Novel Investigation of Multivariable Conformable Calculus for Modeling Scientific Phenomena

Antonio Di Crescenzo,Silvestre Paredes,Zailan Siri,Inmaculada Martínez,Mohammed K. A. Kaabar,Francisco Martínez

doi:10.1155/2021/3670176

Antonio Di Crescenzo, Silvestre Paredes + Show 4 more

Open Access

https://doi.org/10.1155/2021/3670176

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Abstract

New investigation on the conformable version (CoV) of multivariable calculus is proposed. The conformable derivative (CoD) of a real-valued function (RVF) of several variables (SVs) and all related properties are investigated. An extension to vector-valued functions (VVFs) of several real variables (SRVs) is studied in this work. The CoV of chain rule (CR) for functions of SVs is also introduced. At the end, the CoV of implicit function theorem (IFThm) for SVs is established. All results in this work can be potentially applied in studying various modeling scenarios in physical oceanography such as Stommel’s box model of thermohaline circulation and other related models where all our results can provide a new analysis and computational tool to investigate these models or their modified formulations.

Highlights

Many definitions of derivative have been proposed based on two categorizations: global and local types
Several approximate analytical and numerical methods have been recently developed to solve nonlocal fractional and local differential equations that are encountered while modeling various scientific phenomena such as the generalized Riccati expansion for solving the nonlinear KPP equation via fractional derivative (FrDr) [1] and the trigonometric quintic B-spline method for solving the nonlinear telegraph equation constructed via conformable version (CoV) [2]. e nonlinear Klein–Fock–Gordon equation has been analytically solved via two novel techniques such as the methods of generalized Riccati expansion and generalized exponential function [3]
According to the mathematical investigation of conformable derivative (CoD) in [20], it is clear that CoD cannot be addressed as a fractional derivative. erefore, in our study, we have addressed CoD as a modified form of usual derivative which has some applications in physics and engineering due to the fact that the measurements in physics are local. erefore, this definition is highly applicable in theoretical physics

Summary

Introduction

Many definitions of derivative have been proposed based on two categorizations: global (nonlocal) and local types. While there are some recent studies concerning the mathematical analysis of conformable calculus such as the multivariable conformable calculus [15] that was introduced in 2018, the behavior of conformable derivatives of functions in arbitrary Banach spaces [24] that was investigated in 2021, the differential geometry of curves [25] that was investigated in 2019 in the senses of conformable derivatives and integrals, and the behavioral framework for the conformable linear differential systems’ stability [26] that was carefully studied in 2020 to utilize the importance of CoV in modeling scenarios of control theory and power electronics, our results in this work provide a comprehensive investigation of α-derivative of a function of SVs and all related properties, the CoV of CR for functions of SVs, and the CoV of IF m involving many numerical examples to validate our obtained results. According to the best of our knowledge, our original investigation in this article provides an essential mathematical analysis tool for researchers working on modeling phenomena in physics and engineering in the sense of conformable calculus because all theorems and properties in this work will be needed in such modeling scenarios.

Fundamental Notions

The Chain Rule

Conformable Implicit Function Theorem

Conclusion

Conflicts of Interest