibrium are established by exploiting the complete monotonicity of one and two parameter Mittag-Leffler functions. The method and the analysis employed in this article can be used for the study of more general systems of fractional order differential equations.
文摘This paper extends the homotopy perturbation Sumudu transform method (HPSTM) to solve linear and nonlinear fractional Klein-Gordon equations. To illustrate the reliability of the method, some examples are presented. The convergence of the HPSTM solutions to the exact solutions is shown. As a novel application of homotopy perturbation sumudu transform method, the presented work showed some essential difference with existing similar application four classical examples also highlighted the significance of this work.
基金Natural Science Foundation of Shanghai China (No. 10ZR1400100)
文摘Practical stabilities for linear fractional impulsive hybrid systems are investigated in detail.The transformation from a linear fractional differential system to a fractional impulsive hybrid system is interpreted.With the help of the Mittag-Leffler functions for matrix-type,several practical stability criteria for fractional impulsive hybrid systems are derived.Finally,a numerical example is provided to illustrate the effectiveness of the results.
基金Doctoral Foundation of Education Ministry of China(20134219120003)the National Natural Science Foundation of China(61473338)
文摘In this paper, we establish a Lyapunov-type inequality for fractional differential periodic boundary-value problems. As applications, a necessary condition is obtained to ensure the existence and uniqueness of nontrivial solutions to this problem.
基金Supported by the NSFC(11371027)Supported by the Starting Research Fund for Doctors of Anhui University(023033190249)+1 种基金Supported by the NNSF of China,Tian Yuan Special Foundation(11326115)Supported by the Special Research Fund for the Doctoral Program of the Ministry of Education of China(20123401120001)
文摘In this paper, we discuss the Laplace transform of the Caputo fractional dierence and the fractional discrete Mittag-Leer functions. On these bases, linear and nonlinear fractional initial value problems are solved by the Laplace transform method.
文摘In this paper,we introduce mon-symbolic method to obtain the generating functions of the hybrid class of Hermite-associated Laguerre and its associated polynomials.We obtain the series definitions of these hybrid special polynomials.Also,we derive the double lacunary generating functions of the Hermite-Laguerre polynomials and the Hermite-Laguerre-Wright polynomials.Further,we find multiplicative and derivative operators for the Hermite-Laguerre-Wright polynomials which helps to find the symbolic differential equation of the Hermite-Laguerre-Wright polynomials.Some concluding remarks are also given.
文摘Fractional calculus and special functions have contributed a lot to mathematical physics and its various branches. The great use of mathematical physics in distinguished astrophysical problems has attracted astronomers and physicists to pay more attention to available mathematical tools that can be widely used in solving several problems of astrophysics/physics. In view of the great importance and usefulness of kinetic equations in certain astrophysical problems, the authors derive a generalized fractional kinetic equation involving the Lorenzo-Hartley function, a generalized function for fractional calculus. The fractional kinetic equation discussed here can be used to investigate a wide class of known （and possibly also new） fractional kinetic equations, hitherto scattered in the literature. A compact and easily computable solution is established in terms of the Lorenzo-Hartley function. Special cases, involving the generalized Mittag-Leffler function and the R-function, are considered. The obtained results imply the known results more precisely.
