Sunil Dutt Purohit

On fractional partial differential equations related to quantum mechanics

In this paper, we investigate the solutions of generalized fractional partial differential equations involving the Caputo time-fractional derivative and the Liouville space-fractional derivatives. The solutions of these equations are obtained by employing the joint Laplace and Fourier transforms. Several special cases as solutions of one-dimensional non-homogeneous fractional equations occurring in quantum mechanics are presented in the concluding section. The results given earlier by Debnath (2003 Fract. Calc. Appl. Anal. 6 119?55), Saxena et al (2010 Appl. Math. Comput. 216 1412?7) and Pagnini and Mainardi (2010 J. Comput. Appl. Math. 233 1590?5) follow as special cases of our findings.

On Generalized Fractional Kinetic Equations Involving Generalized Bessel Function of the First Kind

We develop a new and further generalized form of the fractional kinetic equation involving generalized Bessel function of the first kind. The manifold generality of the generalized Bessel function of the first kind is discussed in terms of the solution of the fractional kinetic equation in the paper. The results obtained here are quite general in nature and capable of yielding a very large number of known and (presumably) new results.

Solutions of Fractional Partial Differential Equations of Quantum Mechanics

The aim of 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. A Math. Theor., 44 (4) (2011), 045202] follow as special cases of our findings.

CERTAIN NEW INTEGRAL FORMULAS INVOLVING THE GENERALIZED BESSEL FUNCTIONS

Abstract. A remarkably large number of integral formulas involving avariety of special functions have been developed by many authors. Alsomany integral formulas involving various Bessel functions have been pre-sented. Very recently, Choi and Agarwal derived two generalized integralformulas associated with the Bessel function J ν (z) of the first kind, whichare expressed in terms of the generalized (Wright) hypergeometric func-tions. In the present sequel to Choi and Agarwal’s work, here, in thispaper, we establish two new integral formulas involving the generalizedBessel functions, which are also expressed in terms of the generalized(Wright) hypergeometric functions. Some interesting special cases of ourtwo main results are presented. We also point out that the results pre-sented here, being of general character, are easily reducible to yield manydiverse new and known integral formulas involving simpler functions. 1. Introduction and preliminariesA remarkably large number of integral formulas involving a variety of spe-cial functions have been developed by many authors (see, e.g., [5], [7] and [9];for a very recent work, see also [6]). Many integral formulas involving prod-ucts of Bessel functions have been developed and play an important role inseveral physical problems. In fact, Bessel functions are associated with a widerange of problems in diverse areas of mathematical physics, for example, thosein acoustics, radio physics, hydrodynamics, and atomic and nuclear physics.These connections of Bessel functions with various other research areas haveled many researchers to the field of special functions. Among many propertiesof Bessel functions, they also have investigated some possible extensions of theBessel functions. A useful generalization w

On Fractional Integral Inequalities Involving Hypergeometric Operators

Here we aim at establishing certain new fractional integral inequalities involving the Gauss hypergeometric function for synchronous functions which are related to the Chebyshev functional. Several special cases as fractional integral inequalities involving Saigo, Erdelyi-Kober, and Riemann-Liouville type fractional integral operators are presented in the concluding section. Further, we also consider their relevance with other related known results.

On a q-Extension of the Leibniz Rule via Weyl Type of q- Derivative Operator

In the present paper we define a q-extension of the Leibniz rule for q-derivatives via Weyl type q-derivative operator. Expansions and summation formulae for the gener- alized basic hypergeometric functions of one and more variables are deduced as the appli- cations of the main result.

Generalized fractional integrals of product of two H-functions and a general class of polynomials

The purpose of this paper is to compute two unified fractional integrals involving the product of two H-functions, a general class of polynomials and Appell function . These integrals are further applied in proving two theorems on Saigo–Maeda fractional integral operators. Some consequent results and special cases are also pointed out in the concluding section.

On q-Analogues of Sumudu Transform

Abstract The present paper introduces q-analogues of the Sumudu transform and derives some distinct properties, for example its convergence conditions and certain interesting connection theorems involving q-Laplace transforms. Furthermore, certain fundamental properties of q-Sumudu transforms like, linearity, shifting theorems, differentiation and integration etc. have also been investigated. An attempt has also been made to obtain the convolution theorem for the q-Sumudu transform of a function which can be expressed as a convergent infinite series.

Certain Chebyshev Type Integral Inequalities Involving Hadamard’s Fractional Operators

We establish certain new fractional integral inequalities for the differentiable functions whose derivatives belong to the space , related to the weighted version of the Chebyshev functional, involving Hadamard’s fractional integral operators. As an application, particular results have been also established.

Chebyshev Type Integral Inequalities Involving the Fractional Hypergeometric Operators

By making use of the fractional hypergeometric operators, we establish certain new fractional integral inequalities for synchronous functions which are related to the weighted version of the Chebyshev functional. Some consequent results and special cases of the main results are also pointed out.