Pratibha Manohar
Sknau, Statistics, Mathematics and Computer Science, Faculty Member
ABSTRACT In this paper, we introduce and study a Mittag-Leffler-type function of two variables E 1 (x, y) and a generalization of Mittag-Leffler-type function of one variable as limiting case of E 1 (x, y), which includes several... more
ABSTRACT In this paper, we introduce and study a Mittag-Leffler-type function of two variables E 1 (x, y) and a generalization of Mittag-Leffler-type function of one variable as limiting case of E 1 (x, y), which includes several Mittag-Leffler-type functions of one variable as its special cases. Here, we first obtain the domain of convergence of E 1 (x, y), considering all possible cases. Next, we give two differential equations for E 1 (x, y) and one differential equation for for some particular values of the parameters. We further obtain two integral representations and Mellin–Barnes contour integral representation of E 1 (x, y). We also obtain the Laplace transform of one and two dimensions of E 1 (x, y) and its fractional integral and derivative. Next, we define an integral operator with E 1 (x, y) as a kernel and show that it is bounded on the Lebesgue measurable space L(a, b). Finally, we introduce one more Mittag-Leffler-type function of two variables.
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Mukhyamantri Jal Swavlamban Abhiyan is run by the Chief Minister to conserve and harvest rain water and make villages self-reliant even during drought periods. The purpose of this mission on water conservation is to make villages... more
Mukhyamantri Jal Swavlamban Abhiyan is run by the Chief Minister to conserve and harvest rain water and make villages self-reliant even during drought periods. The purpose of this mission on water conservation is to make villages self-sufficient in water use and thus provide a permanent solution to the demand of water besides ensuring storage and conservation of water. In this project, various hydrological components of the Balesar block have been studied with the help of satellite derived information and watershed management has been analyzed. To support the abhiyan through this project, we studied the status of water available through ground water, storage of runoff water and present as well as future water requirement through the data of year 2015. We have also plotted several maps that will help in selection of locations to create new watersheds for storage of runoff water.
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We use generalized differential transform method GDTM to derive the solution of spacetime fractional telegraph equation in closed form. The space and time fractional derivatives are considered in Caputo sense and the solution is obtained... more
We use generalized differential transform method GDTM to derive the solution of spacetime fractional telegraph equation in closed form. The space and time fractional derivatives are considered in Caputo sense and the solution is obtained in terms of Mittag-Leffler functions.
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In recent years the generalized differential transform method is introduced and applied to solve various fractional differential equations involving two variables. In these works, we observe that a basic result giving the law of exponents... more
In recent years the generalized differential transform method is introduced and applied to solve various fractional differential equations involving two variables. In these works, we observe that a basic result giving the law of exponents for Caputo fractional derivatives has been used, which does not hold true. In the present paper, we provide corrected form of the said result as Theorem 3.1. We, next develop three-dimensional generalized differential transform method and apply this method to solve some space-fractional, time-fractional and space-time fractional diffusion equations in two space variables with variable coefficients.
In this paper, we first establish a generalized Taylor's formula for composite fractional derivative and then develop the generalized differential transform method for composite fractional derivative. As an application, we solve... more
In this paper, we first establish a generalized Taylor's formula for composite fractional derivative and then develop the generalized differential transform method for composite fractional derivative. As an application, we solve fractional Fokker–Planck Equation (FFPE) with space derivative considered as composite fractional derivative and time derivative considered as Caputo fractional derivative.
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ABSTRACT In the present paper we use generalized differential transform method to derive analytical solution of linear and non-linear space-time fractional reaction-diffusion equations on a finite domain. The space and time fractional... more
ABSTRACT In the present paper we use generalized differential transform method to derive analytical solution of linear and non-linear space-time fractional reaction-diffusion equations on a finite domain. The space and time fractional derivatives are considered in Caputo sense. Some examples are given and it has been observed that the generalized differential transform method is very effective and convenient and overcomes the difficulty of Adomian decomposition method and homotopy perturbation method.
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ABSTRACT We aim at studying distribution of mixed sum of two independent random variables. We establish a theorem which gives probability density function of sum of doubly infinite and finite independent random variables (i.r.v.) and... more
ABSTRACT We aim at studying distribution of mixed sum of two independent random variables. We establish a theorem which gives probability density function of sum of doubly infinite and finite independent random variables (i.r.v.) and distribution of sum of an infinite and a finite i.r.v. is given in the form of corollary. As an application of these results we obtain distribution of sum of bilateral Chi, bilateral Rayleigh, bilateral Laplace and standard normal variates with generalized trapezoidal variate, respectively. We also give some graphs of these distributions.
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Increasing temperature and declining and erratic rainfall is one of the greatest global challenges. This study presents the trend analysis of temperature and rainfall in five divisional headquarters of Rajasthan, namely, Bikaner, Jaipur,... more
Increasing temperature and declining and erratic rainfall is one of the greatest global challenges. This study presents the trend analysis of temperature and rainfall in five divisional headquarters of Rajasthan, namely, Bikaner, Jaipur, Jodhpur, Kota, and Udaipur. The historic data of minimum and maximum temperature and rainfall for a period of 49 years from 1971 to 2019 were collected from Climate Research and Services, India Meteorological Department, Pune. Detection of trends and change in magnitude was done using the Mann–Kendall (MK) test and Sen’s slope, respectively. The results of the study indicated a significant increase in both minimum and maximum temperature over time for all the five stations. However, rainfall showed a nonsignificant increasing trend for Kota and Udaipur district, whereas Bikaner, Jaipur, and Jodhpur detected a negative trend.
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We establish some generalized inequalities for the gamma function using the properties of logarithmically convex/concave functions.
In the present paper, we use generalized differential transform method (GDTM) to derive solutions of some linear and nonlinear space-time frac-tional Fokker-Planck equations (FPE) in closed form. The space and time fractional derivatives... more
In the present paper, we use generalized differential transform method (GDTM) to derive solutions of some linear and nonlinear space-time frac-tional Fokker-Planck equations (FPE) in closed form. The space and time fractional derivatives are considered in Caputo sense and the solutions are obtained in terms of Mittag-Leffler functions.
In the present paper, we define a generalized composite fractional derivative and obtain some results, which include the image of power function, Laplace transform and composition of Riemann-Liouville fractional integral with the... more
In the present paper, we define a generalized composite fractional derivative and obtain some results, which include the image of power function, Laplace transform and composition of Riemann-Liouville fractional integral with the generalized composite fractional derivative. We also obtain the closed form solution of a generalized fractional free electron laser equation with this fractional derivative by using Adomian decomposition method.
We investigate a pair of partial differential equations of H-function of two variables for suitably constrained values of parameters. The partial differential equations for corresponding G-function of two variables and the generalized... more
We investigate a pair of partial differential equations of H-function of two variables for suitably constrained values of parameters. The partial differential equations for corresponding G-function of two variables and the generalized Kampé de Fériet function are obtained as particular cases and that of Appell’s functions and H-function of one variable are found to be in agreement with those available in literature
We aim at studying distribution of mixed sum of two independent random variables. We establish a theorem which gives probability density function of sum of doubly infinite and finite independent random variables (i.r.v.) and distribution... more
We aim at studying distribution of mixed sum of two independent random variables. We establish a theorem which gives probability density function of sum of doubly infinite and finite independent random variables (i.r.v.) and distribution of sum of an infinite and a finite i.r.v. is given in the form of corollary. As an application of these results we obtain distribution of sum of bilateral Chi, bilateral Rayleigh, bilateral Laplace and standard normal variates with generalized trapezoidal variate, respectively. We also give some graphs of these distributions.
Research Interests:
ABSTRACT In the present paper we solve space-time fractional diffusion-wave equations with three space variables, using matrix method. Here, in particular we give solutions to classical, time-fractional, space-fractional and space-time... more
ABSTRACT In the present paper we solve space-time fractional diffusion-wave equations with three space variables, using matrix method. Here, in particular we give solutions to classical, time-fractional, space-fractional and space-time fractional diffusion equations and classical, time-fractional, space-fractional and space-time fractional wave equations with different combinations of time and space fractional derivatives. A set of MATLAB routines for the implementation of the method to solve these examples has also been developed.