In this article, series of novel bi-SOaH-functionalized ILs were synthesized using simple, efficient and economic procedure. Hammer method had been used to determine the acidity order of these ionic liquids, and the a...In this article, series of novel bi-SOaH-functionalized ILs were synthesized using simple, efficient and economic procedure. Hammer method had been used to determine the acidity order of these ionic liquids, and the acidifies of bi-SOaH-functionalized ILs were stronger than that of traditional single-SOaH-functionalized ILs. Their catalytic activities in the synthesis of N-（3-phenyl）-3- oxo-1-（phenylpropyl）acetamide were investigated and they were consistent with their acidities.展开更多
In this article,an efficient,simple and environmentally friendly approach to the synthesis of diacetals(diketals) pentaerythritol using SOH-functionalized ionic liquids(ILs) as catalysts was reported.The ILs show high...In this article,an efficient,simple and environmentally friendly approach to the synthesis of diacetals(diketals) pentaerythritol using SOH-functionalized ionic liquids(ILs) as catalysts was reported.The ILs show high catalytic activity and reusability with good to excellent yields of the desired products.Hammett method has been used to determine the acidity order of these ionic liquids and the results are consistent with the catalytic activities observed in acetalization reaction.Maximum product yield of 93%was observed on using[PSPy][OTf]as catalyst and it can be reused at least 8 times without obvious activity loss.展开更多
The object of this article is to study and develop the generalized fractional calcu- lus operators given by Saigo and Maeda in 1996. We establish generalized fractional calculus formulas involving the product of R-fun...The object of this article is to study and develop the generalized fractional calcu- lus operators given by Saigo and Maeda in 1996. We establish generalized fractional calculus formulas involving the product of R-function, Appell function F3 and a general class of poly- nomials. The results obtained provide unification and extension of the results given by Saxena et al. [13], Srivastava and Grag [17], Srivastava et al. [20], and etc. The results are obtained in compact form and are useful in preparing some tables of operators of fractional calculus. On account of the general nature of the Saigo-Maeda operators, R-function, and a general class of polynomials a large number of new and known results involving Saigo fractional calculus operators and several special functions notably H-function, /-function, Mittag-Leffier function, generalized Wright hypergeometric function, generalized Bessel-Maitland function follow as special cases of our main findings.展开更多
The flow near a wall suddenly set in motion for a viscoelastic fluid with the generalized Oldroyd-B model is studied. The fractional calculus approach is used in the constitutive relationship of fluid model. Exact ana...The flow near a wall suddenly set in motion for a viscoelastic fluid with the generalized Oldroyd-B model is studied. The fractional calculus approach is used in the constitutive relationship of fluid model. Exact analytical solutions of velocity and stress are obtained by using the discrete Laplace transform of the sequential fractional derivative and the Fox H-function. The obtained results indicate that some well known solutions for the Newtonian fluid, the generalized second grade fluid as well as the ordinary Oldroyd-B fluid, as limiting cases, are included in our solutions.展开更多
The generalized fractional elastic models govern the stochastic motion of several many-body systems,e.g., polymers, membranes, and growing interfaces. This paper focuses on the exact formulations and their asymptotic ...The generalized fractional elastic models govern the stochastic motion of several many-body systems,e.g., polymers, membranes, and growing interfaces. This paper focuses on the exact formulations and their asymptotic behaviors of the average of the solutions of the generalized fractional elastic models. So we directly analyze the Cauchy problem of the averaged generalized elastic model involving time fractional derivative and the convolution integral of a radially symmetric friction kernel with space fractional Laplacian.展开更多
To better describe the phenomenon of non-Fourier heat conduction, the fractional Cattaneo heat equation is introduced from the generalized Cattaneo model with two fractional derivatives of different orders. The anomal...To better describe the phenomenon of non-Fourier heat conduction, the fractional Cattaneo heat equation is introduced from the generalized Cattaneo model with two fractional derivatives of different orders. The anomalous heat conduction under the Neumann boundary condition in a semi-infinity medium is investigated. Exact solutions are obtained in series form of the H-function by using the Laplace transform method. Finally, numerical examples are presented graphically when different kinds of surface temperature gradient are given. The effects of fractional parameters are also discussed.展开更多
In this paper, we presents some new exact solutions corresponding to three unsteady flow problems of a generalized Jeffrey fluid produced by a flat plate between two side walls perpendicular to the plate. The fraction...In this paper, we presents some new exact solutions corresponding to three unsteady flow problems of a generalized Jeffrey fluid produced by a flat plate between two side walls perpendicular to the plate. The fractional calculus approach is used in the governing equations. The exact solutions are established by means of the Fourier sine transform and N-transform. The series solutions of velocity field and associated shear stress in terms of Fox H-function, satisfying all imposed initial and boundary conditions, have been obtained. The similar solutions for ordinary Jeffrey fluid, performing the same motion, appear as limiting case of the solutions are also obtained. Also, the obtained results are analyzed graphically through various pertinent parameters.展开更多
Let s and z be complex variables, Γ(s) be the Gamma function, and for any complex v be the generalized Pochhammer symbol. Wright Type Hypergeometric Function is defined (Virchenko et al. [1]), as: where which is a di...Let s and z be complex variables, Γ(s) be the Gamma function, and for any complex v be the generalized Pochhammer symbol. Wright Type Hypergeometric Function is defined (Virchenko et al. [1]), as: where which is a direct generalization of classical Gauss Hypergeometric Function 2F1(a,b;c;z). The principal aim of this paper is to study the various properties of this Wright type hypergeometric function 2R1(a,b;c;τ;z);which includes differentiation and integration, representation in terms of pFq and in terms of Mellin-Barnes type integral. Euler (Beta) transforms, Laplace transform, Mellin transform, Whittaker transform have also been obtained;along with its relationship with Fox H-function and Wright hypergeometric function.展开更多
In the present paper, the authors introduce a new integral transform which yields a number of potentially useful （known or new） integral transfoms as its special cases. Many fundamental results about this new integr...In the present paper, the authors introduce a new integral transform which yields a number of potentially useful （known or new） integral transfoms as its special cases. Many fundamental results about this new integral transform, which are established in this paper, in- clude （for example） existence theorem, Parseval-type relationship and inversion formula. The relationship between the new integral transform with the H-function and the H-transform are characterized by means of some integral identities. The introduced transform is also used to find solution to a certain differential equation. Some illustrative examples are also given.展开更多
The aimof this article is to investigate the solutions of generalized fractional partial differential equations involving Hilfer time fractional derivative and the space fractional generalized Laplace operators,occurr...The aimof this article is to investigate the solutions of generalized fractional partial differential equations involving Hilfer time fractional derivative and the space fractional generalized Laplace operators,occurring in quantum mechanics.The solutions of these equations are obtained by employing the joint Laplace and Fourier transforms,in terms of the Fox’s H-function.Several special cases as solutions of one dimensional non-homogeneous fractional equations occurring in the quantum mechanics are presented.The results given earlier by Saxena et al.[Fract.Calc.Appl.Anal.,13(2)(2010),pp.177-190]and Purohit and Kalla[J.Phys.AMath.Theor.,44(4)(2011),045202]follow as special cases of our findings.展开更多
Let G be a finite group andНbe a Hartley set of G.In this paper,we prove the existence and conjugacy ofН-injectors of G and describe the characterization of injectors via radicals.As applications,some known results ...Let G be a finite group andНbe a Hartley set of G.In this paper,we prove the existence and conjugacy ofН-injectors of G and describe the characterization of injectors via radicals.As applications,some known results are directly followed.展开更多
In this paper, we study the semi-boundless mixed problem for time-fractional telegraph equation. We are able to use the integral transform method （the Fourier sin and cos transforms） to obtain the solution.
The Krätzel function has many applications in applied analysis,so this function is used as a base to create a density function which will be called the Krätzel density.This density is applicable in chemical ...The Krätzel function has many applications in applied analysis,so this function is used as a base to create a density function which will be called the Krätzel density.This density is applicable in chemical physics,Hartree–Fock energy,helium isoelectric series,statistical mechanics,nuclear energy generation,etc.,and also connected to Bessel functions.The main properties of this newfamily are studied,showing in particular that it may be generated via mixtures of gamma random variables.Some basic statistical quantities associated with this density function such as moments,Mellin transform,and Laplace transform are obtained.Connection of Krätzel distribution to reaction rate probability integral in physics,inverse Gaussian density in stochastic processes,Tsallis statistics and superstatistics in non-extensive statistical mechanics,Mellin convolutions of products and ratios thereby to fractional integrals,synthetic aperture radar,and other areas are pointed out in this article.Finally,we extend the Krätzel density using the pathway model of Mathai,and some applications are also discussed.The new probability model is fitted to solar radiation data.展开更多
基金the National Natural Science Foundation ofChina(Nos.21003049,21073064)the Fundamental Research Funds for the Central Universities for financial support
文摘In this article, series of novel bi-SOaH-functionalized ILs were synthesized using simple, efficient and economic procedure. Hammer method had been used to determine the acidity order of these ionic liquids, and the acidifies of bi-SOaH-functionalized ILs were stronger than that of traditional single-SOaH-functionalized ILs. Their catalytic activities in the synthesis of N-（3-phenyl）-3- oxo-1-（phenylpropyl）acetamide were investigated and they were consistent with their acidities.
基金supported by National 863 High-Tech Research and Development Program of China(No. 2007AA05Z101)
文摘In this article,an efficient,simple and environmentally friendly approach to the synthesis of diacetals(diketals) pentaerythritol using SOH-functionalized ionic liquids(ILs) as catalysts was reported.The ILs show high catalytic activity and reusability with good to excellent yields of the desired products.Hammett method has been used to determine the acidity order of these ionic liquids and the results are consistent with the catalytic activities observed in acetalization reaction.Maximum product yield of 93%was observed on using[PSPy][OTf]as catalyst and it can be reused at least 8 times without obvious activity loss.
基金NBHM Department of Atomic Energy,Government of India,Mumbai for the finanicai assistance under PDF sanction no.2/40(37)/2014/R&D-II/14131
文摘The object of this article is to study and develop the generalized fractional calcu- lus operators given by Saigo and Maeda in 1996. We establish generalized fractional calculus formulas involving the product of R-function, Appell function F3 and a general class of poly- nomials. The results obtained provide unification and extension of the results given by Saxena et al. [13], Srivastava and Grag [17], Srivastava et al. [20], and etc. The results are obtained in compact form and are useful in preparing some tables of operators of fractional calculus. On account of the general nature of the Saigo-Maeda operators, R-function, and a general class of polynomials a large number of new and known results involving Saigo fractional calculus operators and several special functions notably H-function, /-function, Mittag-Leffier function, generalized Wright hypergeometric function, generalized Bessel-Maitland function follow as special cases of our main findings.
基金The project supported by the National Natural Science Foundation of China(10272067)the Doctoral Program Foundation of the Education Ministry of China(20030422046)+1 种基金the Natural Science Foundation of Shandong Province,China(Y2006A 14)the Research Foundation of Shandong University at Weihai.
文摘The flow near a wall suddenly set in motion for a viscoelastic fluid with the generalized Oldroyd-B model is studied. The fractional calculus approach is used in the constitutive relationship of fluid model. Exact analytical solutions of velocity and stress are obtained by using the discrete Laplace transform of the sequential fractional derivative and the Fox H-function. The obtained results indicate that some well known solutions for the Newtonian fluid, the generalized second grade fluid as well as the ordinary Oldroyd-B fluid, as limiting cases, are included in our solutions.
基金Supported by the Program for New Century Excellent Talents in University under Grant No.NCET-09-0438the National Natural Science Foundation of China under Grant Nos.11271173 and 11101330+1 种基金the Starting Research Fund from the Xi’an University of Technology under Grant No.108-211206the Scientific Research Program Funded by Shaanxi Provincial Education Department under Grant No.2013JK0581
文摘The generalized fractional elastic models govern the stochastic motion of several many-body systems,e.g., polymers, membranes, and growing interfaces. This paper focuses on the exact formulations and their asymptotic behaviors of the average of the solutions of the generalized fractional elastic models. So we directly analyze the Cauchy problem of the averaged generalized elastic model involving time fractional derivative and the convolution integral of a radially symmetric friction kernel with space fractional Laplacian.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 11102102, 11072134, and 91130017)the Natural Science Foundation of Shandong Province, China (Grant No. ZR2009AQ014)the Independent Innovation Foundation of Shandong University, China (Grant No. 2010ZRJQ002)
文摘To better describe the phenomenon of non-Fourier heat conduction, the fractional Cattaneo heat equation is introduced from the generalized Cattaneo model with two fractional derivatives of different orders. The anomalous heat conduction under the Neumann boundary condition in a semi-infinity medium is investigated. Exact solutions are obtained in series form of the H-function by using the Laplace transform method. Finally, numerical examples are presented graphically when different kinds of surface temperature gradient are given. The effects of fractional parameters are also discussed.
文摘In this paper, we presents some new exact solutions corresponding to three unsteady flow problems of a generalized Jeffrey fluid produced by a flat plate between two side walls perpendicular to the plate. The fractional calculus approach is used in the governing equations. The exact solutions are established by means of the Fourier sine transform and N-transform. The series solutions of velocity field and associated shear stress in terms of Fox H-function, satisfying all imposed initial and boundary conditions, have been obtained. The similar solutions for ordinary Jeffrey fluid, performing the same motion, appear as limiting case of the solutions are also obtained. Also, the obtained results are analyzed graphically through various pertinent parameters.
文摘Let s and z be complex variables, Γ(s) be the Gamma function, and for any complex v be the generalized Pochhammer symbol. Wright Type Hypergeometric Function is defined (Virchenko et al. [1]), as: where which is a direct generalization of classical Gauss Hypergeometric Function 2F1(a,b;c;z). The principal aim of this paper is to study the various properties of this Wright type hypergeometric function 2R1(a,b;c;τ;z);which includes differentiation and integration, representation in terms of pFq and in terms of Mellin-Barnes type integral. Euler (Beta) transforms, Laplace transform, Mellin transform, Whittaker transform have also been obtained;along with its relationship with Fox H-function and Wright hypergeometric function.
文摘In the present paper, the authors introduce a new integral transform which yields a number of potentially useful （known or new） integral transfoms as its special cases. Many fundamental results about this new integral transform, which are established in this paper, in- clude （for example） existence theorem, Parseval-type relationship and inversion formula. The relationship between the new integral transform with the H-function and the H-transform are characterized by means of some integral identities. The introduced transform is also used to find solution to a certain differential equation. Some illustrative examples are also given.
基金The author thanks the referees for his/her suggestions,which improved the presentation of this paper.Also,the author thanks Professor S.L.Kalla for his valuable suggestions and criticisms.
文摘The aimof this article is to investigate the solutions of generalized fractional partial differential equations involving Hilfer time fractional derivative and the space fractional generalized Laplace operators,occurring in quantum mechanics.The solutions of these equations are obtained by employing the joint Laplace and Fourier transforms,in terms of the Fox’s H-function.Several special cases as solutions of one dimensional non-homogeneous fractional equations occurring in the quantum mechanics are presented.The results given earlier by Saxena et al.[Fract.Calc.Appl.Anal.,13(2)(2010),pp.177-190]and Purohit and Kalla[J.Phys.AMath.Theor.,44(4)(2011),045202]follow as special cases of our findings.
文摘Let G be a finite group andНbe a Hartley set of G.In this paper,we prove the existence and conjugacy ofН-injectors of G and describe the characterization of injectors via radicals.As applications,some known results are directly followed.
文摘In this paper, we study the semi-boundless mixed problem for time-fractional telegraph equation. We are able to use the integral transform method （the Fourier sin and cos transforms） to obtain the solution.
文摘The Krätzel function has many applications in applied analysis,so this function is used as a base to create a density function which will be called the Krätzel density.This density is applicable in chemical physics,Hartree–Fock energy,helium isoelectric series,statistical mechanics,nuclear energy generation,etc.,and also connected to Bessel functions.The main properties of this newfamily are studied,showing in particular that it may be generated via mixtures of gamma random variables.Some basic statistical quantities associated with this density function such as moments,Mellin transform,and Laplace transform are obtained.Connection of Krätzel distribution to reaction rate probability integral in physics,inverse Gaussian density in stochastic processes,Tsallis statistics and superstatistics in non-extensive statistical mechanics,Mellin convolutions of products and ratios thereby to fractional integrals,synthetic aperture radar,and other areas are pointed out in this article.Finally,we extend the Krätzel density using the pathway model of Mathai,and some applications are also discussed.The new probability model is fitted to solar radiation data.
