Certain new weighted estimates proposing generalized proportional fractional operator in another sense

Saima Rashid,Zakia Hammouch,A. A. El-Deeb,Yu-Ming Chu,Thabet Abdeljawad

doi:10.1186/s13662-020-02935-z

Abstract

The present work investigates the applicability and effectiveness of generalized proportional fractional integral (mathcal{GPFI}) operator in another sense. We aim to derive novel weighted generalizations involving a family of positive functions n (nin mathbb{N}) for this recently proposed operator. As applications of this operator, we can generate notable outcomes for Riemann–Liouville (mathcal{RL}) fractional, generalized mathcal{RL}-fractional operator, conformable fractional operator, Katugampola fractional integral operator, and Hadamard fractional integral operator by changing the domain. The proposed strategy is vivid, explicit, and it can be used to derive new solutions for various fractional differential equations applied in mathematical physics. Certain remarkable consequences of the main theorems are also figured.

Highlights

1 Introduction Fractional calculus [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20] is genuinely viewed to be the real-world framework, and it has wide applications in mathematics, physics, biology, medicine, and many other natural and social sciences [21,22,23,24,25,26,27,28,29,30,31,32,33,34,35], for instance, a correspondence structure that contains indulgent interfacing, dependent parts that are used to accomplish a bound together with a goal of transmitting and getting signals, can be depicted using complex framework models [36,37,38,39,40,41,42,43]. This structure is considered as a stunning framework, and the units that make the entire system are seen as the centers of the complex framework
In [54, 55], Rashid et al proposed a different novel fractional approach having an exponential function in its kernel which comes into existence in the theory of fractional calculus, which is known as GPF I operators in another sense
We demonstrate a novel fractional operator which is known as the GPF I operator of a function in another sense proposed by Jarad et al [54] and Rashid et al [55], independently

Summary

Introduction

Fractional calculus [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20] is genuinely viewed to be the real-world framework, and it has wide applications in mathematics, physics, biology, medicine, and many other natural and social sciences [21,22,23,24,25,26,27,28,29,30,31,32,33,34,35], for instance, a correspondence structure that contains indulgent interfacing, dependent parts that are used to accomplish a bound together with a goal of transmitting and getting signals, can be depicted using complex framework models [36,37,38,39,40,41,42,43]. (2) If σ = 1, the left- and right-sided generalized RL-fractional integral operator reduces to the operator defined in [73]. (4) If φ(y1) = ln y1, the left- and right-sided generalized proportional Hadamard fractional integral operator reduces to the operator given in [74].

Results

Conclusion