A Study on the H Function I Function and Their Applications in Boundary Value Problems

Abstract

The generalized Hypergeometric function was given by C.F. Gauss in 1812. Considering suitable representation there was constructed few more extensions of generalized hypergeometric functions are as Mac-Robert s E-function from 1937, more successful however, are subsequent generalized in terms of single Mellin-Barne s type contour integral is G-function from 1941 and the H-function introduced by Charles Fox in 1961. The even more general is I-function defined by Saxena V.P. in 1982. newlineThe application of these functions is in various fields such as in solving different integral and differential equations. Many expansion and summation formulae are defined through these functions. Transform of many functions and various new results using transforms are found with the help of them. It covers wider range and gives general, deeper and useful results directly applicable in various problems of physical, biological, engineering and earth sciences such as fluid flow, diffusion in porous media, kinematics in viscoelastic media, diffusion process in complex systems, propagation of seismic waves , anomalous diffusion and turbulence, etc. newlineThese applications prompted me to take this study entitled A study on H-function, I-function and their applications in boundary value problems . The first chapter is devoted to the general introduction of the subject. This chapter throws light on the topic such as Fox s H-function, H-function of two variables, Saxena s I-function of one variable, I-function of two variables defined by Goyal and Agrawal, Inayat-Hussain and#119867; -function, General class of polynomials, General class of multivariable polynomials of first and second kind, Appell s function. newlineThe second chapter is divided into four sections. In the first section, we have derived three derivatives of I-function of one variable. In the second section, we established an integral with newlinethe product of I-function of one variable, Bessel s function and General class of polynomials.