# Novel Investigation of Multivariable Conformable Calculus for Modeling Scientific Phenomena

Authors: Mohammed K. A. Kaabar; Francisco Martínez; Inmaculada Martínez; Zailan Siri; Silvestre Paredes
Journal: Journal of Mathematics
 (2021)

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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.


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## Body

## 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 α-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 Section2. Then, the α-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 functionf:0,∞⟶R, the αth order CoD can be written as(1)Tαft=limϵ⟶0ft+εt1−α−ftε,
∀t>0, α∈0,1. If f is α-differentiable function αDF in some 0,a, a>0, and limt⟶0+Tαft exists; then, it is expressed as(2)Tαf0=limt⟶0+Tαft.Theorem 1 (See [10]).
Iff:0,∞⟶R is αDF at t0>0, α∈0,1, then f is continuous function (CF) at t0.Theorem 2 (See [10]).
Assuming thatα∈0,1 and f,g are αDFs at a point t>0, we have(i)
Tαaf+bg=aTαf+bTαg, ∀a,b∈R.(ii)
Tαtp=ptp−α, ∀p∈R.(iii)
Tαλ=0, ∀ constant functions ft=λ(iv)
Tαfg=fTαg+gTαf(v)
Tαf/g=gTαf−fTαg/g2(vi)
If we suppose thatf is differentiable, then Tαft=t1−αdf/dtt.
From Definition1, the CoD of some functions are expressed as(i)
Tα1=0(ii)
Tαsinat=at1−αcosat(iii)
Tαcosat=−at1−αsinat(iv)
Tαeat=at1−αeat, a∈R.Definition 2 (See [11]).
The left CoD beginning froma of function f:a,∞⟶R of order α∈0,1 is expressed as(3)Tαaft=limε⟶0ft+εt−a1−α−ftε,t>a.
Fora=0, it is expressed as Tαft. If f is αDF in some a,b, then we set(4)Tαafa=limt⟶a+Tαaft.Theorem 3 (See [12]).
Suppose thatf,g:a,∞⟶R are left αDFs, where α∈0,1. Let us assume that ht=fgt, ht isαDF∀t≠a and gt≠0, and we get(5)Tαaht=Tαafgt⋅Tαagt⋅gtα−1.
Ift=a, then we obtain(6)Tαaha=limt⟶a+Tαafgt⋅Tαagt⋅gtα−1.Theorem 4 (Rolle’s theorem (RT) [10]).
Suppose thata>0, α∈0,1, and function f:a,b⟶R satisfies(i)
f is CF on a,b(ii)
f is αDF on a,b(iii)
fa=fb
Then,∃c∈a,b, ∋Tαfc=0.Theorem 5 (Mean value theorem (MVT) [10]).
Assume thata>0, α∈0,1, and function f:a,b⟶R satisfies(i)
f is CF on a,b(ii)
f is αDF on a,b
Then,∃c∈a,b, ∋ we have(7)Tαfc=fb−fabα/α−aα/α.Theorem 6 (Modified mean value theorem (MMVT) [27]).
Assume thata>0, α∈0,1, and function f:a,b⟶R satisfies(i)
f is CF on a,b(ii)
f is αDFon a,b
Then,∃c∈a,b, ∋ we have(8)Tαfcc1−α/α=fb−fab/α−a/α.Theorem 7 (See [13]).
Suppose thata>0, α∈0,1, and function f:a,b⟶R satisfies(i)
f is CF on a,b(ii)
f is αDF on a,b
Then, we get(i)
IfTαft>0∀t∈a,b, then f is increasing ona,b(ii)
IfTαft<0∀t∈a,b, then f is decreasing ona,b
Let us express the CoV of partial derivative (PDr) of a real-valued function (RVF) with SVs as follows.Definition 3 (See [14, 15]).
Assume thatf is a RVF with n variables, and there is a point a=a1,…,an∈Rn, where its ith component is positive. Then, the limit is written as(9)limε⟶0fa1,…,ai+εai1−α,…,an−fa1,…,anε.
If the above limit exists, theith CoV of PDr of f of the order α∈0,1 at a, represented by ∂α/∂xiαfa.


## 3.α-Derivative of a RVF of SVs

Definition 4.
Suppose thatf is a RVF with n variables x1,…,xn, and α∈0,1. Then, we say that f is αDF at a=a1,…,an∈Rn, each ai>0, if any of the three conditions which are equivalent to each other is verified:(i)
There is a linear transformationL:Rn⟶R, such that(10)limh⟶0fa1+h1a11−α,…,an+hnan1−α−fa1,…,an−Lh‖h‖=0,Whereh=h1,…,hn, ‖h‖=h1+⋯+2hn2, and α∈0,1.(ii)
There is a linear transformationL:Rn⟶R and a function ε:h⟶εh, such that(11)fa1+h1a11−α,…,an+hnan1−α−fa1,…,an=Lh+εh‖h‖,Andlimh⟶0εh=0.(iii)
There is a linear transformationL:Rn⟶R and n functions εi:h⟶εih∀(12)i=1,2,…n,∋fa1+h1a11−α,…,an+hnan1−α−fa1,…,an=Lh+∑i=1nεihhi,Andlimεih⟶0h=0 for i=1,2,…,n.
The linear transformationL:Rn⟶R is defined by Lh=∑i=1nαihi, with h=h1,…,hn and α1,…,αn∈R. This linear transformation is denoted by Dαfa, which is called CoD of f of the order α∈0,1 at a.Remark 1.
The equivalence of conditions (i) and (ii) is immediate, since(13)limh⟶0εh=0↔εhh=o‖h‖.
To see the equivalence between conditions (ii) and (iii), we take(14)εi=εhhih,εh=1h∑i=1nεihhi.
Ashi/‖h‖≤1, then we have the following:(i)
Iflimεh⟶0h=0, then limεih⟶0h=0(ii)
Iflimεih⟶0h=0 for i=1,…,n, then we obtain(15)limh⟶0εh≤limh⟶01‖h‖∑i=1nεih≤limh⟶0∑i=1nεih=0.i.e., limh⟶0εh=0. Hence, the conditions (ii) and (iii) are equivalent.Example 1.
Consider a functionf defined by fx,y=ex−2cosy and a point a,b∈R2, with a>0 and b>0; then, Dαfa,bh1,h2=h1a1−αeα+2h2b1−αsinb.
Solution: to prove this, let us note that(16)limh1,h2⟶0,0fa+a1−αh1,b+b1−αh2−fa,b−Lh1,h2h1,h2=limh1,h2⟶0,0ea+a1−αh1−2cosb+b1−αh2−ea−2cosb−h1a1−αea+2h2b1−αsinbh12+h22≤limh1⟶0ea+a1−αh1−ea−h1a1−αeah1−2limh2⟶0cosb+b1−αh2−cosb+b1−αsinbh2=limh1⟶0ea+a1−αh1−eah1−a1−αea−2limh2⟶0cosb+b1−αh2−cosbh2+b1−αsinb=a1−αea−a1−αea−2−b1−αsinb+b1−αsinb=0.Theorem 8.
If a RVFf with n variables αDF at a=a1,…,an∈Rn, each ai>0, then f is CF at a∈Rn.Proof.
Sincef is αDF at a, we can write the following:(17)fa1+h1a11−α,…,an+hnan1−α−fa1,…,an=∑i=1nαihi+oh.
By taking the limits of the two sides of the equality ash⟶0, we have(18)limh⟶0fa1+h1a11−α,…,an+hnan1−α=fa1,…,an.
Hence,f is CF at a∈Rn.Theorem 9.
If a RVFf with n variables is αDF at a=a1,…,an∈Rn, each ai>0, then ∂α/∂xiαfa exists for 1≤i≤n and the CoD of f of the order α∈0,1 is expressed as(19)Dαfah=∑i=1n∂α∂xiαfahi,where h=h1,…,hn.Proof.
By settinghj=0,∀j≠i in the formula (12), we have(20)fa1,..,ai+hiai1−α+⋯+an−fa1,…,an=αihi+εihhi.
By multiplying by1/hi, we can write(21)fa1,..,ai+hiai1−α+⋯+an−fa1,…,anhi=αi+εih.
By taking the limits of the two sides of the equality ashi⟶0, we have(22)αi=∂α∂xiαfa,∀i=1,2,…,n.
Finally, by substituting the values aboveαi in the formula Dαfah=∑i=1nαihi, the result is followed.Corollary 1.
If a RVFf with n variables is αDF at a=a1,…,an∈Rn, each ai>0, then Dαfa is unique.Remark 2.
If a RVFf with n variables is αDF at a=a1,…,an∈Rn, each ai>0, then the CoV of gradient of f of the order α∈0,1 at a is(23)∇αfa=∂α∂x1αfa,…,∂α∂xnαfa.
Also, the matrix form (MF) of equation (19) is given as follows:(24)Dαfah=∇αfa⋅h=∂α∂x1αfa,…,∂α∂xnαfa⋅h1⋮hn.Theorem 10.
Letα∈0,1, f,g:X⟶R be a RVF defined in an open set (OS): X⊂Rn, ∋∀x=x1,…,xn∈X, each xi>0, and a point a=a1,…,an∈X. If f,g are αDF at a, then we have(i)
Dαλf+μga=λDαfa+μDαga, ∀λ,μ∈R.(ii)
Dαfga=Dαfa⋅ga+faDαgaProof.
(i) follows from Definition4; thus, it follows the proof of (i).
For (ii), letA=a1+h1a11−α,…,an+hnan1−α, and then, we have(25)limh⟶0fgA−fga−Dαfa⋅ga+fa⋅Dαgah‖h‖=limh⟶0fA−fa−Dαfah‖h‖⋅ga+fa⋅gA−ga−Dαgah‖h‖+limh⟶0fA−fa⋅gA−ga‖h‖=0+0+limh⟶0Dαfah⋅Dαgah‖h‖=limh⟶0Dαfah‖h‖⋅Dαgah‖h‖⋅‖h‖=0.Theorem 11.
Letα∈0,1, f:X⟶R be a RVF defined in an OS: X⊂Rn, ∋∀x=x1,…,xn∈X, each xi>0, and a point a=a1,…,an∈X. If the function f has all CoV of PDrs of the order α at each point of a neighbourhood of the point a, Ua, with Ua⊂X, and they are continuous at a, then f is αDF at a.Proof.
See Theorem 2.1 proof in [27].Remark 3.
The above theorem allows defining the space of RVFs withn variables by having continuous CoV of PDrs of order α∈0,1 in a domain X⊂Rn, which can be denoted by CαX,R.
Finally, we can easily extend all of the above results to the VVFs of SRVs.Theorem 12.
Assume thatα∈0,1, f:X⟶Rm be a VVF defined in an OS: X⊂Rn, ∋∀x=x1,…,xn∈X, each xi>0, and the point a=a1,…,an∈X. The function f is αDFat a iff its components are αDF at a, and if these components are f1,f2,…,fm, then the components of Dαfa are the α-derivatives Dαfja, for j=1,2,…,m, i.e.,(26)f=f1,f2,…,fm⇒Dαfa=Dαf1a,Dαf2a,…,Dαfma.Proof.
It is similarly proven as same as traditional calculus by applyingDα instead of derivative.Remark 4.
Assume thatα∈0,1, f:X⟶Rm is a VVF defined in an OS: X⊂Rn, ∋∀x=x1,…,xn∈X, each xi>0, and a point a=a1,…,an∈X. If function f is α-differentiable at a, then ∂α/∂xiαfja exists for i=1,2,…,n, j=1,2,…,m, and the CoV of α-Jacobian of f of order α∈0,1 at a is expressed as(27)Jαfa=∂α∂x1αf1a…∂α∂xnαf1a⋮…⋮∂α∂x1αfma…∂α∂xnαfma.


## 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 thatt∈R and x=x1,…,xn∈Rn. If ft=f1t,…,fnt is αDF at a>0∋α∈0,1and a RVF g, with n variables x1,…,xn, is αDF at fa∈Rn, all fia>0α∈0,1. Then, the composition g∘f is α-differential at a and(28)Tαg∘fa=∑i=1n∂α∂xiαgfa⋅fiaα−1⋅Tαfia.Proof.
Assumeg∈C1Ufa,R, ∋Ufa is the point fa neighborhood. Suppose that ht=g∘ft=gft. By setting u=a+εa1−α in Definition 1, we see that(29)Tαha=limt⟶aht−hat−aa1−α=limt⟶agft−gfat−aa1−α.
Without loss of generality (WLOG),Ufa is assumed to be an open ball (OB), represented by Bfa,r. Since f is a CF, then along with points f1a,…,fna and f1t,…,fnt, the points f1a,f2t,…,fnt,…, f1a,f2a,…,fnt and lines connecting them must also be the ball Bfa,r. In fact, using the classical MVT for differentiable functions (DFs) of one variable is in the following computation [28]:(30)ht−hat−aa1−α=gft−gfat−aa1−α=gf1t,…,fnt−gf1a,f2t,…,fntt−aa1−α+gf1a,f2t,…,fnt−gf1a,f2a,f3t,…,fntt−aa1−α+⋯+gf1a,f2a,…,fnt−gf1a,…,fnat−aa1−α=∂∂x1gc1,f2t,…,fntf1t−f1at−aa1−α=∂∂x2gf1a,c2,…,fntf2t−f2at−aa1−α+⋯+∂∂x2gf1a,c2,…,fntf2t−f2at−aa1−α+⋯+∂∂xngf1a,f2a,…,cnfnt−fnat−aa1−α,where ci is between fia and fit∀i=1,2,…,n.
By taking limits ast⟶a and using the continuity of PDrs of g as well as the fact that ci⟶fia∀i=1,2,…,n, equation (29) is expressed as(31)Tαha=limt⟶aht−hat−aa1−α=limt⟶agft−gfat−aa1−α=limt⟶∞∂∂x1gc1,f2tf1t−f1at−aa1−α+⋯+∂∂x2gf1a,c2,…,fntf2t−f2at−aa1−α+⋯+∂∂xngf1a,f2a1−α,…,cnfnt−fnat−aa1−α+∂∂x1gfa⋅f11a⋅a1−α+∂∂x2gfa⋅f21a⋅a1−α⋅f1aα−1⋅f11a⋅a1−α=∂∂x1gfa⋅f1a1−α⋅f1a1−α⋅f1aα−1⋅f11a⋅a1−α+∂∂x2gfa⋅f2a1−α⋅f2aα−1⋅f21a⋅a1−α+⋯+∂∂xngfa⋅f2a1−α⋅f2aα−1⋅fn1a⋅a1−α=∂α∂x1αgfa⋅f1aα−1⋅Tαf1α+∂α∂x2αgfa⋅f2aα−1⋅Tαf2a+⋯+∂α∂xnαgfa⋅fnaα−1⋅Tαf1α.
Our proof is completely done.Remark 5.
The MF of equation (28) is expressed as(32)Dαg⋅fah=∂α∂x1αgfa,…,∂α∂xnαgfa⋅f1aα−1000⋱000fnaα−1⋅Tαf1a⋮Tαfna⋅h.Theorem 14 (CR).
Letx=x1,…,xn∈Rn and y=y1,…,ym∈Rm. If fx1,…,xn=f1x1,…,xn,…,fmx1,…,xn is a αDF at a=a1,…,an∈Rn, each ai>0∋α∈0,1, and a RVF g with variables y1,…,ym is αDF at fa∈Rn, where all fia>0∋α∈0,1. Then, the composition g∘f is α-differentiable at a, and we have(33)∂α∂xiαg∘fa=∑j=1m∂α∂yjαgfa⋅fjaα−1⋅∂α∂xiαfja∀i=1,2,…,n.Proof.
Proof. According to PDr’s definition and Theorem14, the following is implied:Remark 6.
The MF of equation (33) is expressed as(34)Dαg∘fah=∂α∂y1αgfa,…,∂α∂ymαgfa⋅f1aα−1000⋱000fmaα−1⋅∂α∂x1αf1a…∂α∂xnαf1a…⋯⋯∂α∂x1αfna…∂α∂xnαfna⋅h1⋮hn.


## 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 thatα∈0,1, F:X⟶R is a RVF defined in an OS: X⊂Rn+1, ∋∀x1,…,xn,y∈X, each xi,y>0, and point a1,…,an,b∈X. Suppose that(i)
Fa1,…,an,b=0(ii)
F∈CαX,R(iii)
∂α/∂yαFa1,…,an,b≠0
Then, there is a neighbourhoodU⊂Rn of a1,…,an, such that there is a unique function (UF) y=gx1,…,xn that satisfies the following:
ga1,…,an=b and Fx1,…,xn,gx1,…,xn=0,∀x1,…,xn∈U.
Finally,y=gx1,…,xn is Cα in U, and for every i=1,2,…,n, we have(35)∂α∂xiαgx1,…,xn=−∂α/∂xiαFx1,…,xn,gx1,…,xn∂α/∂yαFx1,…,xn,gx1,…,xn⋅gx1,…,xnα−1.Proof.
WLOG,X is assumed to be an OB, represented by Ba1,…,an,b,ε0. Let ρ∈0,ε0. If we call δ=ε02−ρ2, it is verified that(36)x1,…,xn−a1,…,an<δandy−b<ρ⇒x1,…,xn,y∈Ba1,…,an,b,ε0.
Note that in particular ify−b<ρ, then a1,…,an,y∈Ba1,…,an,b,ε0.
Since∂α/∂yαFa1,…,an,b≠0, it is assumed to be positive (otherwise, −F is considered instead of F. From fact that Fa1,…,an,b=0, it follows that(37)Fa1,…,an,b−ρ>0,Fa1,…,an,b−ρ<0.
By the continuity ofF at a1,…,an,b−ρ and a1,…,an,b+ρ, there exists δ′∈0,δ, such that(38)x1,…,xn−a1,…,an<δ′⇒Fx1,…,xn,b−ρ>0andFx1,…,xn,b+ρ<0.
Since the functiony↦Fx1,…,xn,y is CF on the interval b−ρ,b+ρ, ∀x1,…,xn∈Ba1,…,an,δ′, and via the classical Bolzano’s theorem (BThm), it implies that ∃yx∈b−ρ,b+ρ∋Fx1,…,xn,yx=0, for each x=x1,…,xn. Then, y’s value is unique, since a function, whose derivative is positive, has more than zero. On the other hand, by having U=Ba1,…,an,δ′, for each x1,…,xn∈U, there exists a unique y∋Fx1,…,xn,y=0, we can write y=gx1,…,xn, and then, this function will be proven to be CF on Ba1,…,an,δ′. The continuity of the function g at the point a1,…,an is clear since for each ρ>0, ∃ a value δ′>0∋:(39)x1,…,xn−a1,…,an<δ′⇒b−yx<ρ⇔b−gx1,…,xn<ρ.
The functiong continuity will be proven at any point x1,…,xn∈Ba1,…,an,δ′ by simply substituting Ba1,…,an,δ′ for an OB: Bx1,…,x is contained in Ba1,…,an,δ′.
Finally, let us now show formula (35). By applying Theorem 14 to equation,(40)Fx1,…,xn,y=0,we have∂α∂xiαFx,gx+∂α∂yαFx,gx⋅gxα−1⋅∂α∂xiαgx=0.
∀i=1,2,…,n, ∋x=x1,…,xn. Solving ∂α/∂xiαgx, we get (35). In addition, the formula (35) on the right side is continuous, so the continuity of CoV of PDrs ∂α/∂xiαgx∀, i=1,2,…,n, follows.
Theorem15 will provide us a help in computing CoV of PDrs of implicit function of SVs.Example 2.
Consider(41)Fx,y,z=x3+3y2+4xz2−3yz2−5=0.
This equation’s one solution is1,1,1. F is obviously in Cα which is OB, represented by B1,1,1,ε0, with x,y,z>0, for some α∈0,1, since(42)∂α∂zαF1,1,1=8xz2−α−6yz2−α1,1,1=2≠0.
Theorem15 implies that there is a neighbourhood,U⊂R2, of 1,1∋∃ a UF: z=gx,y satisfies the following:g1,1=1 and Fx,y,gx,y=0,∀x,y∈U.
Moreover,z=gx,y is Cα in U, and we have(43)∂α∂xαgx,y=−3x+4z2x1−α8x−6yz,∂α∂yαgx,y=−6y−3z2y1−α8x−6yz.
Finally, we obtain∂α/∂xαg1,1=−7/2 and ∂α/∂xαg1,1=−3/2.
Finally, CoV of IFThm for a system of several equations and SRVs is obtained.Theorem 16 (Conformable general implicit function theorem).
Letα∈0,1, F:X⟶Rm be a VVF defined in an OS: X⊂Rn+m, ∋∀x;y=x1,…,xn;y1,…,ym∈X, each xi,yj>0, and point a;b=a1,…,an;b1,…,bm∈X. Assume that(i)
Fa;b=0(ii)
F∈CαX,Rm(iii)
detJyαFa;b≠0
Then, there is a neighbourhood,U⊂Rn, of a∋∃a UF: f:U⟶Rm,x↦y=fx that satisfies fa=b and Fx;fx=0,∀x∈U.
Finally,y=fx is class Cα in U, and for every i=1,2,…,n, we have(44)∂αf∂xiαt=−fα−1−1⋅JyαF−1⋅∂αF∂xiαt,where(45)∂αf∂xiα=∂αf1∂xiα,…,∂αfm∂xiα,fα−1=f1α−1…0⋮…⋮0…fmα−1,JyαF=∂αF1∂y1α…∂αF1∂ymα………∂αFm∂y1α…∂αFm∂ymα,∂αF∂xiα=∂αF1∂xiα,…,∂αFm∂xiα.Proof.
The proof of existence and uniqueness of the implicit function is done similar to the traditional multivariable calculus by applying mathematical induction onq 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 as(46)Fx;y=0orF1x1,…,xn;y1,…,ym=0⋮Fmx1,…,xn;y1,…,ym=0,which satisfies hypotheses (i)–(iii) of the theorem; then, this system is defined in a neighbourhood,U⊂Rn, of a, the implicit function y=fx class Cα in U, such that fa=b which satisfies equation (1), i.e.,(47)Fx;fx=0orF1x;f1x,…,fmx=0…Fmx;f1x,…,fmx=0.
By applying CoV of CR to the above equation, we have(48)∂αF1∂xiα+∂αF1∂y1α⋅f1α−1⋅∂αf1∂xiα+⋯+∂αF1∂ymα⋅fmα−1⋅∂αfm∂xiα=0⋮∂αFm∂xiα+∂αFm∂y1α⋅f1α−1⋅∂αf1∂xiα+⋯+∂αFm∂ymα⋅fmα−1⋅∂αfm∂xiα=0or∂αF∂xiα+∑j=1m∂αF∂yjα⋅fjα−1⋅∂αfj∂xiα=0,for all i=1,2,…,n.
In addition, the MF of equation (48) is given as follows:(49)∂αF1∂xiα⋮∂αFm∂xiα=−∂αF1∂y1α…∂αF1∂ymα………∂αFm∂y1α…∂αFm∂ymα⋅f1α−1…0⋮…⋮0…fmα−1⋅∂αf1∂xiα⋮∂αFm∂xiαor∂αF∂xiαt=−JyαF⋅fα−1⋅∂αf∂xiαt.
SinceJyαF and fα−1 are regular matrices by hypothesis, we have(50)∂αf1∂xiα⋮∂αFm∂xiα=−f1α−1…0⋮…⋮0…fmα−1−1⋅∂αF1∂y1α…∂αF1∂ymα………∂αFm∂y1α…∂αFm∂ymα−1⋅∂αF1∂xiα⋮∂αFm∂xiαor∂αf∂xiαt=−fα−1−1⋅JyαF−1⋅∂αF∂xiαt,which completes the proof.
We will now show that Theorem16 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:(51)F1x,y,z,w=x2+y2+z2+w2−6=0,F2x,y,z,w=x2−y2+z2−w2=0.
One solution of this equation isx,y,z,w=1,1,2,2. Clearly, F=F1,F2 is in Cα, which is OB: B1,1,2,2,ε0, with x,y,z,w>0, for some α∈0,1, since(52)detJz,wαF2,2,1,1=det2z2−α2w2−α2z2−α−2w2−α1,1,2,2=−322∝≠0.
Theorem16 implies that there is a neighbourhood,U⊂R2, of 2,2∋∃a UF: f=f1,f2 given by(53)z=f1x,y,w=f2x,y,that satisfies(54)f11,1=2,f21,1=2,F1x,y,f1x,y,f2x,y=0,F2x,y,f1x,y,f2x,y=0,∀x,y∈U.
Moreover,f=f1,f2 is class Cα in U, and we have(55)∂αf1∂xα∂αf2∂xα=−zα−100wα−1−1⋅2z2−α2w2−α2z2−α−2w2−α−1⋅2x2−α2x2−α=−14⋅z1−α00w1−α⋅zα−2zα−2wα−2−wα−2⋅2x2−α2x2−α=−14⋅z−1z−1w−1−w−1⋅2x2−α2x2−α=−x2−αz0∂αf1∂yα∂αf2∂yα=−zα−100wα−1−1⋅2z2−α2w2−α2z2−α−2w2−α−1⋅2y2−α−2y2−α=−14⋅z1−α00w1−α⋅zα−2zα−2wα−2−wα−2⋅2y2−α−2y2−α=−14⋅z−1z−1w−1−w−1⋅2y2−α−2y2−α=0−y2−αw.
Finally, we have(56)∂αf1∂xα∂αf2∂xα1,1,2,2=−120,∂αf1∂yα∂αf2∂yα1,1,2,2=0−12.
According 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. Theα-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.



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