Discrete Fractional Inequalities Pertaining a Fractional Sum Operator with Some Applications on Time Scales

Zareen A Khan,Kamal Shah,Sivasubramanian Srikandan

doi:10.1155/2021/8734535

Abstract

This content replicates some discrete nonlinear fractional inequalities by virtue of the fractional sum operator Ψ ¯ on time scales. Through the recognition of the principle of discrete fractional calculus, we are able to recover the precise estimates for unknown functions of inequalities of the Gronwall type. The resultant inequalities are of unique structure comparative with the latest reviewing disclosures and can be described as a complementary tool for numerically testing the solutions of discrete partial differential equations. The foremost consequences are probably confirmed via handling of assessment procedure and technique of mean value speculation. We display few examples of the proposed inequalities to represent the incentives of our effort.

Highlights

Introduction and EssentialsFractional calculus and its conceptual applications have acquired a huge amount of potential in terms of the reality that fractional operators are becoming a valuable asset with more specificity and success in demonstrating a few complicated discoveries in numerous seemingly diverse and wide fields of science and in many areas, such as fluid flow, physics, chaos, image analysis, virology, and financial economy [1, 2]
A few years earlier, fractional differential equations and dynamic systems have been validated as being significant gadgets in showing a few marvels in different parts of applied and pure sciences
The aim of the paper is to impose discrete fractional sum equations in order to build up a procedure to comprehend certain equations and to extract corresponding Gronwall sort of inequality

Summary

Introduction and Essentials

Fractional calculus and its conceptual applications have acquired a huge amount of potential in terms of the reality that fractional operators are becoming a valuable asset with more specificity and success in demonstrating a few complicated discoveries in numerous seemingly diverse and wide fields of science and in many areas, such as fluid flow, physics, chaos, image analysis, virology, and financial economy [1, 2]. A few years earlier, fractional differential equations and dynamic systems have been validated as being significant gadgets in showing a few marvels in different parts of applied and pure sciences. They draw enormous importance in research-oriented fields The essentially identical to discrete hypothesis by a fractional sum of order Ψ > 0 was identified due to Miller and Ross [13] through solution based on linear differential equation, and many key aspects of proposed operator were tested. The remaining section is considered in accomplishing the theoretical examination specifications

Preliminaries

Result

Boundedness and Uniqueness

Concluding Remarks