Novel Investigation of Multivariable Conformable Calculus for Modeling Scientific Phenomena

Kaabar, Mohammed K. A.; Martínez, Francisco; Martínez, Inmaculada; Siri, Zailan; Paredes, Silvestre

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

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Volume 2021 | Article ID 3670176 | https://doi.org/10.1155/2021/3670176

Novel Investigation of Multivariable Conformable Calculus for Modeling Scientific Phenomena

Mohammed K. A. Kaabar,^1,2Francisco Martínez,³Inmaculada Martínez,³Zailan Siri,¹and Silvestre Paredes³

Academic Editor: Antonio Di Crescenzo

Received11 Sept 2021

Accepted29 Oct 2021

Published23 Nov 2021

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.

1. Introduction

Many definitions of derivative have been proposed based on two categorizations: global (nonlocal) and local types. In the first one, the nonlocal fractional derivative (FrDr) is represented via integral transformation or any other related transformations where nonlocality is seen in this type along with a memory. The Riemann–Liouville and Caputo fractional definitions are considered as the most commonly known fractional definitions. 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 FrDr [1] and the trigonometric quintic B-spline method for solving the nonlinear telegraph equation constructed via CoV [2]. The 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]. Fractional vector (FV) calculus has been recently introduced in [4] to represent the spherical coordinates framework using the fractional derivative and integral of Caputo and Riemann–Liouville senses, respectively. FV calculus can be used effectively in modeling processes in fractal media and other fractional dynamical systems such as hydrodynamical and electrodynamical systems [4]. Fractional calculus is a powerful tool in modeling the potential flow past a sphere of an inviscid fluid [5] which is a very important research problem because this problem particularly investigates the Laplacian in three-dimensional space (spherical coordinates) which is used in governing various physical and mechanical systems in heat conduction, elasticity, Newtonian gravitational potential, ideal fluid flow, and electrostatics [6]. In addition, this problem has been applied in finding the stream function solutions for the stokes flow inside viscous sphere in an inviscid extensional flow [7]. The basics of these essential notions are mentioned in [8, 9]. The local definition is based on certain incremental ratios. A new local derivative definition was initially formulated by Khalil et al. [10, 11], which is called conformable derivative (CoD). The main aim of CoD is to avoid some obstacles of solving nonlocal fractional differential equation where the analytical solutions can be very complicated to obtain. Consequently, several research works have mathematically analyzed some essential notions [10, 12–16]. The Schrodinger–Hirota equation and modified KdV–Zakharov–Kuznetsov equation have been solved with the help of generalized CoDs [17]. CoDs have been employed in investigating the wick-type stochastic nonlinear evolution equations via the improved technique of Kudryashov [18] (see also the wick-type stochastic KdV equation formulated in the context of generalized CoDs [19]). According to the mathematical investigation of CoD in [20], it is clear that CoD cannot be addressed as a fractional derivative. Therefore, 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. Therefore, this definition is highly applicable in theoretical physics. CoD can still be helpful in many related modeling scenarios.

The CoV of analytic functions’ theory has been proposed in [21]. In addition, new results are investigated on the contour conformable integral in [22, 23]. Thus, the definition of the contour conformable integral has been utilized in [23].

Studying conformable derivative and integral is essential in various fields of natural sciences and engineering. The need for local derivative is highly appreciated in multidisciplinary sciences. While the nonlocal fractional derivatives can provide a good explanation to the dynamics of certain systems, particularly in modeling epidemic diseases, the difficulty of obtaining exact or analytical solutions to the problems formulated in the senses of nonlocal fractional derivatives can make the investigation of fractional-order systems a real challenge for researchers. Thus, researchers have paid more attention to conformable derivative and other related local derivatives in modeling scientific phenomena. 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 -derivative of a function of SVs and all related properties, the CoV of CR for functions of SVs, and the CoV of IFThm 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.

The article consists of the following sections: essential notions of the CoV of calculus are mentioned in Section 2. Then, the -derivative of RVF of SVs is investigated, and all its main properties are established in Section 3. Furthermore, these results are extended to the VVFs of SRVs. In addition, the CR for functions of SVs is introduced in two particular cases in Section 4. In the last part, the CoV of IFThm for SVs is obtained in Section 5 by first establishing the conformable theorem of existence and regularity of the implicit function for single equation. Second, this result is extended to a system of several equations and SRVs. Some concluding remarks are specified in Section 6.

2. Fundamental Notions

Definition 1 (See [10]). For a function , the order CoD can be written as, . If is -differentiable function in some , , and exists; then, it is expressed as

Theorem 1 (See [10]). If is at , , then is continuous function (CF) at .

Theorem 2 (See [10]). Assuming that and are at a point , we have(i), .(ii), .(iii), constant functions (iv)(v)(vi)If we suppose that is differentiable, then .From Definition 1, the CoD of some functions are expressed as(i)(ii)(iii)(iv), .

Definition 2 (See [11]). The left CoD beginning from of function of order is expressed asFor , it is expressed as . If is in some , then we set

Theorem 3 (See [12]). Suppose that are left , where . Let us assume that , is and , and we getIf , then we obtain

Theorem 4 (Rolle’s theorem (RT) [10]). Suppose that , , and function satisfies(i) is CF on (ii) is on (iii)Then, , .

Theorem 5 (Mean value theorem (MVT) [10]). Assume that , , and function satisfies(i) is CF on (ii) is on Then, , we have

Theorem 6 (Modified mean value theorem (MMVT) [27]). Assume that , , and function satisfies(i) is CF on (ii) is on Then, , we have

Theorem 7 (See [13]). Suppose that , , and function satisfies(i) is CF on (ii) is on Then, we get(i)If , then is increasing on(ii)If , then is decreasing onLet us express the CoV of partial derivative (PDr) of a real-valued function (RVF) with SVs as follows.

Definition 3 (See [14, 15]). Assume that is a RVF with variables, and there is a point , where its component is positive. Then, the limit is written asIf the above limit exists, the CoV of PDr of of the order at , represented by .

3. -Derivative of a RVF of SVs

Definition 4. Suppose that is a RVF with variables , and . Then, we say that is at , each , if any of the three conditions which are equivalent to each other is verified:(i)There is a linear transformation , such that Where , , and .(ii)There is a linear transformation and a function , such that And .(iii)There is a linear transformation and functions And for .The linear transformation is defined by , with and . This linear transformation is denoted by , which is called CoD of of the order at .

Remark 1. The equivalence of conditions (i) and (ii) is immediate, sinceTo see the equivalence between conditions (ii) and (iii), we takeAs , then we have the following:(i)If , then (ii)If for , then we obtaini.e., . Hence, the conditions (ii) and (iii) are equivalent.

Example 1. Consider a function defined by and a point , with and ; then, .
Solution: to prove this, let us note that

Theorem 8. If a RVF with variables at , each , then is CF at .

Proof. Since is at , we can write the following:By taking the limits of the two sides of the equality as , we haveHence, is CF at .

Theorem 9. If a RVF with variables is at , each , then exists for and the CoD of of the order is expressed aswhere .

Proof. By setting in the formula (12), we haveBy multiplying by , we can writeBy taking the limits of the two sides of the equality as , we haveFinally, by substituting the values above in the formula , the result is followed.

Corollary 1. If a RVF with variables is at , each , then is unique.

Remark 2. If a RVF with variables is at , each , then the CoV of gradient of of the order at isAlso, the matrix form (MF) of equation (19) is given as follows:

Theorem 10. Let , be a RVF defined in an open set (OS): , , each , and a point . If are at , then we have(i), (ii)

Proof. (i) follows from Definition 4; thus, it follows the proof of (i).
For (ii), let , and then, we have

Theorem 11. Let , be a RVF defined in an OS: , , each , and a point . If the function has all CoV of PDrs of the order at each point of a neighbourhood of the point , , with , and they are continuous at , then is at .

Proof. See Theorem 2.1 proof in [27].

Remark 3. The above theorem allows defining the space of RVFs with variables by having continuous CoV of PDrs of order in a domain , which can be denoted by .
Finally, we can easily extend all of the above results to the VVFs of SRVs.

Theorem 12. Assume that , be a VVF defined in an OS: , , each , and the point . The function is at iff its components are at , and if these components are then the components of are the -derivatives , for , i.e.,

Proof. It is similarly proven as same as traditional calculus by applying instead of derivative.

Remark 4. Assume that , is a VVF defined in an OS: , , each , and a point . If function is -differentiable at , then exists for , , and the CoV of -Jacobian of of order at is expressed as

4. The Chain Rule

Let us prove the CR for the functions of SVs in 2 particular cases. For the proof’s purpose, the continuity’s hypothesis of PDrs is given [28].

Theorem 13 (CR). Suppose that and . If is at and a RVF , with variables , is at , all . Then, the composition is -differential at and

Proof. Assume, is the point neighborhood. Suppose that . By setting in Definition 1, we see thatWithout loss of generality (WLOG), is assumed to be an open ball (OB), represented by . Since is a CF, then along with points and , the points ,…, and lines connecting them must also be the ball . In fact, using the classical MVT for differentiable functions (DFs) of one variable is in the following computation [28]:where is between and .
By taking limits as and using the continuity of PDrs of as well as the fact that , equation (29) is expressed asOur proof is completely done.

Remark 5. The MF of equation (28) is expressed as

Theorem 14 (CR). Let and . If is a at , each , and a RVF with variables is at , where all . Then, the composition is -differentiable at , and we have

Proof. Proof. According to PDr’s definition and Theorem 14, the following is implied:

Remark 6. The MF of equation (33) is expressed as

5. Conformable Implicit Function Theorem

In this section, the CoV of IFThm for SVs is obtained. First, we establish the conformable theorem of existence and regularity of the implicit function for the case of a single equation.

Theorem 15 (Conformable implicit function theorem for the case of a single equation). Suppose that , is a RVF defined in an OS: , , each , and point . Suppose that(i)(ii)(iii)Then, there is a neighbourhood of , such that there is a unique function (UF) that satisfies the following:
and .
Finally, is in , and for every , we have

Proof. WLOG, is assumed to be an OB, represented by . Let . If we call , it is verified thatNote that in particular if , then .
Since , it is assumed to be positive (otherwise, is considered instead of From fact that , it follows thatBy the continuity of at and , there exists , such thatSince the function is CF on the interval , , and via the classical Bolzano’s theorem (BThm), it implies that , for each . Then, ’s value is unique, since a function, whose derivative is positive, has more than zero. On the other hand, by having , for each , there exists a unique , we can write , and then, this function will be proven to be CF on . The continuity of the function at the point is clear since for each , a value :The function continuity will be proven at any point by simply substituting for an OB: is contained in .
Finally, let us now show formula (35). By applying Theorem 14 to equation, , . Solving , we get (35). In addition, the formula (35) on the right side is continuous, so the continuity of CoV of PDrs , , follows.
Theorem 15 will provide us a help in computing CoV of PDrs of implicit function of SVs.

Example 2. ConsiderThis equation’s one solution is . is obviously in which is OB, represented by , with , for some , sinceTheorem 15 implies that there is a neighbourhood, of a UF: satisfies the following: and .
Moreover, is in , and we haveFinally, we obtain and .
Finally, CoV of IFThm for a system of several equations and SRVs is obtained.

Theorem 16 (Conformable general implicit function theorem). Let , be a VVF defined in an OS: , , each , and point . Assume that(i)(ii)(iii)Then, there is a neighbourhood, of a UF: that satisfies and .
Finally, is class in , and for every , we havewhere

Proof. The proof of existence and uniqueness of the implicit function is done similar to the traditional multivariable calculus by applying mathematical induction on and using the conformable implicit function theorem for several variables [28].
Let us now show formula (44). Assume that a system with several equations and SRVs is expressed aswhich satisfies hypotheses (i)–(iii) of the theorem; then, this system is defined in a neighbourhood, of , the implicit function class in , such that which satisfies equation (1), i.e.,By applying CoV of CR to the above equation, we havefor all .
In addition, the MF of equation (48) is given as follows:Since and are regular matrices by hypothesis, we havewhich completes the proof.
We will now show that Theorem 16 can be used to compute CoV of PDrs of systems with several equations and SRVs.

Example 3. Consider a system of two equations and two real variables:One solution of this equation is . Clearly, is in , which is OB: , with , for some , sinceTheorem 16 implies that there is a neighbourhood, of a UF: given bythat satisfiesMoreover, is class in , and we haveFinally, we haveAccording to all of our numerical examples in this work, with the help of our investigation to all proposed theorems and properties in this article, the numerical examples show that all our obtained results are valid and efficient. As seen in all these examples, the computations are simple and cost-efficient, which are important when scientific phenomena are modelled in the sense of conformable calculus. All in all, the simplicity of computations is always needed in modeling applications in comparison to other complicated computations that are needed using other approaches in other fractional operators. All our obtained results in this work are accurate because our results coincide with the usual integer-order results.

6. Conclusion

A new investigation on the CoV of multivariable calculus has been discussed in this work in detail. The -derivative of a function of SVs and all related properties have been investigated. The CoV of CR for functions of SVs has also been studied. The CoV of IFThm has been presented in the last part of our results, and numerical experiments have been conducted to support our theoretical results. The findings of this investigations show that all results formulated via CoD coincide with the ones in the traditional integer case. All results in this work can be potentially applied in modeling oceanographic phenomena such as Stommel’s box model of thermohaline circulation and other related models where this study’s analysis can be further extended or generalized in future works for various related physical models.

Data Availability

No data were used to support this study.

Conflicts of Interest

All authors declare that they have no conflicts of interest.

Authors’ Contributions

Mohammed K. A. Kaabar involved in actualization, developed methodology, performed formal analysis, validated and investigated the study, prepared the initial draft, and supervised and edited the original draft. Francisco Martínez, Inmaculada Martínez, and Silvestre Paredes involved in actualization, developed methodology, performed formal analysis, validated and investigated the study, and prepared the initial draft. Zailan Siri developed methodology, performed formal analysis, validated and investigated the study, and prepared the initial draft. All authors read and approved the final version.

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Copyright © 2021 Mohammed K. A. Kaabar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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