On Multi-Index Mittag–Leffler Function of Several Variables and Fractional Differential Equations

B B Jaimini,Manju Sharma,D L Suthar,S D Purohit,Zakia Hammouch

doi:10.1155/2021/5458037

Abstract

In this paper, we have studied a unified multi-index Mittag–Leffler function of several variables. An integral operator involving this Mittag–Leffler function is defined, and then, certain properties of the operator are established. The fractional differential equations involving the multi-index Mittag–Leffler function of several variables are also solved. Our results are very general, and these unify many known results. Some of the results are concluded at the end of the paper as special cases of our primary results.

Highlights

Mittag–Leffler (M-L) functions have demonstrated their special connection to fractional calculus, with a particular emphasis on fractional calculus problems arising from implementations
Several new special functions and implementations have been discovered over the last few decades. e advancement of research in the new era of special functions and their applications in mathematical modelling continues to attract many scientists from various disciplines
We have studied here, the integral operator involving the function defined by (3), as follows:

Summary

Introduction

Mittag–Leffler (M-L) functions have demonstrated their special connection to fractional calculus, with a particular emphasis on fractional calculus problems arising from implementations. E advancement of research in the new era of special functions and their applications in mathematical modelling continues to attract many scientists from various disciplines (see recent papers; [1,2,3,4,5,6,7,8,9,10,11,12,13]). E Mittag–Leffler function is extended to multi-index function in the following form [14, 15]: Ec,K􏼂 α1, β1􏼁, . A multivariable extension of Mittag–Leffler function widely studied by Gautam [18], and by Saxena et al Motivated by the work on these functions, we consider here the subsequent multivariable and multi-index Mittag–Leffler function: E((cρ(1rr)),)(,.l.r.),(ρ(mr));β1,...,βm 􏼂z1,. We have studied here, the integral operator involving the function defined by (3), as follows:.

Results

Main Results

Conclusion