Research ArticleNovel Investigation of Multivariable Conformable Calculus forModeling Scientific Phenomena

MohammedK A Kaabar 12 FranciscoMartınez3 InmaculadaMartınez3 Zailan Siri 1

and Silvestre Paredes3

1Institute of Mathematical Sciences Faculty of Science University of Malaya Kuala Lumpur 50603 Malaysia2Gofa Camp Near Gofa Industrial College and German Adebabay Nifas Silk-Lafto 26649 Addis Ababa Ethiopia3Department of Applied Mathematics and Statistics Technological University of Cartagena Cartagena 30203 Spain

Correspondence should be addressed to Mohammed K A Kaabar mohammedkaabarwsuedu andZailan Siri zailansiriumedumy

Received 11 September 2021 Accepted 29 October 2021 Published 23 November 2021

Academic Editor Antonio Di Crescenzo

Copyright copy 2021 Mohammed K A Kaabar et al is is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium provided the original work isproperly cited

New investigation on the conformable version (CoV) of multivariable calculus is proposede conformable derivative (CoD) of areal-valued function (RVF) of several variables (SVs) and all related properties are investigated An extension to vector-valuedfunctions (VVFs) of several real variables (SRVs) is studied in this work e CoV of chain rule (CR) for functions of SVs is alsointroduced At the end the CoV of implicit function theorem (IFm) for SVs is established All results in this work can bepotentially applied in studying various modeling scenarios in physical oceanography such as Stommelrsquos box model of ther-mohaline circulation and other related models where all our results can provide a new analysis and computational tool toinvestigate these models or their modified formulations

1 Introduction

Many definitions of derivative have been proposed based ontwo categorizations global (nonlocal) and local types In thefirst one the nonlocal fractional derivative (FrDr) is rep-resented via integral transformation or any other relatedtransformations where nonlocality is seen in this type alongwith a memory e RiemannndashLiouville and Caputo frac-tional definitions are considered as the most commonlyknown fractional definitions Several approximate analyticaland numerical methods have been recently developed tosolve nonlocal fractional and local differential equations thatare encountered while modeling various scientific phe-nomena such as the generalized Riccati expansion forsolving the nonlinear KPP equation via FrDr [1] and thetrigonometric quintic B-spline method for solving thenonlinear telegraph equation constructed via CoV [2] enonlinear KleinndashFockndashGordon equation has been analyti-cally solved via two novel techniques such as the methods ofgeneralized Riccati expansion and generalized exponential

function [3] Fractional vector (FV) calculus has been re-cently introduced in [4] to represent the spherical coordi-nates framework using the fractional derivative and integralof Caputo and RiemannndashLiouville senses respectively FVcalculus can be used effectively in modeling processes infractal media and other fractional dynamical systems such ashydrodynamical and electrodynamical systems [4] Frac-tional calculus is a powerful tool in modeling the potentialflow past a sphere of an inviscid fluid [5] which is a veryimportant research problem because this problem particu-larly investigates the Laplacian in three-dimensional space(spherical coordinates) which is used in governing variousphysical and mechanical systems in heat conduction elas-ticity Newtonian gravitational potential ideal fluid flowand electrostatics [6] In addition this problem has beenapplied in finding the stream function solutions for thestokes flow inside viscous sphere in an inviscid extensionalflow [7] e basics of these essential notions are mentionedin [8 9] e local definition is based on certain incrementalratios A new local derivative definition was initially

HindawiJournal of MathematicsVolume 2021 Article ID 3670176 12 pageshttpsdoiorg10115520213670176

formulated by Khalil et al [10 11] which is called con-formable derivative (CoD) e main aim of CoD is to avoidsome obstacles of solving nonlocal fractional differentialequation where the analytical solutions can be very com-plicated to obtain Consequently several research workshave mathematically analyzed some essential notions[10 12ndash16] e SchrodingerndashHirota equation and modifiedKdVndashZakharovndashKuznetsov equation have been solved withthe help of generalized CoDs [17] CoDs have beenemployed in investigating the wick-type stochastic nonlinearevolution equations via the improved technique ofKudryashov [18] (see also the wick-type stochastic KdVequation formulated in the context of generalized CoDs[19]) According to the mathematical investigation of CoDin [20] it is clear that CoD cannot be addressed as afractional derivative erefore in our study we haveaddressed CoD as a modified form of usual derivative whichhas some applications in physics and engineering due to thefact that the measurements in physics are local ereforethis definition is highly applicable in theoretical physicsCoD can still be helpful in many related modeling scenarios

e CoV of analytic functionsrsquo theory has been proposedin [21] In addition new results are investigated on thecontour conformable integral in [22 23] us the defini-tion of the contour conformable integral has been utilized in[23]

Studying conformable derivative and integral is essentialin various fields of natural sciences and engineering eneed for local derivative is highly appreciated in multidis-ciplinary sciences While the nonlocal fractional derivativescan provide a good explanation to the dynamics of certainsystems particularly in modeling epidemic diseases thedifficulty of obtaining exact or analytical solutions to theproblems formulated in the senses of nonlocal fractionalderivatives can make the investigation of fractional-ordersystems a real challenge for researchers us researchershave paid more attention to conformable derivative andother related local derivatives in modeling scientific phe-nomena While there are some recent studies concerning themathematical analysis of conformable calculus such as themultivariable conformable calculus [15] that was introducedin 2018 the behavior of conformable derivatives of functionsin arbitrary Banach spaces [24] that was investigated in 2021the differential geometry of curves [25] that was investigatedin 2019 in the senses of conformable derivatives and inte-grals and the behavioral framework for the conformablelinear differential systemsrsquo stability [26] that was carefullystudied in 2020 to utilize the importance of CoV inmodelingscenarios of control theory and power electronics our re-sults in this work provide a comprehensive investigation ofα-derivative of a function of SVs and all related propertiesthe CoV of CR for functions of SVs and the CoV of IFminvolving many numerical examples to validate our obtainedresults According to the best of our knowledge our originalinvestigation in this article provides an essential mathe-matical analysis tool for researchers working on modelingphenomena in physics and engineering in the sense ofconformable calculus because all theorems and properties inthis work will be needed in such modeling scenarios

e article consists of the following sections essentialnotions of the CoV of calculus are mentioned in Section 2en the α-derivative of RVF of SVs is investigated and allits main properties are established in Section 3 Further-more these results are extended to the VVFs of SRVs Inaddition the CR for functions of SVs is introduced in twoparticular cases in Section 4 In the last part the CoV ofIFm for SVs is obtained in Section 5 by first establishingthe conformable theorem of existence and regularity of theimplicit function for single equation Second this result isextended to a system of several equations and SRVs Someconcluding remarks are specified in Section 6

2 Fundamental Notions

Definition 1 (See [10]) For a function f [0infin)⟶ R theαth order CoD can be written as

Tαf( 1113857(t) limϵ⟶0

f t + εt1minus α1113872 1113873 minus f(t)

ε (1)

forall tgt 0 α isin (0 1] If f is α-differentiable function(α DF) in some (0 a) agt 0 and lim

t⟶0+(Tαf)(t) exists then

it is expressed as

Tαf( 1113857(0) limt⟶0+

Tαf( 1113857(t) (2)

Theorem 1 (See [10]) If f [0infin)⟶ R is α DF at t0 gt 0α isin (0 1] then f is continuous function (CF) at t0

Theorem 2 (See [10]) Assuming that α isin (0 1] and f g areα DF s at a point tgt 0 we have

(i) Tα(af + bg) a (Tαf) + b (Tαg) foralla b isin R(ii) Tα(tp) ptpminus α forallp isin R(iii) Tα(λ) 0 forall constant functions f(t) λ(iv) Tα(fg) f(Tαg) + g(Tαf)

(v) Tα(fg) g(Tαf) minus f(Tαg)g2

(vi) If we suppose that f is differentiable then(Tαf)(t) t1minus αdfdt(t)

From Definition 1 the CoD of some functions areexpressed as

(i) Tα(1) 0(ii) Tα(sin(at)) at1minus α cos(at)

(iii) Tα(cos(at)) minus at1minus α sin(at)

(iv) Tα(eat) at1minus αeat a isin R

Definition 2 (See [11]) e left CoD beginning from a offunction f [ainfin)⟶ R of order α isin (0 1] is expressed as

Taαf( 1113857(t) lim

ε⟶0

f t + ε(t minus a)1minus α

1113872 1113873 minus f(t)

ε tgt a (3)

For a 0 it is expressed as (Tαf)(t) If f is α DF insome (a b) then we set

2 Journal of Mathematics

Taαf( 1113857(a) lim

t⟶a+T

aαf( 1113857(t) (4)

Theorem 3 (See [12]) Suppose that f g (ainfin)⟶ R areleft α DF s where α isin (0 1] Let us assume thath(t) f(g(t)) h(t) is α DF forall tne a and g(t) ne 0 and we get

Taαh( 1113857(t) T

aαf( 1113857(g(t)) middot T

aαg( 1113857(t) middot (g(t))

αminus 1 (5)

If t a then we obtain

Taαh( 1113857(a) lim

t⟶a+T

aαf( 1113857(g(t)) middot T

aαg( 1113857(t) middot (g(t))

αminus 1

(6)

Theorem 4 (Rollersquos theorem (RT) [10]) Suppose that agt 0α isin (0 1] and function f [a b]⟶ R satisfies

(i) f is CF on [a b]

(ii) f is α DF on (a b)

(iii) f(a) f(b)

en exist c isin (a b) ni(Tαf)(c) 0

Theorem 5 (Mean value theorem (MVT) [10]) Assume thatagt 0 α isin (0 1] and function f [a b]⟶ R satisfies

(i) f is CF on [a b]

(ii) f is α DF on (a b)

en exist c isin (a b) ni we have

Tαf( 1113857(c) f(b) minus f(a)

bαα minus a

αα (7)

Theorem 6 (Modified mean value theorem (MMVT)[27]) Assume that agt 0 α isin (0 1] and functionf [a b]⟶ R satisfies

(i) f is CF on [a b]

(ii) f is α DF on (a b)

en exist c isin (a b) ni we have

Tαf( 1113857(c)

c1minus αα

f(b) minus f(a)

(bα) minus (aα) (8)

Theorem 7 (See [13]) Suppose that agt 0 α isin (0 1] andfunction f [a b]⟶ R satisfies

(i) f is CF on [a b]

(ii) f is α DF on (a b)

en we get

(i) If (Tαf)(t)gt 0 forall t isin (a b) then f is increasingon[a b]

(ii) If (Tαf)(t)lt 0 forallt isin (a b) then f is decreasingon[a b]

Let us express the CoV of partial derivative (PDr) of areal-valued function (RVF) with SVs as follows

Definition 3 (See [14 15]) Assume that f is a RVF with n

variables and there is a point a (a1 an) isin Rn whereits ith component is positive en the limit is written as

limε⟶0

f a1 ai + εai1minus α

an1113872 1113873 minus f a1 an( 1113857

ε (9)

If the above limit exists the ith CoV of PDr of f of theorder α isin (0 1] at a represented by zαzxα

i f(a)

3 α-Derivative of a RVF of SVs

Definition 4 Suppose that f is a RVF with n variablesx1 xn and α isin (0 1] en we say that f is α DF ata (a1 an) isin Rn each ai gt 0 if any of the three con-ditions which are equivalent to each other is verified

(i) ere is a linear transformation L Rn⟶ R suchthat

limh⟶0

f a1 + h1a11minus α

an + hnan1minus α

1113872 1113873 minus f a1 an( 1113857 minus L(h)

h 0 (10)

Where h (h1 hn) h h21+middotmiddotmiddot+h2

n

1113969 and

α isin (0 1](ii) ere is a linear transformation L Rn⟶ R and a

function ε h⟶ ε(h) such that

f a1 + h1a11minus α

an + hnan1minus α

1113872 1113873 minus f a1 an( 1113857 L(h) + ε(h)h (11)

And limh⟶0ε(h) 0

Journal of Mathematics 3

(iii) ere is a linear transformation L Rn⟶ R and n

functions εi h⟶ εi(h) forall

i 1 2 n ni

f a1 + h1a11minus α

an + hnan1minus α

1113872 1113873 minus f a1 an( 1113857 L(h) + 1113944n

i1εi(h)hi

(12)

And lim εih⟶ 0(h) 0 for i 1 2 n

e linear transformation L Rn⟶ R is defined byL(h) 1113936

ni1 αihi with h (h1 hn) and α1 αn isin R

is linear transformation is denoted by Dαf(a) which iscalled CoD of f of the order α isin (0 1] at a

Remark 1 e equivalence of conditions (i) and (ii) isimmediate since

limh⟶0

ε(h) 0harrε(h)h o(h) (13)

To see the equivalence between conditions (ii) and (iii)we take

εi ε(h)hi

h

ε(h) 1

h1113944

ni1εi(h)hi (14)

As |hih|le 1 then we have the following

(i) If limεh⟶0(h) 0 then lim εi h⟶ 0(h) 0(ii) If lim εi h⟶ 0(h) 0 for i 1 n then we obtain

limh⟶0

ε(h)le limh⟶0

1h

1113944

n

i1εi(h)

le lim

h⟶01113944

n

i1εi(h)

0

(15)

ie limh⟶0ε(h) 0 Hence the conditions (ii) and (iii) areequivalent

Example 1 Consider a function f defined by f(x y) ex minus

2cosy and a point (a b) isin R2 with agt 0 and bgt 0 thenDαf(a b)(h1 h2) h1a

1minus αeα + 2h2b1minus α sin b

Solution to prove this let us note that

limh1 h2( )⟶(00)

f a + a1minus α

h1 b + b1minus α

h21113872 1113873 minus f(a b) minus L h1 h2( 1113857

h1 h2( 1113857

limh1 h2( )⟶(00)

ea+a1minus αh1

minus 2 cos b + b1minus α

h21113872 1113873 minus ea

minus 2 cos b( 1113857 minus h1a1minus α

ea

+ 2h2b1minus α sin b1113872 1113873

h21 + h

22

1113969

le limh1⟶0

ea+a1minus αh1

minus ea

minus h1a1minus α

ea

h1minus 2 lim

h2⟶0

cos b + b1minus α

h21113872 1113873 minus cos b + b1minus α sin b

h2

limh1⟶0

ea+a1minus αh1

minus ea

h1minus a

1minus αe

a⎛⎝ ⎞⎠ minus 2 limh2⟶0

cos b + b1minus α

h21113872 1113873 minus cos b

h2+ b

1minus α sin b⎛⎝ ⎞⎠ a1minus α

ea

minus a1minus α

ea

1113872 1113873

minus 2 minus b1minus α sin b + b

1minus α sin b1113872 1113873 0

(16)

Theorem 8 If a RVF f with n variables α DF ata (a1 an) isin Rn each ai gt 0 then f is CF at a isin Rn

Proof Since f is α DF at a we can write the following

f a1 + h1a11minus α

an + hnan1minus α

1113872 1113873 minus f a1 an( 1113857 1113944n

i1αihi + o(h) (17)

4 Journal of Mathematics

By taking the limits of the two sides of the equality ash⟶ 0 we have

limh⟶0

f a1 + h1a11minus α

an + hnan1minus α

1113872 1113873 f a1 an( 1113857

(18)

Hence f is CF at a isin Rn

Theorem 9 If a RVF f with n variables is α DF ata (a1 an) isin Rn each ai gt 0 then zαzxα

i f(a) exists for1le ile n and the CoD of f of the order α isin (0 1] is expressedas

Dαf(a)(h) 1113944

n

i1

zα

zxαi

f(a)hi (19)

where h (h1 hn)

Proof By setting hj 0foralljne i in the formula (12) we have

f a1 ai + hiai1minus α

+ middot middot middot + an1113872 1113873 minus f a1 an( 1113857 αihi + εi(h)hi

(20)

By multiplying by 1hi we can write

f a1 ai + hiai1minus α

+ middot middot middot + an1113872 1113873 minus f a1 an( 1113857

hi

αi + εi(h)

(21)

By taking the limits of the two sides of the equality ashi⟶ 0 we have

αi zα

zxαi

f(a) foralli 1 2 n (22)

Finally by substituting the values above αi in the formulaDαf(a)(h) 1113936

ni1 αihi the result is followed

Corollary 1 If a RVF f with n variables is α DF ata (a1 an) isin Rn each ai gt 0 then Dαf(a) is unique

Remark 2 If a RVF f with n variables is α DF ata (a1 an) isin Rn each ai gt 0 then the CoV of gradientof f of the order α isin (0 1] at a is

nablaαf(a) zα

zxα1

f(a) zα

zxαn

f(a)1113888 1113889 (23)

Also the matrix form (MF) of equation (19) is given asfollows

Dαf(a)(h) nablaαf(a) middot h

zα

zxα1

f(a) zα

zxαn

f(a)1113888 1113889 middot

h1

⋮

hn

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(24)

Theorem 10 Let α isin (0 1] f g X⟶ R be a RVF definedin an open set (OS) X sub Rn ni forall x (x1 xn) isin X eachxi gt 0 and a point a (a1 an) isin X If f g are α DF at athen we have

(i) Dα(λf + μg)(a) λDα(f)(a) + μDα(g)(a)forall λ μ isin R

(ii) Dα(fg)(a) Dα(f)(a) middot g(a) + f(a)Dα(g)(a)

Proof (i) follows fromDefinition 4 thus it follows the proofof (i)

For (ii) let A (a1 + h1a1minus α1 an + hna1minus α

n ) and thenwe have

limh⟶0

((fg)(A) minus (fg)(a)) minus Dαf(a) middot g(a) + f(a) middot D

αg(a)( 1113857(h)

h

limh⟶0

(f(A) minus f(a)) minus Dαf(a)(h)

hmiddot g(a) + f(a) middot

(g(A) minus g(a)) minus Dαg(a)(h)

h1113888 1113889 + lim

h⟶0

(f(A) minus f(a)) middot (g(A) minus g(a))

h

0 + 0 + limh⟶0

Dαf(a)(h)( 1113857 middot D

αg(a)(h)( 1113857

h lim

h⟶0D

αf(a)

hh

1113888 11138891113888 1113889 middot Dαg(a)

hh

1113888 11138891113888 1113889 middot h 0

(25)

Theorem 11 Let α isin (0 1] f X⟶ R be a RVF defined inan OS X sub Rn niforall x (x1 xn) isin X each xi gt 0 and apoint a (a1 an) isin X If the function f has all CoV ofPDrs of the order α at each point of a neighbourhood of thepoint a U(a) with U(a) sub X and they are continuous at athen f is α DF at a

Proof See eorem 21 proof in [27]

Remark 3 e above theorem allows defining the space ofRVFs with n variables by having continuous CoV of PDrs oforder α isin (0 1] in a domain X sub Rn which can be denotedby Cα(X R)

Finally we can easily extend all of the above results to theVVFs of SRVs

Theorem 12 Assume that α isin (0 1] f X⟶ Rm be a VVFdefined in an OS X sub Rn niforallx (x1 xn) isin X each

Journal of Mathematics 5

xi gt 0 and the point a (a1 an) isin X e function f isα DF at a iff its components are α DF at a and if these

components are f1 f2 fm then the components ofDαf(a) are the α-derivativesDαfj(a) for j 1 2 m ie

f f1 f2 fm( 1113857rArr Dαf(a) D

αf1(a) D

αf2(a) D

αfm(a)( 1113857 (26)

Proof It is similarly proven as same as traditional calculusby applying Dα instead of derivative

Remark 4 Assume that α isin (0 1] f X⟶ Rm is a VVFdefined in an OS X sub Rn niforall x (x1 xn) isin X eachxi gt 0 and a point a (a1 an) isin X If function f isα-differentiable at a then zαzxα

i fj(a) exists fori 1 2 n j 1 2 m and the CoV of α-Jacobian off of order α isin (0 1] at a is expressed as

Jαf(a)

zα

zxα1f1(a)

zα

zxαn

f1(a)

⋮ ⋮

zα

zxα1fm(a)

zα

zxαn

fm(a)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(27)

4 The Chain Rule

Let us prove the CR for the functions of SVs in 2 particularcases For the proofrsquos purpose the continuityrsquos hypothesis ofPDrs is given [28]

Theorem 13 (CR) Suppose that t isin R andx (x1 xn) isin Rn If f(t) (f1(t) fn(t)) is α DF

at agt 0 ni α isin (0 1] and a RVF g with n variablesx1 xn is α DF at f(a) isin Rn all fi(a)gt 0 α isin (0 1] en the composition g ∘f is α-differential at a and

Tαg ∘ f( 1113857(a) 1113944n

i1

zα

zxαi

g(f(a)) middot fi(a)( 1113857αminus 1

middot Tαfi( 1113857(a)

(28)

Proof Assumeg isin C1(U(f(a)) R) niU(f(a)) is the pointf(a) neighborhood Suppose thath(t) (g ∘ f)(t) g(f(t)) By setting u a + εa1minus α inDefinition 1 we see that

Tαh( 1113857(a) limt⟶a

h(t) minus h(a)

t minus aa1minus α

limt⟶a

g(f(t)) minus g(f(a))

t minus aa1minus α

(29)

Without loss of generality (WLOG) U(f(a)) is assumedto be an open ball (OB) represented by B(f(a) r) Since f isa CF then along with points (f1(a) fn(a)) and(f1(t) fn(t)) the points(f1(a) f2(t) fn(t)) (f1(a) f2(a) fn(t))

and lines connecting them must also be the ball B(f(a) r)In fact using the classical MVT for differentiable functions(DFs) of one variable is in the following computation [28]

h(t) minus h(a)

t minus aa1minus α

g(f(t)) minus g(f(a))

t minus aa1minus α

g f1(t) fn(t)( 1113857 minus g f1(a) f2(t) fn(t)( 1113857

t minus aa1minus α

+g f1(a) f2(t) fn(t)( 1113857 minus g f1(a) f2(a) f3(t) fn(t)( 1113857

t minus aa1minus α

+ middot middot middot

+g f1(a) f2(a) fn(t)( 1113857 minus g f1(a) fn(a)( 1113857

t minus aa1minus α

z

zx1g c1 f2(t) fn(t)( 1113857

f1(t) minus f1(a)

t minus aa1minus α

z

zx2g f1(a) c2 fn(t)( 1113857

f2(t) minus f2(a)

t minus aa1minus α

+ middot middot middot

+z

zx2g f1(a) c2 fn(t)( 1113857

f2(t) minus f2(a)

t minus aa1minus α

+ middot middot middot

+z

zxn

g f1(a) f2(a) cn( 1113857fn(t) minus fn(a)

t minus aa1minus α

(30)

6 Journal of Mathematics

where ci is between fi(a) and fi(t) forall i 1 2 n By taking limits as t⟶ a and using the continuity ofPDrs of g as well as the fact that ci⟶ fi(a) foralli 1 2 n equation (29) is expressed as

Tαh( 1113857(a) limt⟶a

h(t) minus h(a)

t minus aa1minus α

limt⟶a

g(f(t)) minus g(f(a))

t minus aa1minus α

limt⟶infin

z

zx1g c1 f2(t)( 1113857

f1(t) minus f1(a)

t minus aa1minus α

+ middot middot middot1113888 1113889

+z

zx2g f1(a) c2 fn(t)( 1113857

f2(t) minus f2(a)

t minus aa1minus α

+ middot middot middot

+z

zxn

g f1(a) f2(a)1minus α

cn1113872 1113873fn(t) minus fn(a)

t minus aa1minus α

1113889 +z

zx1g(f(a)) middot f

(1)1 (a) middot a

1minus α

+z

zx2g(f(a)) middot f

(1)2 (a) middot a

1minus αmiddot f1(a)

αminus 1middot f

(1)1 (a) middot a

1minus α

z

zx1g(f(a)) middot f1(a)

1minus αmiddot f1(a)

1minus αmiddot f1(a)

αminus 1middot f

(1)1 (a) middot a

1minus α+

z

zx2g(f(a)) middot f2(a)

1minus α

middot f2(a)αminus 1

middot f(1)2 (a) middot a

1minus α+ middot middot middot +

z

zxn

g(f(a)) middot f2(a)1minus α

middot f2(a)αminus 1

middot f(1)n (a) middot a

1minus α

zα

zxα1

g(f(a)) middot f1(a)αminus 1

middot Tαf1( 1113857(α) +zα

zxα2

g(f(a)) middot f2(a)αminus 1

middot Tαf2( 1113857(a) + middot middot middot

+zα

zxαn

g(f(a)) middot fn(a)αminus 1

middot Tαf1( 1113857(α)

(31)

Our proof is completely done Remark 5 e MF of equation (28) is expressed as

Dα(g middot f)(a)(h)

zα

zxα1

g(f(a)) zα

zxαn

g(f(a))1113888 1113889

middot

f1(a)αminus 1 0 0

0 ⋱ 0

0 0 fn(a)αminus 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠middot

Tαf1( 1113857(a)

⋮

Tαfn( 1113857(a)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠middot h

(32)

Theorem 14 (CR) Let x (x1 xn) isin Rn andy (y1 ym) isin Rm If f(x1 xn) (f1(x1

xn) fm(x1 xn)) is a α DF at a (a1 an) isin Rneach ai gt 0 ni α isin (0 1] and a RVF g with variables y1 ym

is α DF at f(a) isin Rn where allfi(a)gt 0 niα isin (0 1]en thecomposition g ∘ f is α-differentiable at a and we have

zα

zxαi

(g ∘ f)(a) 1113944m

j1

zα

zyαj

g(f(a)) middot fj(a)αminus 1

middotzα

zxαi

fj(a)

forall i 1 2 n

(33)

Journal of Mathematics 7

Proof Proof According to PDrrsquos definition and eorem14 the following is implied

Remark 6 e MF of equation (33) is expressed as

Dα(g ∘ f)(a)(h)

zα

zyα1

g(f(a)) zα

zyαm

g(f(a))1113888 1113889 middot

f1(a)αminus 1 0 0

0 ⋱ 0

0 0 fm(a)αminus 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠middot

zα

zxα1f1(a)

zα

zxαn

f1(a)

middot middot middot middot middot middot

zα

zxα1fn(a)

zα

zxαn

fn(a)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

middot

h1

⋮

hn

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (34)

5 Conformable Implicit Function Theorem

In this section the CoV of IFm for SVs is obtained Firstwe establish the conformable theorem of existence andregularity of the implicit function for the case of a singleequation

Theorem 15 (Conformable implicit function theorem forthe case of a single equation) Suppose that α isin (0 1]F X⟶ R is a RVF defined in an OS X sub Rn+1 ni forall(x1 xn y) isin X each xi ygt 0 and point(a1 an b) isin X Suppose that

(i) F(a1 an b) 0(ii) F isin Cα(X R)

(iii) zαzyαF(a1 an b)ne 0

en there is a neighbourhood U sub Rn of (a1 an)such that there is a unique function (UF) y g(x1 xn)

that satisfies the followingg(a1 an) b and F(x1 xn g(x1 xn))

0 forall(x1 xn) isin UFinally y g(x1 xn) is Cα in U and for every

i 1 2 n we have

zα

zxαi

g x1 xn( 1113857 minuszαzx

αi F x1 xn( 1113857 g x1 xn( 1113857( 1113857( 1113857

zαzy

αF x1 xn( 1113857 g x1 xn( 1113857( 1113857( 1113857 middot g x1 xn( 1113857( 1113857

αminus 1 (35)

Proof WLOG X is assumed to be an OB represented byB((a1 an b) ε0) Let ρ isin (0 ε0) If we call δ

ε20 minus ρ21113969

it is verified that

x1 xn( 1113857 minus a1 an( 1113857

lt δ and |y minus b|lt ρ1113960 1113961rArr x1 xn y( 1113857 isin B a1 an b( 1113857 ε0( 1113857 (36)

Note that in particular if |y minus b|lt ρ then(a1 an y) isin B((a1 an b) ε0)

Since zαzyαF(a1 an b)ne 0 it is assumed to bepositive (otherwise minus F is considered instead of F) From factthat F(a1 an b) 0 it follows that

F a1 an b minus ρ( 1113857gt 0

F a1 an b minus ρ( 1113857lt 0(37)

By the continuity of F at (a1 an b minus ρ) and(a1 an b + ρ) there exists δprime isin (0 δ) such that

x1 xn( 1113857 minus a1 an( 1113857

lt δprimerArr F x1 xn b minus ρ( 1113857gt 0 andF x1 xn b + ρ( 1113857lt 01113858 1113859 (38)

Since the function y↦F(x1 xn y) is CF on the in-terval [b minus ρ b + ρ ] forall (x1 xn) isin B((a1 an) δprime) andvia the classical Bolzanorsquos theorem (Bm) it implies thatexistyx isin ( b minus ρ b + ρ) niF(x1 xn yx) 0 for eachx (x1 xn) en yrsquos value is unique since a functionwhose derivative is positive has more than zero On the other

hand by having U B((a1 an) δprime) for each(x1 xn) isin U there exists a unique y

niF(x1 xn y) 0 we can write y g(x1 xn) andthen this function will be proven to be CF onB((a1 an) δprime)e continuity of the function g at the point(a1 an) is clear since for each ρgt 0 exist a value δprime gt 0 ni

8 Journal of Mathematics

x1 xn( 1113857 minus a1 an( 1113857

lt δprimerArr b minus yx

11138681113868111386811138681113868111386811138681113868

lt ρhArr b minus g x1 xn( 11138571113868111386811138681113868

1113868111386811138681113868lt ρ(39)

e function g continuity will be proven at any point(x1 xn) isin B((a1 an) δprime) by simply substitutingB((a1 an) δprime) for an OB B((x1 x)) is contained inB((a1 an) δprime)

Finally let us now show formula (35) By applyingeorem 14 to equation

F x1 xn y( 1113857 0 we have

zα

zxαi

F(x g(x)) +zα

zyα F(x g(x)) middot g(x)

αminus 1middot

zα

zxαi

g(x) 0

(40)

forall i 1 2 n nix (x1 xn) Solving zαzxαi g(x)

we get (35) In addition the formula (35) on the right side iscontinuous so the continuity of CoV of PDrs zαzxα

i g(x) foralli 1 2 n follows

eorem 15 will provide us a help in computing CoV ofPDrs of implicit function of SVs

Example 2 Consider

F(x y z) x3

+ 3y2

+ 4xz2

minus 3yz2

minus 5 0 (41)

is equationrsquos one solution is (1 1 1) F is obviously inCα which is OB represented by B((1 1 1) ε0) withx y zgt 0 for some α isin (0 1] since

zα

zzα F(1 1 1) 8xz

2minus αminus 6yz

2minus α1113872 11138731113961

(111) 2ne 0 (42)

eorem 15 implies that there is aneighbourhood U sub R2 of (1 1) niexist a UF z g(x y)

satisfies the followingg(1 1) 1 andF(x y g(x y)) 0 forall(x y) isin U

Moreover z g(x y) is Cα in U and we have

zα

zxα g(x y) minus

3x + 4z2

1113872 1113873x1minus α

(8x minus 6y)z

zα

zyα g(x y) minus

6y minus 3z2

1113872 1113873y1minus α

(8x minus 6y)z

(43)

Finally we obtain zαzxαg(1 1) minus (72) andzαzxαg(1 1) minus (32)

Finally CoV of IFm for a system of several equationsand SRVs is obtained

Theorem 16 (Conformable general implicit functiontheorem) Let α isin (0 1] F X⟶ Rm be a VVF defined inan OS X sub Rn+m ni forall (x y) (x1 xn y1 ym) isin Xeach xi yj gt 0 and point(a b) (a1 an b1 bm) isin X Assume that

(i) F(a b) 0(ii) F isin Cα(X Rm )

(iii) det[JαyF(a b)]ne 0

en there is a neighbourhood U sub Rn of a niexist a UFf U⟶ Rm x↦ y f(x) that satisfies f(a) b andF(x f(x) ) 0 forallx isin U

Finally y f(x) is class Cα in U and for everyi 1 2 n we have

zαfzxα

i

1113890 1113891

t

minus fαminus 11113872 1113873

minus 1middot J

αyF1113872 1113873

minus 1middot

zαFzxα

i

1113890 1113891

t

(44)

where

zαf

zxαi

1113890 1113891 zαf1

zxαi

zαfm

zxαi

1113888 1113889

fαminus 1

fαminus 11 0

⋮ ⋮

0 fαminus 1m

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

JαyF

zαF1

zyα1

zαF1

zyαm

zαFm

zyα1

zαFm

zyαm

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

zαF

zxαi

1113890 1113891 zαF1

zxαi

zαFm

zxαi

1113888 1113889

(45)

Proof e proof of existence and uniqueness of the implicitfunction is done similar to the traditional multivariablecalculus by applying mathematical induction on q and usingthe conformable implicit function theorem for severalvariables [28]

Let us now show formula (44) Assume that a systemwith several equations and SRVs is expressed as

F(x y) 0 or

F1 x1 xn y1 ym( 1113857 0

⋮

Fm x1 xn y1 ym( 1113857 0

⎧⎪⎪⎨

⎪⎪⎩ (46)

which satisfies hypotheses (i)ndash(iii) of the theorem then thissystem is defined in a neighbourhood U sub Rn of a theimplicit function y f(x) class Cα in U such that f(a) bwhich satisfies equation (1) ie

F(x f(x)) 0 or

F1 x f1(x) fm(x)( 1113857 0

Fm x f1(x) fm(x)( 1113857 0

⎧⎪⎪⎨

⎪⎪⎩(47)

By applying CoV of CR to the above equation we have

Journal of Mathematics 9

zαF1

zxαi

+zαF1

zyα1

middot fαminus 11 middot

zαf1

zxαi

+ middot middot middot +zαF1

zyαm

middot fαminus 1m middot

zαfm

zxαi

0

⋮

zαFm

zxαi

+zαFm

zyα1

middot fαminus 11 middot

zαf1

zxαi

+ middot middot middot +zαFm

zyαm

middot fαminus 1m middot

zαfm

zxαi

0

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

orzαF

zxαi

+ 1113944m

j1

zαF

zyαj

middot fαminus 1j middot

zαfj

zxαi

0

(48)

for all i 1 2 n In addition the MF of equation (48) is given as follows

zαF1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus

zαF1

zyα1

zαF1

zyαm

zαFm

zyα1

zαFm

zyαm

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

middot

fαminus 11 0

⋮ ⋮

0 fαminus 1m

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠middot

zαf1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

orzαFzxα

i

1113890 1113891

t

minus JαyF middot fαminus 1

middotzαfzxα

i

1113890 1113891

t

(49)

Since JαyF and fαminus 1 are regular matrices by hypothesis we

have

zαf1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus

fαminus 11 0

⋮ ⋮

0 fαminus 1m

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus 1

middot

zαF1

zyα1

zαF1

zyαm

zαFm

zyα1

zαFm

zyαm

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus 1

middot

zαF1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

orzαfzxα

i

1113890 1113891

t

minus fαminus 11113872 1113873

minus 1middot J

αyF1113872 1113873

minus 1middot

zαFzxα

i

1113890 1113891

t

(50)

which completes the proofWe will now show that eorem 16 can be used to

compute CoV of PDrs of systems with several equations andSRVs

Example 3 Consider a system of two equations and two realvariables

F1(x y z w) x2

+ y2

+ z2

+ w2

minus 6 0

F2(x y z w) x2

minus y2

+ z2

minus w2

0

⎧⎨

⎩ (51)

One solution of this equation is(x y z w) (1 1

2

radic

2

radic) Clearly F (F1 F2) is in Cα

which is OB B((1 12

radic

2

radic) ε0) with x y z wgt 0 for

some α isin (0 1] since

10 Journal of Mathematics

det JαzwF((

2

radic

2

radic 1 1))1113960 1113961 det

2z2minus α 2w2minus α

2z2minus α minus 2w2minus α⎛⎝ ⎞⎠⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

(112

radic

2

radic)

minus322propne 0 (52)

eorem 16 implies that there is aneighbourhood U sub R2 of (

2

radic

2

radic) niexist a UF f (f1 f2)

given by

z f1(x y)

w f2(x y)1113896 (53)

that satisfies

f1(1 1) 2

radic

f2(1 1) 2

radic

⎧⎨

⎩

F1 x y f1(x y) f2(x y)( 1113857 0

F2 x y f1(x y) f2(x y)( 1113857 0forall(x y) isin U1113896

(54)

Moreover f (f1 f2) is class Cα in U and we have

zαf1

zxα

zαf2

zxα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ minus

zαminus 1 0

0 wαminus 1⎛⎝ ⎞⎠

minus 1

middot2z2minus α 2w2minus α

2z2minus α minus 2w2minus α⎛⎝ ⎞⎠

minus 1

middot2x

2minus α

2x2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz1minus α 0

0 w1minus α

⎛⎜⎝ ⎞⎟⎠ middotzαminus 2

zαminus 2

wαminus 2

minus wαminus 2

⎛⎜⎝ ⎞⎟⎠

middot2x

2minus α

2x2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz

minus 1z

minus 1

wminus 1

minus wminus 1

⎛⎜⎝ ⎞⎟⎠ middot2x

2minus α

2x2minus α

⎛⎜⎝ ⎞⎟⎠ minus

x2minus α

z

0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

zαf1

zyα

zαf2

zyα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minuszαminus 1 0

0 wαminus 1⎛⎝ ⎞⎠

minus 1

middot2z2minus α 2w2minus α

2z2minus α minus 2w2minus α⎛⎝ ⎞⎠

minus 1

middot2y

2minus α

minus 2y2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz1minus α 0

0 w1minus α

⎛⎜⎝ ⎞⎟⎠ middotzαminus 2

zαminus 2

wαminus 2

minus wαminus 2

⎛⎜⎝ ⎞⎟⎠

middot2y

2minus α

minus 2y2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz

minus 1z

minus 1

wminus 1

minus wminus 1

⎛⎜⎝ ⎞⎟⎠ middot2y

2minus α

minus 2y2minus α

⎛⎜⎝ ⎞⎟⎠

0

minusy2minus α

w

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(55)

Finally we have

zαf1

zxα

zαf2

zxα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(112

radic

2

radic)

minus12

radic

0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

zαf1

zyα

zαf2

zyα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(112

radic

2

radic)

0

minus12

radic

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(56)

According to all of our numerical examples in this workwith the help of our investigation to all proposed theoremsand properties in this article the numerical examples showthat all our obtained results are valid and efficient As seen inall these examples the computations are simple and cost-

efficient which are important when scientific phenomenaare modelled in the sense of conformable calculus All in allthe simplicity of computations is always needed in modelingapplications in comparison to other complicated compu-tations that are needed using other approaches in otherfractional operators All our obtained results in this work areaccurate because our results coincide with the usual integer-order results

6 Conclusion

Anew investigation on the CoV ofmultivariable calculus hasbeen discussed in this work in detail e α-derivative of afunction of SVs and all related properties have been in-vestigatede CoV of CR for functions of SVs has also beenstudied e CoV of IFm has been presented in the lastpart of our results and numerical experiments have beenconducted to support our theoretical results e findings ofthis investigations show that all results formulated via CoD

Journal of Mathematics 11

coincide with the ones in the traditional integer case Allresults in this work can be potentially applied in modelingoceanographic phenomena such as Stommelrsquos box model ofthermohaline circulation and other related models wherethis studyrsquos analysis can be further extended or generalizedin 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

Authorsrsquo Contributions

MohammedK A Kaabar involved in actualization developedmethodology performed formal analysis validated and in-vestigated the study prepared the initial draft and supervisedand edited the original draft Francisco Martınez InmaculadaMartınez and Silvestre Paredes involved in actualizationdeveloped methodology performed formal analysis validatedand investigated the study and prepared the initial draftZailan Siri developed methodology performed formal anal-ysis validated and investigated the study and prepared theinitial draft All authors read and approved the final version

References

[1] M M A Khater M S Mohamed and R A M Attia ldquoOnsemi analytical and numerical simulations for a mathematicalbiological model the time-fractional nonlinear Kolmogor-ovndashPetrovskiindashPiskunov (KPP) equationrdquo Chaos Solitons ampFractals vol 144 Article ID 110676 2021

[2] M A Khater K S Nisar and M S Mohamed ldquoNumericalinvestigation for the fractional nonlinear space-time telegraphequation via the trigonometric Quintic B-spline schemerdquoMathematical Methods in the Applied Sciences vol 44 no 6pp 4598ndash4606 2021

[3] M M A Khater M S Mohamed and S K Elagan ldquoDiverseaccurate computational solutions of the nonlinearKleinndashFockndashGordon equationrdquo Results in Physics vol 23Article ID 104003 2021

[4] M F Li J R Ren and T Zhu ldquoFractional vector calculus andfractional special functionrdquo arXiv preprint10012889 2010

[5] D J Wollkind and B J Dichone Comprehensive AppliedMathematical Modeling in the Natural and Engineering Sci-ences Springer International Publishing Berlin Germany2017

[6] P J Olver Introduction to Partial Differential EquationsSpringer International Publishing Berlin Germany 2014

[7] J D Krehbiel and J B Freund ldquoStokes flow inside a sphere inan inviscid extensional flowrdquo Zeitschrift fur AngewandteMathematik und Physik vol 68 no 4 pp 68ndash81 2017

[8] A A Kilbas H M Srivastava and J J Trujillo ldquoeory andapplications of fractional differential equationsrdquo in North-Holland Mathematics StudiesVol 204 Elsevier AmsterdamNetherlands 2006

[9] K S Miller An Introduction to Fractional Calculus andFractional Differential Equations John Wiley and SonsHoboken NJ USA 1993

[10] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014

[11] R Khalil M Al Horani and M Abu Hammad ldquoGeometricmeaning of conformable derivative via fractional cordsrdquo eJournal of Mathematics and Computer Science vol 19 no 4pp 241ndash245 2019

[12] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journalof Computational and Applied Mathematics vol 279pp 57ndash66 2015

[13] O S Iyiola and E R Nwaeze ldquoSome new results on the newconformable fractional calculus with application usingDprimeAlambert approachrdquo Progress in Fractional Differentiationand Applications vol 2 no 2 pp 115ndash122 2016

[14] A Atangana D Baleanu and A Alsaedi ldquoNew properties ofconformable derivativerdquo Open Mathematics vol 13 pp 57ndash63 2015

[15] N Yazici and U Gozutok ldquoMultivariable conformablefractional calculusrdquo Filomat vol 32 no 2 pp 45ndash53 2018

[16] F Martınez I Martınez and S Paredes ldquoConformableEulerprimes theorem on homogeneous functionsrdquo ComputationalMathematical Methods vol 1 no 5 pp 1ndash11 2018

[17] A A Hyder and A H Soliman ldquoExact solutions of space-timelocal fractal nonlinear evolution equations a generalizedconformable derivative approachrdquo Results in Physics vol 17Article ID 103135 2020

[18] A A Hyder ldquoWhite noise theory and general improvedKudryashov method for stochastic nonlinear evolutionequations with conformable derivativesrdquo Advances in Dif-ference Equations vol 202019 pages 2020

[19] A H Soliman and A A Hyder ldquoClosed-form solutions ofstochastic KdV equation with generalized conformable deriv-ativesrdquo Physica Scripta vol 95 no 6 Article ID 065219 2020

[20] V E Tarasov ldquoNo violation of the Leibniz rule No fractionalderivativerdquo Communications in Nonlinear Science and Nu-merical Simulation vol 18 no 11 pp 2945ndash2948 2013

[21] R Khalil M Al Horani A Yousef and M SababhehldquoFractional analytic functionsrdquo Far East Journal of Mathe-matical Sciences vol 103 no 1 pp 113ndash123 2018

[22] S Uccedilar and N Y Ozgur ldquoComplex conformable derivativerdquoArabian Journal of Geosciences vol 12 no 201 2019

[23] F Martınez I Martınez I Martınez M K A Kaabar andS Paredes ldquoNew results on complex conformable integralrdquoAIMS Mathematics vol 5 no 6 pp 7695ndash7710 2020

[24] H Kiskinov M Petkova A Zahariev and M VeselinovaldquoSome results about conformable derivatives in Banach spacesand an application to the partial differential equationsrdquo inAIPConference Proceedingsvol 2333 no 1 AIP Publishing LLCArticle ID 120002 2021

[25] U Gozutok H A Ccediloban and Y Sagıroglu ldquoFrenet framewith respect to conformable derivativerdquo Filomat vol 33no 6 pp 1541ndash1550 2019

[26] J C Mayo-Maldonado G Fernandez-Anaya and O F Ruiz-Martinez ldquoStability of conformable linear differential systemsa behavioural framework with applications in fractional-ordercontrolrdquo IET Control eory amp Applications vol 14 no 18pp 2900ndash2913 2020

[27] M Al Horani and R Khalil ldquoTotal fractional differential withapplications to exact fractional differential equationsrdquo In-ternational Journal of Computer Mathematics vol 95 no 6-7pp 1444ndash1452 2018

[28] J E Marsden and M J Hoffman Vector Calculus WH Freeman and Company New York NY USA 4th edition1996

12 Journal of Mathematics

Taαf( 1113857(a) lim

t⟶a+T

aαf( 1113857(t) (4)

Theorem 3 (See [12]) Suppose that f g (ainfin)⟶ R areleft α DF s where α isin (0 1] Let us assume thath(t) f(g(t)) h(t) is α DF forall tne a and g(t) ne 0 and we get

Taαh( 1113857(t) T

aαf( 1113857(g(t)) middot T

aαg( 1113857(t) middot (g(t))

αminus 1 (5)

If t a then we obtain

Taαh( 1113857(a) lim

t⟶a+T

aαf( 1113857(g(t)) middot T

aαg( 1113857(t) middot (g(t))

αminus 1

(6)

Theorem 4 (Rollersquos theorem (RT) [10]) Suppose that agt 0α isin (0 1] and function f [a b]⟶ R satisfies

(i) f is CF on [a b]

(ii) f is α DF on (a b)

(iii) f(a) f(b)

en exist c isin (a b) ni(Tαf)(c) 0

Theorem 5 (Mean value theorem (MVT) [10]) Assume thatagt 0 α isin (0 1] and function f [a b]⟶ R satisfies

(i) f is CF on [a b]

(ii) f is α DF on (a b)

en exist c isin (a b) ni we have

Tαf( 1113857(c) f(b) minus f(a)

bαα minus a

αα (7)

Theorem 6 (Modified mean value theorem (MMVT)[27]) Assume that agt 0 α isin (0 1] and functionf [a b]⟶ R satisfies

(i) f is CF on [a b]

(ii) f is α DF on (a b)

en exist c isin (a b) ni we have

Tαf( 1113857(c)

c1minus αα

f(b) minus f(a)

(bα) minus (aα) (8)

Theorem 7 (See [13]) Suppose that agt 0 α isin (0 1] andfunction f [a b]⟶ R satisfies

(i) f is CF on [a b]

(ii) f is α DF on (a b)

en we get

(i) If (Tαf)(t)gt 0 forall t isin (a b) then f is increasingon[a b]

(ii) If (Tαf)(t)lt 0 forallt isin (a b) then f is decreasingon[a b]

Let us express the CoV of partial derivative (PDr) of areal-valued function (RVF) with SVs as follows

Definition 3 (See [14 15]) Assume that f is a RVF with n

variables and there is a point a (a1 an) isin Rn whereits ith component is positive en the limit is written as

limε⟶0

f a1 ai + εai1minus α

an1113872 1113873 minus f a1 an( 1113857

ε (9)

If the above limit exists the ith CoV of PDr of f of theorder α isin (0 1] at a represented by zαzxα

i f(a)

3 α-Derivative of a RVF of SVs

Definition 4 Suppose that f is a RVF with n variablesx1 xn and α isin (0 1] en we say that f is α DF ata (a1 an) isin Rn each ai gt 0 if any of the three con-ditions which are equivalent to each other is verified

(i) ere is a linear transformation L Rn⟶ R suchthat

limh⟶0

f a1 + h1a11minus α

an + hnan1minus α

1113872 1113873 minus f a1 an( 1113857 minus L(h)

h 0 (10)

Where h (h1 hn) h h21+middotmiddotmiddot+h2

n

1113969 and

α isin (0 1](ii) ere is a linear transformation L Rn⟶ R and a

function ε h⟶ ε(h) such that

f a1 + h1a11minus α

an + hnan1minus α

1113872 1113873 minus f a1 an( 1113857 L(h) + ε(h)h (11)

And limh⟶0ε(h) 0

Journal of Mathematics 3

(iii) ere is a linear transformation L Rn⟶ R and n

functions εi h⟶ εi(h) forall

i 1 2 n ni

f a1 + h1a11minus α

an + hnan1minus α

1113872 1113873 minus f a1 an( 1113857 L(h) + 1113944n

i1εi(h)hi

(12)

And lim εih⟶ 0(h) 0 for i 1 2 n

e linear transformation L Rn⟶ R is defined byL(h) 1113936

ni1 αihi with h (h1 hn) and α1 αn isin R

is linear transformation is denoted by Dαf(a) which iscalled CoD of f of the order α isin (0 1] at a

Remark 1 e equivalence of conditions (i) and (ii) isimmediate since

limh⟶0

ε(h) 0harrε(h)h o(h) (13)

To see the equivalence between conditions (ii) and (iii)we take

εi ε(h)hi

h

ε(h) 1

h1113944

ni1εi(h)hi (14)

As |hih|le 1 then we have the following

(i) If limεh⟶0(h) 0 then lim εi h⟶ 0(h) 0(ii) If lim εi h⟶ 0(h) 0 for i 1 n then we obtain

limh⟶0

ε(h)le limh⟶0

1h

1113944

n

i1εi(h)

le lim

h⟶01113944

n

i1εi(h)

0

(15)

ie limh⟶0ε(h) 0 Hence the conditions (ii) and (iii) areequivalent

Example 1 Consider a function f defined by f(x y) ex minus

2cosy and a point (a b) isin R2 with agt 0 and bgt 0 thenDαf(a b)(h1 h2) h1a

1minus αeα + 2h2b1minus α sin b

Solution to prove this let us note that

limh1 h2( )⟶(00)

f a + a1minus α

h1 b + b1minus α

h21113872 1113873 minus f(a b) minus L h1 h2( 1113857

h1 h2( 1113857

limh1 h2( )⟶(00)

ea+a1minus αh1

minus 2 cos b + b1minus α

h21113872 1113873 minus ea

minus 2 cos b( 1113857 minus h1a1minus α

ea

+ 2h2b1minus α sin b1113872 1113873

h21 + h

22

1113969

le limh1⟶0

ea+a1minus αh1

minus ea

minus h1a1minus α

ea

h1minus 2 lim

h2⟶0

cos b + b1minus α

h21113872 1113873 minus cos b + b1minus α sin b

h2

limh1⟶0

ea+a1minus αh1

minus ea

h1minus a

1minus αe

a⎛⎝ ⎞⎠ minus 2 limh2⟶0

cos b + b1minus α

h21113872 1113873 minus cos b

h2+ b

1minus α sin b⎛⎝ ⎞⎠ a1minus α

ea

minus a1minus α

ea

1113872 1113873

minus 2 minus b1minus α sin b + b

1minus α sin b1113872 1113873 0

(16)

Theorem 8 If a RVF f with n variables α DF ata (a1 an) isin Rn each ai gt 0 then f is CF at a isin Rn

Proof Since f is α DF at a we can write the following

f a1 + h1a11minus α

an + hnan1minus α

1113872 1113873 minus f a1 an( 1113857 1113944n

i1αihi + o(h) (17)

4 Journal of Mathematics

By taking the limits of the two sides of the equality ash⟶ 0 we have

limh⟶0

f a1 + h1a11minus α

an + hnan1minus α

1113872 1113873 f a1 an( 1113857

(18)

Hence f is CF at a isin Rn

Theorem 9 If a RVF f with n variables is α DF ata (a1 an) isin Rn each ai gt 0 then zαzxα

i f(a) exists for1le ile n and the CoD of f of the order α isin (0 1] is expressedas

Dαf(a)(h) 1113944

n

i1

zα

zxαi

f(a)hi (19)

where h (h1 hn)

Proof By setting hj 0foralljne i in the formula (12) we have

f a1 ai + hiai1minus α

+ middot middot middot + an1113872 1113873 minus f a1 an( 1113857 αihi + εi(h)hi

(20)

By multiplying by 1hi we can write

f a1 ai + hiai1minus α

+ middot middot middot + an1113872 1113873 minus f a1 an( 1113857

hi

αi + εi(h)

(21)

By taking the limits of the two sides of the equality ashi⟶ 0 we have

αi zα

zxαi

f(a) foralli 1 2 n (22)

Finally by substituting the values above αi in the formulaDαf(a)(h) 1113936

ni1 αihi the result is followed

Corollary 1 If a RVF f with n variables is α DF ata (a1 an) isin Rn each ai gt 0 then Dαf(a) is unique

Remark 2 If a RVF f with n variables is α DF ata (a1 an) isin Rn each ai gt 0 then the CoV of gradientof f of the order α isin (0 1] at a is

nablaαf(a) zα

zxα1

f(a) zα

zxαn

f(a)1113888 1113889 (23)

Also the matrix form (MF) of equation (19) is given asfollows

Dαf(a)(h) nablaαf(a) middot h

zα

zxα1

f(a) zα

zxαn

f(a)1113888 1113889 middot

h1

⋮

hn

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(24)

Theorem 10 Let α isin (0 1] f g X⟶ R be a RVF definedin an open set (OS) X sub Rn ni forall x (x1 xn) isin X eachxi gt 0 and a point a (a1 an) isin X If f g are α DF at athen we have

(i) Dα(λf + μg)(a) λDα(f)(a) + μDα(g)(a)forall λ μ isin R

(ii) Dα(fg)(a) Dα(f)(a) middot g(a) + f(a)Dα(g)(a)

Proof (i) follows fromDefinition 4 thus it follows the proofof (i)

For (ii) let A (a1 + h1a1minus α1 an + hna1minus α

n ) and thenwe have

limh⟶0

((fg)(A) minus (fg)(a)) minus Dαf(a) middot g(a) + f(a) middot D

αg(a)( 1113857(h)

h

limh⟶0

(f(A) minus f(a)) minus Dαf(a)(h)

hmiddot g(a) + f(a) middot

(g(A) minus g(a)) minus Dαg(a)(h)

h1113888 1113889 + lim

h⟶0

(f(A) minus f(a)) middot (g(A) minus g(a))

h

0 + 0 + limh⟶0

Dαf(a)(h)( 1113857 middot D

αg(a)(h)( 1113857

h lim

h⟶0D

αf(a)

hh

1113888 11138891113888 1113889 middot Dαg(a)

hh

1113888 11138891113888 1113889 middot h 0

(25)

Theorem 11 Let α isin (0 1] f X⟶ R be a RVF defined inan OS X sub Rn niforall x (x1 xn) isin X each xi gt 0 and apoint a (a1 an) isin X If the function f has all CoV ofPDrs of the order α at each point of a neighbourhood of thepoint a U(a) with U(a) sub X and they are continuous at athen f is α DF at a

Proof See eorem 21 proof in [27]

Remark 3 e above theorem allows defining the space ofRVFs with n variables by having continuous CoV of PDrs oforder α isin (0 1] in a domain X sub Rn which can be denotedby Cα(X R)

Finally we can easily extend all of the above results to theVVFs of SRVs

Theorem 12 Assume that α isin (0 1] f X⟶ Rm be a VVFdefined in an OS X sub Rn niforallx (x1 xn) isin X each

Journal of Mathematics 5

xi gt 0 and the point a (a1 an) isin X e function f isα DF at a iff its components are α DF at a and if these

components are f1 f2 fm then the components ofDαf(a) are the α-derivativesDαfj(a) for j 1 2 m ie

f f1 f2 fm( 1113857rArr Dαf(a) D

αf1(a) D

αf2(a) D

αfm(a)( 1113857 (26)

Proof It is similarly proven as same as traditional calculusby applying Dα instead of derivative

Remark 4 Assume that α isin (0 1] f X⟶ Rm is a VVFdefined in an OS X sub Rn niforall x (x1 xn) isin X eachxi gt 0 and a point a (a1 an) isin X If function f isα-differentiable at a then zαzxα

i fj(a) exists fori 1 2 n j 1 2 m and the CoV of α-Jacobian off of order α isin (0 1] at a is expressed as

Jαf(a)

zα

zxα1f1(a)

zα

zxαn

f1(a)

⋮ ⋮

zα

zxα1fm(a)

zα

zxαn

fm(a)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(27)

4 The Chain Rule

Let us prove the CR for the functions of SVs in 2 particularcases For the proofrsquos purpose the continuityrsquos hypothesis ofPDrs is given [28]

Theorem 13 (CR) Suppose that t isin R andx (x1 xn) isin Rn If f(t) (f1(t) fn(t)) is α DF

at agt 0 ni α isin (0 1] and a RVF g with n variablesx1 xn is α DF at f(a) isin Rn all fi(a)gt 0 α isin (0 1] en the composition g ∘f is α-differential at a and

Tαg ∘ f( 1113857(a) 1113944n

i1

zα

zxαi

g(f(a)) middot fi(a)( 1113857αminus 1

middot Tαfi( 1113857(a)

(28)

Proof Assumeg isin C1(U(f(a)) R) niU(f(a)) is the pointf(a) neighborhood Suppose thath(t) (g ∘ f)(t) g(f(t)) By setting u a + εa1minus α inDefinition 1 we see that

Tαh( 1113857(a) limt⟶a

h(t) minus h(a)

t minus aa1minus α

limt⟶a

g(f(t)) minus g(f(a))

t minus aa1minus α

(29)

Without loss of generality (WLOG) U(f(a)) is assumedto be an open ball (OB) represented by B(f(a) r) Since f isa CF then along with points (f1(a) fn(a)) and(f1(t) fn(t)) the points(f1(a) f2(t) fn(t)) (f1(a) f2(a) fn(t))

and lines connecting them must also be the ball B(f(a) r)In fact using the classical MVT for differentiable functions(DFs) of one variable is in the following computation [28]

h(t) minus h(a)

t minus aa1minus α

g(f(t)) minus g(f(a))

t minus aa1minus α

g f1(t) fn(t)( 1113857 minus g f1(a) f2(t) fn(t)( 1113857

t minus aa1minus α

+g f1(a) f2(t) fn(t)( 1113857 minus g f1(a) f2(a) f3(t) fn(t)( 1113857

t minus aa1minus α

+ middot middot middot

+g f1(a) f2(a) fn(t)( 1113857 minus g f1(a) fn(a)( 1113857

t minus aa1minus α

z

zx1g c1 f2(t) fn(t)( 1113857

f1(t) minus f1(a)

t minus aa1minus α

z

zx2g f1(a) c2 fn(t)( 1113857

f2(t) minus f2(a)

t minus aa1minus α

+ middot middot middot

+z

zx2g f1(a) c2 fn(t)( 1113857

f2(t) minus f2(a)

t minus aa1minus α

+ middot middot middot

+z

zxn

g f1(a) f2(a) cn( 1113857fn(t) minus fn(a)

t minus aa1minus α

(30)

6 Journal of Mathematics

where ci is between fi(a) and fi(t) forall i 1 2 n By taking limits as t⟶ a and using the continuity ofPDrs of g as well as the fact that ci⟶ fi(a) foralli 1 2 n equation (29) is expressed as

Tαh( 1113857(a) limt⟶a

h(t) minus h(a)

t minus aa1minus α

limt⟶a

g(f(t)) minus g(f(a))

t minus aa1minus α

limt⟶infin

z

zx1g c1 f2(t)( 1113857

f1(t) minus f1(a)

t minus aa1minus α

+ middot middot middot1113888 1113889

+z

zx2g f1(a) c2 fn(t)( 1113857

f2(t) minus f2(a)

t minus aa1minus α

+ middot middot middot

+z

zxn

g f1(a) f2(a)1minus α

cn1113872 1113873fn(t) minus fn(a)

t minus aa1minus α

1113889 +z

zx1g(f(a)) middot f

(1)1 (a) middot a

1minus α

+z

zx2g(f(a)) middot f

(1)2 (a) middot a

1minus αmiddot f1(a)

αminus 1middot f

(1)1 (a) middot a

1minus α

z

zx1g(f(a)) middot f1(a)

1minus αmiddot f1(a)

1minus αmiddot f1(a)

αminus 1middot f

(1)1 (a) middot a

1minus α+

z

zx2g(f(a)) middot f2(a)

1minus α

middot f2(a)αminus 1

middot f(1)2 (a) middot a

1minus α+ middot middot middot +

z

zxn

g(f(a)) middot f2(a)1minus α

middot f2(a)αminus 1

middot f(1)n (a) middot a

1minus α

zα

zxα1

g(f(a)) middot f1(a)αminus 1

middot Tαf1( 1113857(α) +zα

zxα2

g(f(a)) middot f2(a)αminus 1

middot Tαf2( 1113857(a) + middot middot middot

+zα

zxαn

g(f(a)) middot fn(a)αminus 1

middot Tαf1( 1113857(α)

(31)

Our proof is completely done Remark 5 e MF of equation (28) is expressed as

Dα(g middot f)(a)(h)

zα

zxα1

g(f(a)) zα

zxαn

g(f(a))1113888 1113889

middot

f1(a)αminus 1 0 0

0 ⋱ 0

0 0 fn(a)αminus 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠middot

Tαf1( 1113857(a)

⋮

Tαfn( 1113857(a)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠middot h

(32)

Theorem 14 (CR) Let x (x1 xn) isin Rn andy (y1 ym) isin Rm If f(x1 xn) (f1(x1

xn) fm(x1 xn)) is a α DF at a (a1 an) isin Rneach ai gt 0 ni α isin (0 1] and a RVF g with variables y1 ym

is α DF at f(a) isin Rn where allfi(a)gt 0 niα isin (0 1]en thecomposition g ∘ f is α-differentiable at a and we have

zα

zxαi

(g ∘ f)(a) 1113944m

j1

zα

zyαj

g(f(a)) middot fj(a)αminus 1

middotzα

zxαi

fj(a)

forall i 1 2 n

(33)

Journal of Mathematics 7

Proof Proof According to PDrrsquos definition and eorem14 the following is implied

Remark 6 e MF of equation (33) is expressed as

Dα(g ∘ f)(a)(h)

zα

zyα1

g(f(a)) zα

zyαm

g(f(a))1113888 1113889 middot

f1(a)αminus 1 0 0

0 ⋱ 0

0 0 fm(a)αminus 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠middot

zα

zxα1f1(a)

zα

zxαn

f1(a)

middot middot middot middot middot middot

zα

zxα1fn(a)

zα

zxαn

fn(a)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

middot

h1

⋮

hn

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (34)

5 Conformable Implicit Function Theorem

In this section the CoV of IFm for SVs is obtained Firstwe establish the conformable theorem of existence andregularity of the implicit function for the case of a singleequation

Theorem 15 (Conformable implicit function theorem forthe case of a single equation) Suppose that α isin (0 1]F X⟶ R is a RVF defined in an OS X sub Rn+1 ni forall(x1 xn y) isin X each xi ygt 0 and point(a1 an b) isin X Suppose that

(i) F(a1 an b) 0(ii) F isin Cα(X R)

(iii) zαzyαF(a1 an b)ne 0

en there is a neighbourhood U sub Rn of (a1 an)such that there is a unique function (UF) y g(x1 xn)

that satisfies the followingg(a1 an) b and F(x1 xn g(x1 xn))

0 forall(x1 xn) isin UFinally y g(x1 xn) is Cα in U and for every

i 1 2 n we have

zα

zxαi

g x1 xn( 1113857 minuszαzx

αi F x1 xn( 1113857 g x1 xn( 1113857( 1113857( 1113857

zαzy

αF x1 xn( 1113857 g x1 xn( 1113857( 1113857( 1113857 middot g x1 xn( 1113857( 1113857

αminus 1 (35)

Proof WLOG X is assumed to be an OB represented byB((a1 an b) ε0) Let ρ isin (0 ε0) If we call δ

ε20 minus ρ21113969

it is verified that

x1 xn( 1113857 minus a1 an( 1113857

lt δ and |y minus b|lt ρ1113960 1113961rArr x1 xn y( 1113857 isin B a1 an b( 1113857 ε0( 1113857 (36)

Note that in particular if |y minus b|lt ρ then(a1 an y) isin B((a1 an b) ε0)

Since zαzyαF(a1 an b)ne 0 it is assumed to bepositive (otherwise minus F is considered instead of F) From factthat F(a1 an b) 0 it follows that

F a1 an b minus ρ( 1113857gt 0

F a1 an b minus ρ( 1113857lt 0(37)

By the continuity of F at (a1 an b minus ρ) and(a1 an b + ρ) there exists δprime isin (0 δ) such that

x1 xn( 1113857 minus a1 an( 1113857

lt δprimerArr F x1 xn b minus ρ( 1113857gt 0 andF x1 xn b + ρ( 1113857lt 01113858 1113859 (38)

Since the function y↦F(x1 xn y) is CF on the in-terval [b minus ρ b + ρ ] forall (x1 xn) isin B((a1 an) δprime) andvia the classical Bolzanorsquos theorem (Bm) it implies thatexistyx isin ( b minus ρ b + ρ) niF(x1 xn yx) 0 for eachx (x1 xn) en yrsquos value is unique since a functionwhose derivative is positive has more than zero On the other

hand by having U B((a1 an) δprime) for each(x1 xn) isin U there exists a unique y

niF(x1 xn y) 0 we can write y g(x1 xn) andthen this function will be proven to be CF onB((a1 an) δprime)e continuity of the function g at the point(a1 an) is clear since for each ρgt 0 exist a value δprime gt 0 ni

8 Journal of Mathematics

x1 xn( 1113857 minus a1 an( 1113857

lt δprimerArr b minus yx

11138681113868111386811138681113868111386811138681113868

lt ρhArr b minus g x1 xn( 11138571113868111386811138681113868

1113868111386811138681113868lt ρ(39)

e function g continuity will be proven at any point(x1 xn) isin B((a1 an) δprime) by simply substitutingB((a1 an) δprime) for an OB B((x1 x)) is contained inB((a1 an) δprime)

Finally let us now show formula (35) By applyingeorem 14 to equation

F x1 xn y( 1113857 0 we have

zα

zxαi

F(x g(x)) +zα

zyα F(x g(x)) middot g(x)

αminus 1middot

zα

zxαi

g(x) 0

(40)

forall i 1 2 n nix (x1 xn) Solving zαzxαi g(x)

we get (35) In addition the formula (35) on the right side iscontinuous so the continuity of CoV of PDrs zαzxα

i g(x) foralli 1 2 n follows

eorem 15 will provide us a help in computing CoV ofPDrs of implicit function of SVs

Example 2 Consider

F(x y z) x3

+ 3y2

+ 4xz2

minus 3yz2

minus 5 0 (41)

is equationrsquos one solution is (1 1 1) F is obviously inCα which is OB represented by B((1 1 1) ε0) withx y zgt 0 for some α isin (0 1] since

zα

zzα F(1 1 1) 8xz

2minus αminus 6yz

2minus α1113872 11138731113961

(111) 2ne 0 (42)

eorem 15 implies that there is aneighbourhood U sub R2 of (1 1) niexist a UF z g(x y)

satisfies the followingg(1 1) 1 andF(x y g(x y)) 0 forall(x y) isin U

Moreover z g(x y) is Cα in U and we have

zα

zxα g(x y) minus

3x + 4z2

1113872 1113873x1minus α

(8x minus 6y)z

zα

zyα g(x y) minus

6y minus 3z2

1113872 1113873y1minus α

(8x minus 6y)z

(43)

Finally we obtain zαzxαg(1 1) minus (72) andzαzxαg(1 1) minus (32)

Finally CoV of IFm for a system of several equationsand SRVs is obtained

Theorem 16 (Conformable general implicit functiontheorem) Let α isin (0 1] F X⟶ Rm be a VVF defined inan OS X sub Rn+m ni forall (x y) (x1 xn y1 ym) isin Xeach xi yj gt 0 and point(a b) (a1 an b1 bm) isin X Assume that

(i) F(a b) 0(ii) F isin Cα(X Rm )

(iii) det[JαyF(a b)]ne 0

en there is a neighbourhood U sub Rn of a niexist a UFf U⟶ Rm x↦ y f(x) that satisfies f(a) b andF(x f(x) ) 0 forallx isin U

Finally y f(x) is class Cα in U and for everyi 1 2 n we have

zαfzxα

i

1113890 1113891

t

minus fαminus 11113872 1113873

minus 1middot J

αyF1113872 1113873

minus 1middot

zαFzxα

i

1113890 1113891

t

(44)

where

zαf

zxαi

1113890 1113891 zαf1

zxαi

zαfm

zxαi

1113888 1113889

fαminus 1

fαminus 11 0

⋮ ⋮

0 fαminus 1m

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

JαyF

zαF1

zyα1

zαF1

zyαm

zαFm

zyα1

zαFm

zyαm

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

zαF

zxαi

1113890 1113891 zαF1

zxαi

zαFm

zxαi

1113888 1113889

(45)

Proof e proof of existence and uniqueness of the implicitfunction is done similar to the traditional multivariablecalculus by applying mathematical induction on q and usingthe conformable implicit function theorem for severalvariables [28]

Let us now show formula (44) Assume that a systemwith several equations and SRVs is expressed as

F(x y) 0 or

F1 x1 xn y1 ym( 1113857 0

⋮

Fm x1 xn y1 ym( 1113857 0

⎧⎪⎪⎨

⎪⎪⎩ (46)

which satisfies hypotheses (i)ndash(iii) of the theorem then thissystem is defined in a neighbourhood U sub Rn of a theimplicit function y f(x) class Cα in U such that f(a) bwhich satisfies equation (1) ie

F(x f(x)) 0 or

F1 x f1(x) fm(x)( 1113857 0

Fm x f1(x) fm(x)( 1113857 0

⎧⎪⎪⎨

⎪⎪⎩(47)

By applying CoV of CR to the above equation we have

Journal of Mathematics 9

zαF1

zxαi

+zαF1

zyα1

middot fαminus 11 middot

zαf1

zxαi

+ middot middot middot +zαF1

zyαm

middot fαminus 1m middot

zαfm

zxαi

0

⋮

zαFm

zxαi

+zαFm

zyα1

middot fαminus 11 middot

zαf1

zxαi

+ middot middot middot +zαFm

zyαm

middot fαminus 1m middot

zαfm

zxαi

0

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

orzαF

zxαi

+ 1113944m

j1

zαF

zyαj

middot fαminus 1j middot

zαfj

zxαi

0

(48)

for all i 1 2 n In addition the MF of equation (48) is given as follows

zαF1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus

zαF1

zyα1

zαF1

zyαm

zαFm

zyα1

zαFm

zyαm

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

middot

fαminus 11 0

⋮ ⋮

0 fαminus 1m

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠middot

zαf1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

orzαFzxα

i

1113890 1113891

t

minus JαyF middot fαminus 1

middotzαfzxα

i

1113890 1113891

t

(49)

Since JαyF and fαminus 1 are regular matrices by hypothesis we

have

zαf1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus

fαminus 11 0

⋮ ⋮

0 fαminus 1m

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus 1

middot

zαF1

zyα1

zαF1

zyαm

zαFm

zyα1

zαFm

zyαm

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus 1

middot

zαF1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

orzαfzxα

i

1113890 1113891

t

minus fαminus 11113872 1113873

minus 1middot J

αyF1113872 1113873

minus 1middot

zαFzxα

i

1113890 1113891

t

(50)

which completes the proofWe will now show that eorem 16 can be used to

compute CoV of PDrs of systems with several equations andSRVs

Example 3 Consider a system of two equations and two realvariables

F1(x y z w) x2

+ y2

+ z2

+ w2

minus 6 0

F2(x y z w) x2

minus y2

+ z2

minus w2

0

⎧⎨

⎩ (51)

One solution of this equation is(x y z w) (1 1

2

radic

2

radic) Clearly F (F1 F2) is in Cα

which is OB B((1 12

radic

2

radic) ε0) with x y z wgt 0 for

some α isin (0 1] since

10 Journal of Mathematics

det JαzwF((

2

radic

2

radic 1 1))1113960 1113961 det

2z2minus α 2w2minus α

2z2minus α minus 2w2minus α⎛⎝ ⎞⎠⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

(112

radic

2

radic)

minus322propne 0 (52)

eorem 16 implies that there is aneighbourhood U sub R2 of (

2

radic

2

radic) niexist a UF f (f1 f2)

given by

z f1(x y)

w f2(x y)1113896 (53)

that satisfies

f1(1 1) 2

radic

f2(1 1) 2

radic

⎧⎨

⎩

F1 x y f1(x y) f2(x y)( 1113857 0

F2 x y f1(x y) f2(x y)( 1113857 0forall(x y) isin U1113896

(54)

Moreover f (f1 f2) is class Cα in U and we have

zαf1

zxα

zαf2

zxα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ minus

zαminus 1 0

0 wαminus 1⎛⎝ ⎞⎠

minus 1

middot2z2minus α 2w2minus α

2z2minus α minus 2w2minus α⎛⎝ ⎞⎠

minus 1

middot2x

2minus α

2x2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz1minus α 0

0 w1minus α

⎛⎜⎝ ⎞⎟⎠ middotzαminus 2

zαminus 2

wαminus 2

minus wαminus 2

⎛⎜⎝ ⎞⎟⎠

middot2x

2minus α

2x2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz

minus 1z

minus 1

wminus 1

minus wminus 1

⎛⎜⎝ ⎞⎟⎠ middot2x

2minus α

2x2minus α

⎛⎜⎝ ⎞⎟⎠ minus

x2minus α

z

0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

zαf1

zyα

zαf2

zyα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minuszαminus 1 0

0 wαminus 1⎛⎝ ⎞⎠

minus 1

middot2z2minus α 2w2minus α

2z2minus α minus 2w2minus α⎛⎝ ⎞⎠

minus 1

middot2y

2minus α

minus 2y2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz1minus α 0

0 w1minus α

⎛⎜⎝ ⎞⎟⎠ middotzαminus 2

zαminus 2

wαminus 2

minus wαminus 2

⎛⎜⎝ ⎞⎟⎠

middot2y

2minus α

minus 2y2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz

minus 1z

minus 1

wminus 1

minus wminus 1

⎛⎜⎝ ⎞⎟⎠ middot2y

2minus α

minus 2y2minus α

⎛⎜⎝ ⎞⎟⎠

0

minusy2minus α

w

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(55)

Finally we have

zαf1

zxα

zαf2

zxα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(112

radic

2

radic)

minus12

radic

0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

zαf1

zyα

zαf2

zyα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(112

radic

2

radic)

0

minus12

radic

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(56)

According to all of our numerical examples in this workwith the help of our investigation to all proposed theoremsand properties in this article the numerical examples showthat all our obtained results are valid and efficient As seen inall these examples the computations are simple and cost-

efficient which are important when scientific phenomenaare modelled in the sense of conformable calculus All in allthe simplicity of computations is always needed in modelingapplications in comparison to other complicated compu-tations that are needed using other approaches in otherfractional operators All our obtained results in this work areaccurate because our results coincide with the usual integer-order results

6 Conclusion

Anew investigation on the CoV ofmultivariable calculus hasbeen discussed in this work in detail e α-derivative of afunction of SVs and all related properties have been in-vestigatede CoV of CR for functions of SVs has also beenstudied e CoV of IFm has been presented in the lastpart of our results and numerical experiments have beenconducted to support our theoretical results e findings ofthis investigations show that all results formulated via CoD

Journal of Mathematics 11

coincide with the ones in the traditional integer case Allresults in this work can be potentially applied in modelingoceanographic phenomena such as Stommelrsquos box model ofthermohaline circulation and other related models wherethis studyrsquos analysis can be further extended or generalizedin 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

Authorsrsquo Contributions

MohammedK A Kaabar involved in actualization developedmethodology performed formal analysis validated and in-vestigated the study prepared the initial draft and supervisedand edited the original draft Francisco Martınez InmaculadaMartınez and Silvestre Paredes involved in actualizationdeveloped methodology performed formal analysis validatedand investigated the study and prepared the initial draftZailan Siri developed methodology performed formal anal-ysis validated and investigated the study and prepared theinitial draft All authors read and approved the final version

References

[1] M M A Khater M S Mohamed and R A M Attia ldquoOnsemi analytical and numerical simulations for a mathematicalbiological model the time-fractional nonlinear Kolmogor-ovndashPetrovskiindashPiskunov (KPP) equationrdquo Chaos Solitons ampFractals vol 144 Article ID 110676 2021

[2] M A Khater K S Nisar and M S Mohamed ldquoNumericalinvestigation for the fractional nonlinear space-time telegraphequation via the trigonometric Quintic B-spline schemerdquoMathematical Methods in the Applied Sciences vol 44 no 6pp 4598ndash4606 2021

[3] M M A Khater M S Mohamed and S K Elagan ldquoDiverseaccurate computational solutions of the nonlinearKleinndashFockndashGordon equationrdquo Results in Physics vol 23Article ID 104003 2021

[4] M F Li J R Ren and T Zhu ldquoFractional vector calculus andfractional special functionrdquo arXiv preprint10012889 2010

[5] D J Wollkind and B J Dichone Comprehensive AppliedMathematical Modeling in the Natural and Engineering Sci-ences Springer International Publishing Berlin Germany2017

[6] P J Olver Introduction to Partial Differential EquationsSpringer International Publishing Berlin Germany 2014

[7] J D Krehbiel and J B Freund ldquoStokes flow inside a sphere inan inviscid extensional flowrdquo Zeitschrift fur AngewandteMathematik und Physik vol 68 no 4 pp 68ndash81 2017

[8] A A Kilbas H M Srivastava and J J Trujillo ldquoeory andapplications of fractional differential equationsrdquo in North-Holland Mathematics StudiesVol 204 Elsevier AmsterdamNetherlands 2006

[9] K S Miller An Introduction to Fractional Calculus andFractional Differential Equations John Wiley and SonsHoboken NJ USA 1993

[10] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014

[11] R Khalil M Al Horani and M Abu Hammad ldquoGeometricmeaning of conformable derivative via fractional cordsrdquo eJournal of Mathematics and Computer Science vol 19 no 4pp 241ndash245 2019

[12] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journalof Computational and Applied Mathematics vol 279pp 57ndash66 2015

[13] O S Iyiola and E R Nwaeze ldquoSome new results on the newconformable fractional calculus with application usingDprimeAlambert approachrdquo Progress in Fractional Differentiationand Applications vol 2 no 2 pp 115ndash122 2016

[14] A Atangana D Baleanu and A Alsaedi ldquoNew properties ofconformable derivativerdquo Open Mathematics vol 13 pp 57ndash63 2015

[15] N Yazici and U Gozutok ldquoMultivariable conformablefractional calculusrdquo Filomat vol 32 no 2 pp 45ndash53 2018

[16] F Martınez I Martınez and S Paredes ldquoConformableEulerprimes theorem on homogeneous functionsrdquo ComputationalMathematical Methods vol 1 no 5 pp 1ndash11 2018

[17] A A Hyder and A H Soliman ldquoExact solutions of space-timelocal fractal nonlinear evolution equations a generalizedconformable derivative approachrdquo Results in Physics vol 17Article ID 103135 2020

[18] A A Hyder ldquoWhite noise theory and general improvedKudryashov method for stochastic nonlinear evolutionequations with conformable derivativesrdquo Advances in Dif-ference Equations vol 202019 pages 2020

[19] A H Soliman and A A Hyder ldquoClosed-form solutions ofstochastic KdV equation with generalized conformable deriv-ativesrdquo Physica Scripta vol 95 no 6 Article ID 065219 2020

[20] V E Tarasov ldquoNo violation of the Leibniz rule No fractionalderivativerdquo Communications in Nonlinear Science and Nu-merical Simulation vol 18 no 11 pp 2945ndash2948 2013

[21] R Khalil M Al Horani A Yousef and M SababhehldquoFractional analytic functionsrdquo Far East Journal of Mathe-matical Sciences vol 103 no 1 pp 113ndash123 2018

[22] S Uccedilar and N Y Ozgur ldquoComplex conformable derivativerdquoArabian Journal of Geosciences vol 12 no 201 2019

[23] F Martınez I Martınez I Martınez M K A Kaabar andS Paredes ldquoNew results on complex conformable integralrdquoAIMS Mathematics vol 5 no 6 pp 7695ndash7710 2020

[24] H Kiskinov M Petkova A Zahariev and M VeselinovaldquoSome results about conformable derivatives in Banach spacesand an application to the partial differential equationsrdquo inAIPConference Proceedingsvol 2333 no 1 AIP Publishing LLCArticle ID 120002 2021

[25] U Gozutok H A Ccediloban and Y Sagıroglu ldquoFrenet framewith respect to conformable derivativerdquo Filomat vol 33no 6 pp 1541ndash1550 2019

[26] J C Mayo-Maldonado G Fernandez-Anaya and O F Ruiz-Martinez ldquoStability of conformable linear differential systemsa behavioural framework with applications in fractional-ordercontrolrdquo IET Control eory amp Applications vol 14 no 18pp 2900ndash2913 2020

[27] M Al Horani and R Khalil ldquoTotal fractional differential withapplications to exact fractional differential equationsrdquo In-ternational Journal of Computer Mathematics vol 95 no 6-7pp 1444ndash1452 2018

[28] J E Marsden and M J Hoffman Vector Calculus WH Freeman and Company New York NY USA 4th edition1996

12 Journal of Mathematics

(iii) ere is a linear transformation L Rn⟶ R and n

functions εi h⟶ εi(h) forall

i 1 2 n ni

f a1 + h1a11minus α

an + hnan1minus α

1113872 1113873 minus f a1 an( 1113857 L(h) + 1113944n

i1εi(h)hi

(12)

And lim εih⟶ 0(h) 0 for i 1 2 n

e linear transformation L Rn⟶ R is defined byL(h) 1113936

ni1 αihi with h (h1 hn) and α1 αn isin R

is linear transformation is denoted by Dαf(a) which iscalled CoD of f of the order α isin (0 1] at a

Remark 1 e equivalence of conditions (i) and (ii) isimmediate since

limh⟶0

ε(h) 0harrε(h)h o(h) (13)

To see the equivalence between conditions (ii) and (iii)we take

εi ε(h)hi

h

ε(h) 1

h1113944

ni1εi(h)hi (14)

As |hih|le 1 then we have the following

(i) If limεh⟶0(h) 0 then lim εi h⟶ 0(h) 0(ii) If lim εi h⟶ 0(h) 0 for i 1 n then we obtain

limh⟶0

ε(h)le limh⟶0

1h

1113944

n

i1εi(h)

le lim

h⟶01113944

n

i1εi(h)

0

(15)

ie limh⟶0ε(h) 0 Hence the conditions (ii) and (iii) areequivalent

Example 1 Consider a function f defined by f(x y) ex minus

2cosy and a point (a b) isin R2 with agt 0 and bgt 0 thenDαf(a b)(h1 h2) h1a

1minus αeα + 2h2b1minus α sin b

Solution to prove this let us note that

limh1 h2( )⟶(00)

f a + a1minus α

h1 b + b1minus α

h21113872 1113873 minus f(a b) minus L h1 h2( 1113857

h1 h2( 1113857

limh1 h2( )⟶(00)

ea+a1minus αh1

minus 2 cos b + b1minus α

h21113872 1113873 minus ea

minus 2 cos b( 1113857 minus h1a1minus α

ea

+ 2h2b1minus α sin b1113872 1113873

h21 + h

22

1113969

le limh1⟶0

ea+a1minus αh1

minus ea

minus h1a1minus α

ea

h1minus 2 lim

h2⟶0

cos b + b1minus α

h21113872 1113873 minus cos b + b1minus α sin b

h2

limh1⟶0

ea+a1minus αh1

minus ea

h1minus a

1minus αe

a⎛⎝ ⎞⎠ minus 2 limh2⟶0

cos b + b1minus α

h21113872 1113873 minus cos b

h2+ b

1minus α sin b⎛⎝ ⎞⎠ a1minus α

ea

minus a1minus α

ea

1113872 1113873

minus 2 minus b1minus α sin b + b

1minus α sin b1113872 1113873 0

(16)

Theorem 8 If a RVF f with n variables α DF ata (a1 an) isin Rn each ai gt 0 then f is CF at a isin Rn

Proof Since f is α DF at a we can write the following

f a1 + h1a11minus α

an + hnan1minus α

1113872 1113873 minus f a1 an( 1113857 1113944n

i1αihi + o(h) (17)

4 Journal of Mathematics

By taking the limits of the two sides of the equality ash⟶ 0 we have

limh⟶0

f a1 + h1a11minus α

an + hnan1minus α

1113872 1113873 f a1 an( 1113857

(18)

Hence f is CF at a isin Rn

Theorem 9 If a RVF f with n variables is α DF ata (a1 an) isin Rn each ai gt 0 then zαzxα

i f(a) exists for1le ile n and the CoD of f of the order α isin (0 1] is expressedas

Dαf(a)(h) 1113944

n

i1

zα

zxαi

f(a)hi (19)

where h (h1 hn)

Proof By setting hj 0foralljne i in the formula (12) we have

f a1 ai + hiai1minus α

+ middot middot middot + an1113872 1113873 minus f a1 an( 1113857 αihi + εi(h)hi

(20)

By multiplying by 1hi we can write

f a1 ai + hiai1minus α

+ middot middot middot + an1113872 1113873 minus f a1 an( 1113857

hi

αi + εi(h)

(21)

By taking the limits of the two sides of the equality ashi⟶ 0 we have

αi zα

zxαi

f(a) foralli 1 2 n (22)

Finally by substituting the values above αi in the formulaDαf(a)(h) 1113936

ni1 αihi the result is followed

Corollary 1 If a RVF f with n variables is α DF ata (a1 an) isin Rn each ai gt 0 then Dαf(a) is unique

Remark 2 If a RVF f with n variables is α DF ata (a1 an) isin Rn each ai gt 0 then the CoV of gradientof f of the order α isin (0 1] at a is

nablaαf(a) zα

zxα1

f(a) zα

zxαn

f(a)1113888 1113889 (23)

Also the matrix form (MF) of equation (19) is given asfollows

Dαf(a)(h) nablaαf(a) middot h

zα

zxα1

f(a) zα

zxαn

f(a)1113888 1113889 middot

h1

⋮

hn

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(24)

Theorem 10 Let α isin (0 1] f g X⟶ R be a RVF definedin an open set (OS) X sub Rn ni forall x (x1 xn) isin X eachxi gt 0 and a point a (a1 an) isin X If f g are α DF at athen we have

(i) Dα(λf + μg)(a) λDα(f)(a) + μDα(g)(a)forall λ μ isin R

(ii) Dα(fg)(a) Dα(f)(a) middot g(a) + f(a)Dα(g)(a)

Proof (i) follows fromDefinition 4 thus it follows the proofof (i)

For (ii) let A (a1 + h1a1minus α1 an + hna1minus α

n ) and thenwe have

limh⟶0

((fg)(A) minus (fg)(a)) minus Dαf(a) middot g(a) + f(a) middot D

αg(a)( 1113857(h)

h

limh⟶0

(f(A) minus f(a)) minus Dαf(a)(h)

hmiddot g(a) + f(a) middot

(g(A) minus g(a)) minus Dαg(a)(h)

h1113888 1113889 + lim

h⟶0

(f(A) minus f(a)) middot (g(A) minus g(a))

h

0 + 0 + limh⟶0

Dαf(a)(h)( 1113857 middot D

αg(a)(h)( 1113857

h lim

h⟶0D

αf(a)

hh

1113888 11138891113888 1113889 middot Dαg(a)

hh

1113888 11138891113888 1113889 middot h 0

(25)

Theorem 11 Let α isin (0 1] f X⟶ R be a RVF defined inan OS X sub Rn niforall x (x1 xn) isin X each xi gt 0 and apoint a (a1 an) isin X If the function f has all CoV ofPDrs of the order α at each point of a neighbourhood of thepoint a U(a) with U(a) sub X and they are continuous at athen f is α DF at a

Proof See eorem 21 proof in [27]

Remark 3 e above theorem allows defining the space ofRVFs with n variables by having continuous CoV of PDrs oforder α isin (0 1] in a domain X sub Rn which can be denotedby Cα(X R)

Finally we can easily extend all of the above results to theVVFs of SRVs

Theorem 12 Assume that α isin (0 1] f X⟶ Rm be a VVFdefined in an OS X sub Rn niforallx (x1 xn) isin X each

Journal of Mathematics 5

xi gt 0 and the point a (a1 an) isin X e function f isα DF at a iff its components are α DF at a and if these

components are f1 f2 fm then the components ofDαf(a) are the α-derivativesDαfj(a) for j 1 2 m ie

f f1 f2 fm( 1113857rArr Dαf(a) D

αf1(a) D

αf2(a) D

αfm(a)( 1113857 (26)

Proof It is similarly proven as same as traditional calculusby applying Dα instead of derivative

Remark 4 Assume that α isin (0 1] f X⟶ Rm is a VVFdefined in an OS X sub Rn niforall x (x1 xn) isin X eachxi gt 0 and a point a (a1 an) isin X If function f isα-differentiable at a then zαzxα

i fj(a) exists fori 1 2 n j 1 2 m and the CoV of α-Jacobian off of order α isin (0 1] at a is expressed as

Jαf(a)

zα

zxα1f1(a)

zα

zxαn

f1(a)

⋮ ⋮

zα

zxα1fm(a)

zα

zxαn

fm(a)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(27)

4 The Chain Rule

Let us prove the CR for the functions of SVs in 2 particularcases For the proofrsquos purpose the continuityrsquos hypothesis ofPDrs is given [28]

Theorem 13 (CR) Suppose that t isin R andx (x1 xn) isin Rn If f(t) (f1(t) fn(t)) is α DF

at agt 0 ni α isin (0 1] and a RVF g with n variablesx1 xn is α DF at f(a) isin Rn all fi(a)gt 0 α isin (0 1] en the composition g ∘f is α-differential at a and

Tαg ∘ f( 1113857(a) 1113944n

i1

zα

zxαi

g(f(a)) middot fi(a)( 1113857αminus 1

middot Tαfi( 1113857(a)

(28)

Proof Assumeg isin C1(U(f(a)) R) niU(f(a)) is the pointf(a) neighborhood Suppose thath(t) (g ∘ f)(t) g(f(t)) By setting u a + εa1minus α inDefinition 1 we see that

Tαh( 1113857(a) limt⟶a

h(t) minus h(a)

t minus aa1minus α

limt⟶a

g(f(t)) minus g(f(a))

t minus aa1minus α

(29)

Without loss of generality (WLOG) U(f(a)) is assumedto be an open ball (OB) represented by B(f(a) r) Since f isa CF then along with points (f1(a) fn(a)) and(f1(t) fn(t)) the points(f1(a) f2(t) fn(t)) (f1(a) f2(a) fn(t))

and lines connecting them must also be the ball B(f(a) r)In fact using the classical MVT for differentiable functions(DFs) of one variable is in the following computation [28]

h(t) minus h(a)

t minus aa1minus α

g(f(t)) minus g(f(a))

t minus aa1minus α

g f1(t) fn(t)( 1113857 minus g f1(a) f2(t) fn(t)( 1113857

t minus aa1minus α

+g f1(a) f2(t) fn(t)( 1113857 minus g f1(a) f2(a) f3(t) fn(t)( 1113857

t minus aa1minus α

+ middot middot middot

+g f1(a) f2(a) fn(t)( 1113857 minus g f1(a) fn(a)( 1113857

t minus aa1minus α

z

zx1g c1 f2(t) fn(t)( 1113857

f1(t) minus f1(a)

t minus aa1minus α

z

zx2g f1(a) c2 fn(t)( 1113857

f2(t) minus f2(a)

t minus aa1minus α

+ middot middot middot

+z

zx2g f1(a) c2 fn(t)( 1113857

f2(t) minus f2(a)

t minus aa1minus α

+ middot middot middot

+z

zxn

g f1(a) f2(a) cn( 1113857fn(t) minus fn(a)

t minus aa1minus α

(30)

6 Journal of Mathematics

where ci is between fi(a) and fi(t) forall i 1 2 n By taking limits as t⟶ a and using the continuity ofPDrs of g as well as the fact that ci⟶ fi(a) foralli 1 2 n equation (29) is expressed as

Tαh( 1113857(a) limt⟶a

h(t) minus h(a)

t minus aa1minus α

limt⟶a

g(f(t)) minus g(f(a))

t minus aa1minus α

limt⟶infin

z

zx1g c1 f2(t)( 1113857

f1(t) minus f1(a)

t minus aa1minus α

+ middot middot middot1113888 1113889

+z

zx2g f1(a) c2 fn(t)( 1113857

f2(t) minus f2(a)

t minus aa1minus α

+ middot middot middot

+z

zxn

g f1(a) f2(a)1minus α

cn1113872 1113873fn(t) minus fn(a)

t minus aa1minus α

1113889 +z

zx1g(f(a)) middot f

(1)1 (a) middot a

1minus α

+z

zx2g(f(a)) middot f

(1)2 (a) middot a

1minus αmiddot f1(a)

αminus 1middot f

(1)1 (a) middot a

1minus α

z

zx1g(f(a)) middot f1(a)

1minus αmiddot f1(a)

1minus αmiddot f1(a)

αminus 1middot f

(1)1 (a) middot a

1minus α+

z

zx2g(f(a)) middot f2(a)

1minus α

middot f2(a)αminus 1

middot f(1)2 (a) middot a

1minus α+ middot middot middot +

z

zxn

g(f(a)) middot f2(a)1minus α

middot f2(a)αminus 1

middot f(1)n (a) middot a

1minus α

zα

zxα1

g(f(a)) middot f1(a)αminus 1

middot Tαf1( 1113857(α) +zα

zxα2

g(f(a)) middot f2(a)αminus 1

middot Tαf2( 1113857(a) + middot middot middot

+zα

zxαn

g(f(a)) middot fn(a)αminus 1

middot Tαf1( 1113857(α)

(31)

Our proof is completely done Remark 5 e MF of equation (28) is expressed as

Dα(g middot f)(a)(h)

zα

zxα1

g(f(a)) zα

zxαn

g(f(a))1113888 1113889

middot

f1(a)αminus 1 0 0

0 ⋱ 0

0 0 fn(a)αminus 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠middot

Tαf1( 1113857(a)

⋮

Tαfn( 1113857(a)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠middot h

(32)

Theorem 14 (CR) Let x (x1 xn) isin Rn andy (y1 ym) isin Rm If f(x1 xn) (f1(x1

xn) fm(x1 xn)) is a α DF at a (a1 an) isin Rneach ai gt 0 ni α isin (0 1] and a RVF g with variables y1 ym

is α DF at f(a) isin Rn where allfi(a)gt 0 niα isin (0 1]en thecomposition g ∘ f is α-differentiable at a and we have

zα

zxαi

(g ∘ f)(a) 1113944m

j1

zα

zyαj

g(f(a)) middot fj(a)αminus 1

middotzα

zxαi

fj(a)

forall i 1 2 n

(33)

Journal of Mathematics 7

Proof Proof According to PDrrsquos definition and eorem14 the following is implied

Remark 6 e MF of equation (33) is expressed as

Dα(g ∘ f)(a)(h)

zα

zyα1

g(f(a)) zα

zyαm

g(f(a))1113888 1113889 middot

f1(a)αminus 1 0 0

0 ⋱ 0

0 0 fm(a)αminus 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠middot

zα

zxα1f1(a)

zα

zxαn

f1(a)

middot middot middot middot middot middot

zα

zxα1fn(a)

zα

zxαn

fn(a)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

middot

h1

⋮

hn

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (34)

5 Conformable Implicit Function Theorem

In this section the CoV of IFm for SVs is obtained Firstwe establish the conformable theorem of existence andregularity of the implicit function for the case of a singleequation

Theorem 15 (Conformable implicit function theorem forthe case of a single equation) Suppose that α isin (0 1]F X⟶ R is a RVF defined in an OS X sub Rn+1 ni forall(x1 xn y) isin X each xi ygt 0 and point(a1 an b) isin X Suppose that

(i) F(a1 an b) 0(ii) F isin Cα(X R)

(iii) zαzyαF(a1 an b)ne 0

en there is a neighbourhood U sub Rn of (a1 an)such that there is a unique function (UF) y g(x1 xn)

that satisfies the followingg(a1 an) b and F(x1 xn g(x1 xn))

0 forall(x1 xn) isin UFinally y g(x1 xn) is Cα in U and for every

i 1 2 n we have

zα

zxαi

g x1 xn( 1113857 minuszαzx

αi F x1 xn( 1113857 g x1 xn( 1113857( 1113857( 1113857

zαzy

αF x1 xn( 1113857 g x1 xn( 1113857( 1113857( 1113857 middot g x1 xn( 1113857( 1113857

αminus 1 (35)

Proof WLOG X is assumed to be an OB represented byB((a1 an b) ε0) Let ρ isin (0 ε0) If we call δ

ε20 minus ρ21113969

it is verified that

x1 xn( 1113857 minus a1 an( 1113857

lt δ and |y minus b|lt ρ1113960 1113961rArr x1 xn y( 1113857 isin B a1 an b( 1113857 ε0( 1113857 (36)

Note that in particular if |y minus b|lt ρ then(a1 an y) isin B((a1 an b) ε0)

Since zαzyαF(a1 an b)ne 0 it is assumed to bepositive (otherwise minus F is considered instead of F) From factthat F(a1 an b) 0 it follows that

F a1 an b minus ρ( 1113857gt 0

F a1 an b minus ρ( 1113857lt 0(37)

By the continuity of F at (a1 an b minus ρ) and(a1 an b + ρ) there exists δprime isin (0 δ) such that

x1 xn( 1113857 minus a1 an( 1113857

lt δprimerArr F x1 xn b minus ρ( 1113857gt 0 andF x1 xn b + ρ( 1113857lt 01113858 1113859 (38)

Since the function y↦F(x1 xn y) is CF on the in-terval [b minus ρ b + ρ ] forall (x1 xn) isin B((a1 an) δprime) andvia the classical Bolzanorsquos theorem (Bm) it implies thatexistyx isin ( b minus ρ b + ρ) niF(x1 xn yx) 0 for eachx (x1 xn) en yrsquos value is unique since a functionwhose derivative is positive has more than zero On the other

hand by having U B((a1 an) δprime) for each(x1 xn) isin U there exists a unique y

niF(x1 xn y) 0 we can write y g(x1 xn) andthen this function will be proven to be CF onB((a1 an) δprime)e continuity of the function g at the point(a1 an) is clear since for each ρgt 0 exist a value δprime gt 0 ni

8 Journal of Mathematics

x1 xn( 1113857 minus a1 an( 1113857

lt δprimerArr b minus yx

11138681113868111386811138681113868111386811138681113868

lt ρhArr b minus g x1 xn( 11138571113868111386811138681113868

1113868111386811138681113868lt ρ(39)

e function g continuity will be proven at any point(x1 xn) isin B((a1 an) δprime) by simply substitutingB((a1 an) δprime) for an OB B((x1 x)) is contained inB((a1 an) δprime)

Finally let us now show formula (35) By applyingeorem 14 to equation

F x1 xn y( 1113857 0 we have

zα

zxαi

F(x g(x)) +zα

zyα F(x g(x)) middot g(x)

αminus 1middot

zα

zxαi

g(x) 0

(40)

forall i 1 2 n nix (x1 xn) Solving zαzxαi g(x)

we get (35) In addition the formula (35) on the right side iscontinuous so the continuity of CoV of PDrs zαzxα

i g(x) foralli 1 2 n follows

eorem 15 will provide us a help in computing CoV ofPDrs of implicit function of SVs

Example 2 Consider

F(x y z) x3

+ 3y2

+ 4xz2

minus 3yz2

minus 5 0 (41)

is equationrsquos one solution is (1 1 1) F is obviously inCα which is OB represented by B((1 1 1) ε0) withx y zgt 0 for some α isin (0 1] since

zα

zzα F(1 1 1) 8xz

2minus αminus 6yz

2minus α1113872 11138731113961

(111) 2ne 0 (42)

eorem 15 implies that there is aneighbourhood U sub R2 of (1 1) niexist a UF z g(x y)

satisfies the followingg(1 1) 1 andF(x y g(x y)) 0 forall(x y) isin U

Moreover z g(x y) is Cα in U and we have

zα

zxα g(x y) minus

3x + 4z2

1113872 1113873x1minus α

(8x minus 6y)z

zα

zyα g(x y) minus

6y minus 3z2

1113872 1113873y1minus α

(8x minus 6y)z

(43)

Finally we obtain zαzxαg(1 1) minus (72) andzαzxαg(1 1) minus (32)

Finally CoV of IFm for a system of several equationsand SRVs is obtained

Theorem 16 (Conformable general implicit functiontheorem) Let α isin (0 1] F X⟶ Rm be a VVF defined inan OS X sub Rn+m ni forall (x y) (x1 xn y1 ym) isin Xeach xi yj gt 0 and point(a b) (a1 an b1 bm) isin X Assume that

(i) F(a b) 0(ii) F isin Cα(X Rm )

(iii) det[JαyF(a b)]ne 0

en there is a neighbourhood U sub Rn of a niexist a UFf U⟶ Rm x↦ y f(x) that satisfies f(a) b andF(x f(x) ) 0 forallx isin U

Finally y f(x) is class Cα in U and for everyi 1 2 n we have

zαfzxα

i

1113890 1113891

t

minus fαminus 11113872 1113873

minus 1middot J

αyF1113872 1113873

minus 1middot

zαFzxα

i

1113890 1113891

t

(44)

where

zαf

zxαi

1113890 1113891 zαf1

zxαi

zαfm

zxαi

1113888 1113889

fαminus 1

fαminus 11 0

⋮ ⋮

0 fαminus 1m

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

JαyF

zαF1

zyα1

zαF1

zyαm

zαFm

zyα1

zαFm

zyαm

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

zαF

zxαi

1113890 1113891 zαF1

zxαi

zαFm

zxαi

1113888 1113889

(45)

Proof e proof of existence and uniqueness of the implicitfunction is done similar to the traditional multivariablecalculus by applying mathematical induction on q and usingthe conformable implicit function theorem for severalvariables [28]

Let us now show formula (44) Assume that a systemwith several equations and SRVs is expressed as

F(x y) 0 or

F1 x1 xn y1 ym( 1113857 0

⋮

Fm x1 xn y1 ym( 1113857 0

⎧⎪⎪⎨

⎪⎪⎩ (46)

which satisfies hypotheses (i)ndash(iii) of the theorem then thissystem is defined in a neighbourhood U sub Rn of a theimplicit function y f(x) class Cα in U such that f(a) bwhich satisfies equation (1) ie

F(x f(x)) 0 or

F1 x f1(x) fm(x)( 1113857 0

Fm x f1(x) fm(x)( 1113857 0

⎧⎪⎪⎨

⎪⎪⎩(47)

By applying CoV of CR to the above equation we have

Journal of Mathematics 9

zαF1

zxαi

+zαF1

zyα1

middot fαminus 11 middot

zαf1

zxαi

+ middot middot middot +zαF1

zyαm

middot fαminus 1m middot

zαfm

zxαi

0

⋮

zαFm

zxαi

+zαFm

zyα1

middot fαminus 11 middot

zαf1

zxαi

+ middot middot middot +zαFm

zyαm

middot fαminus 1m middot

zαfm

zxαi

0

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

orzαF

zxαi

+ 1113944m

j1

zαF

zyαj

middot fαminus 1j middot

zαfj

zxαi

0

(48)

for all i 1 2 n In addition the MF of equation (48) is given as follows

zαF1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus

zαF1

zyα1

zαF1

zyαm

zαFm

zyα1

zαFm

zyαm

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

middot

fαminus 11 0

⋮ ⋮

0 fαminus 1m

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠middot

zαf1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

orzαFzxα

i

1113890 1113891

t

minus JαyF middot fαminus 1

middotzαfzxα

i

1113890 1113891

t

(49)

Since JαyF and fαminus 1 are regular matrices by hypothesis we

have

zαf1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus

fαminus 11 0

⋮ ⋮

0 fαminus 1m

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus 1

middot

zαF1

zyα1

zαF1

zyαm

zαFm

zyα1

zαFm

zyαm

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus 1

middot

zαF1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

orzαfzxα

i

1113890 1113891

t

minus fαminus 11113872 1113873

minus 1middot J

αyF1113872 1113873

minus 1middot

zαFzxα

i

1113890 1113891

t

(50)

which completes the proofWe will now show that eorem 16 can be used to

compute CoV of PDrs of systems with several equations andSRVs

Example 3 Consider a system of two equations and two realvariables

F1(x y z w) x2

+ y2

+ z2

+ w2

minus 6 0

F2(x y z w) x2

minus y2

+ z2

minus w2

0

⎧⎨

⎩ (51)

One solution of this equation is(x y z w) (1 1

2

radic

2

radic) Clearly F (F1 F2) is in Cα

which is OB B((1 12

radic

2

radic) ε0) with x y z wgt 0 for

some α isin (0 1] since

10 Journal of Mathematics

det JαzwF((

2

radic

2

radic 1 1))1113960 1113961 det

2z2minus α 2w2minus α

2z2minus α minus 2w2minus α⎛⎝ ⎞⎠⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

(112

radic

2

radic)

minus322propne 0 (52)

eorem 16 implies that there is aneighbourhood U sub R2 of (

2

radic

2

radic) niexist a UF f (f1 f2)

given by

z f1(x y)

w f2(x y)1113896 (53)

that satisfies

f1(1 1) 2

radic

f2(1 1) 2

radic

⎧⎨

⎩

F1 x y f1(x y) f2(x y)( 1113857 0

F2 x y f1(x y) f2(x y)( 1113857 0forall(x y) isin U1113896

(54)

Moreover f (f1 f2) is class Cα in U and we have

zαf1

zxα

zαf2

zxα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ minus

zαminus 1 0

0 wαminus 1⎛⎝ ⎞⎠

minus 1

middot2z2minus α 2w2minus α

2z2minus α minus 2w2minus α⎛⎝ ⎞⎠

minus 1

middot2x

2minus α

2x2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz1minus α 0

0 w1minus α

⎛⎜⎝ ⎞⎟⎠ middotzαminus 2

zαminus 2

wαminus 2

minus wαminus 2

⎛⎜⎝ ⎞⎟⎠

middot2x

2minus α

2x2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz

minus 1z

minus 1

wminus 1

minus wminus 1

⎛⎜⎝ ⎞⎟⎠ middot2x

2minus α

2x2minus α

⎛⎜⎝ ⎞⎟⎠ minus

x2minus α

z

0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

zαf1

zyα

zαf2

zyα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minuszαminus 1 0

0 wαminus 1⎛⎝ ⎞⎠

minus 1

middot2z2minus α 2w2minus α

2z2minus α minus 2w2minus α⎛⎝ ⎞⎠

minus 1

middot2y

2minus α

minus 2y2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz1minus α 0

0 w1minus α

⎛⎜⎝ ⎞⎟⎠ middotzαminus 2

zαminus 2

wαminus 2

minus wαminus 2

⎛⎜⎝ ⎞⎟⎠

middot2y

2minus α

minus 2y2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz

minus 1z

minus 1

wminus 1

minus wminus 1

⎛⎜⎝ ⎞⎟⎠ middot2y

2minus α

minus 2y2minus α

⎛⎜⎝ ⎞⎟⎠

0

minusy2minus α

w

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(55)

Finally we have

zαf1

zxα

zαf2

zxα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(112

radic

2

radic)

minus12

radic

0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

zαf1

zyα

zαf2

zyα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(112

radic

2

radic)

0

minus12

radic

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(56)

According to all of our numerical examples in this workwith the help of our investigation to all proposed theoremsand properties in this article the numerical examples showthat all our obtained results are valid and efficient As seen inall these examples the computations are simple and cost-

efficient which are important when scientific phenomenaare modelled in the sense of conformable calculus All in allthe simplicity of computations is always needed in modelingapplications in comparison to other complicated compu-tations that are needed using other approaches in otherfractional operators All our obtained results in this work areaccurate because our results coincide with the usual integer-order results

6 Conclusion

Anew investigation on the CoV ofmultivariable calculus hasbeen discussed in this work in detail e α-derivative of afunction of SVs and all related properties have been in-vestigatede CoV of CR for functions of SVs has also beenstudied e CoV of IFm has been presented in the lastpart of our results and numerical experiments have beenconducted to support our theoretical results e findings ofthis investigations show that all results formulated via CoD

Journal of Mathematics 11

coincide with the ones in the traditional integer case Allresults in this work can be potentially applied in modelingoceanographic phenomena such as Stommelrsquos box model ofthermohaline circulation and other related models wherethis studyrsquos analysis can be further extended or generalizedin 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

Authorsrsquo Contributions

MohammedK A Kaabar involved in actualization developedmethodology performed formal analysis validated and in-vestigated the study prepared the initial draft and supervisedand edited the original draft Francisco Martınez InmaculadaMartınez and Silvestre Paredes involved in actualizationdeveloped methodology performed formal analysis validatedand investigated the study and prepared the initial draftZailan Siri developed methodology performed formal anal-ysis validated and investigated the study and prepared theinitial draft All authors read and approved the final version

References

[1] M M A Khater M S Mohamed and R A M Attia ldquoOnsemi analytical and numerical simulations for a mathematicalbiological model the time-fractional nonlinear Kolmogor-ovndashPetrovskiindashPiskunov (KPP) equationrdquo Chaos Solitons ampFractals vol 144 Article ID 110676 2021

[2] M A Khater K S Nisar and M S Mohamed ldquoNumericalinvestigation for the fractional nonlinear space-time telegraphequation via the trigonometric Quintic B-spline schemerdquoMathematical Methods in the Applied Sciences vol 44 no 6pp 4598ndash4606 2021

[3] M M A Khater M S Mohamed and S K Elagan ldquoDiverseaccurate computational solutions of the nonlinearKleinndashFockndashGordon equationrdquo Results in Physics vol 23Article ID 104003 2021

[4] M F Li J R Ren and T Zhu ldquoFractional vector calculus andfractional special functionrdquo arXiv preprint10012889 2010

[5] D J Wollkind and B J Dichone Comprehensive AppliedMathematical Modeling in the Natural and Engineering Sci-ences Springer International Publishing Berlin Germany2017

[6] P J Olver Introduction to Partial Differential EquationsSpringer International Publishing Berlin Germany 2014

[7] J D Krehbiel and J B Freund ldquoStokes flow inside a sphere inan inviscid extensional flowrdquo Zeitschrift fur AngewandteMathematik und Physik vol 68 no 4 pp 68ndash81 2017

[8] A A Kilbas H M Srivastava and J J Trujillo ldquoeory andapplications of fractional differential equationsrdquo in North-Holland Mathematics StudiesVol 204 Elsevier AmsterdamNetherlands 2006

[9] K S Miller An Introduction to Fractional Calculus andFractional Differential Equations John Wiley and SonsHoboken NJ USA 1993

[10] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014

[11] R Khalil M Al Horani and M Abu Hammad ldquoGeometricmeaning of conformable derivative via fractional cordsrdquo eJournal of Mathematics and Computer Science vol 19 no 4pp 241ndash245 2019

[12] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journalof Computational and Applied Mathematics vol 279pp 57ndash66 2015

[13] O S Iyiola and E R Nwaeze ldquoSome new results on the newconformable fractional calculus with application usingDprimeAlambert approachrdquo Progress in Fractional Differentiationand Applications vol 2 no 2 pp 115ndash122 2016

[14] A Atangana D Baleanu and A Alsaedi ldquoNew properties ofconformable derivativerdquo Open Mathematics vol 13 pp 57ndash63 2015

[15] N Yazici and U Gozutok ldquoMultivariable conformablefractional calculusrdquo Filomat vol 32 no 2 pp 45ndash53 2018

[16] F Martınez I Martınez and S Paredes ldquoConformableEulerprimes theorem on homogeneous functionsrdquo ComputationalMathematical Methods vol 1 no 5 pp 1ndash11 2018

[17] A A Hyder and A H Soliman ldquoExact solutions of space-timelocal fractal nonlinear evolution equations a generalizedconformable derivative approachrdquo Results in Physics vol 17Article ID 103135 2020

[18] A A Hyder ldquoWhite noise theory and general improvedKudryashov method for stochastic nonlinear evolutionequations with conformable derivativesrdquo Advances in Dif-ference Equations vol 202019 pages 2020

[19] A H Soliman and A A Hyder ldquoClosed-form solutions ofstochastic KdV equation with generalized conformable deriv-ativesrdquo Physica Scripta vol 95 no 6 Article ID 065219 2020

[20] V E Tarasov ldquoNo violation of the Leibniz rule No fractionalderivativerdquo Communications in Nonlinear Science and Nu-merical Simulation vol 18 no 11 pp 2945ndash2948 2013

[21] R Khalil M Al Horani A Yousef and M SababhehldquoFractional analytic functionsrdquo Far East Journal of Mathe-matical Sciences vol 103 no 1 pp 113ndash123 2018

[22] S Uccedilar and N Y Ozgur ldquoComplex conformable derivativerdquoArabian Journal of Geosciences vol 12 no 201 2019

[23] F Martınez I Martınez I Martınez M K A Kaabar andS Paredes ldquoNew results on complex conformable integralrdquoAIMS Mathematics vol 5 no 6 pp 7695ndash7710 2020

[24] H Kiskinov M Petkova A Zahariev and M VeselinovaldquoSome results about conformable derivatives in Banach spacesand an application to the partial differential equationsrdquo inAIPConference Proceedingsvol 2333 no 1 AIP Publishing LLCArticle ID 120002 2021

[25] U Gozutok H A Ccediloban and Y Sagıroglu ldquoFrenet framewith respect to conformable derivativerdquo Filomat vol 33no 6 pp 1541ndash1550 2019

[26] J C Mayo-Maldonado G Fernandez-Anaya and O F Ruiz-Martinez ldquoStability of conformable linear differential systemsa behavioural framework with applications in fractional-ordercontrolrdquo IET Control eory amp Applications vol 14 no 18pp 2900ndash2913 2020

[27] M Al Horani and R Khalil ldquoTotal fractional differential withapplications to exact fractional differential equationsrdquo In-ternational Journal of Computer Mathematics vol 95 no 6-7pp 1444ndash1452 2018

[28] J E Marsden and M J Hoffman Vector Calculus WH Freeman and Company New York NY USA 4th edition1996

12 Journal of Mathematics

By taking the limits of the two sides of the equality ash⟶ 0 we have

limh⟶0

f a1 + h1a11minus α

an + hnan1minus α

1113872 1113873 f a1 an( 1113857

(18)

Hence f is CF at a isin Rn

Theorem 9 If a RVF f with n variables is α DF ata (a1 an) isin Rn each ai gt 0 then zαzxα

i f(a) exists for1le ile n and the CoD of f of the order α isin (0 1] is expressedas

Dαf(a)(h) 1113944

n

i1

zα

zxαi

f(a)hi (19)

where h (h1 hn)

Proof By setting hj 0foralljne i in the formula (12) we have

f a1 ai + hiai1minus α

+ middot middot middot + an1113872 1113873 minus f a1 an( 1113857 αihi + εi(h)hi

(20)

By multiplying by 1hi we can write

f a1 ai + hiai1minus α

+ middot middot middot + an1113872 1113873 minus f a1 an( 1113857

hi

αi + εi(h)

(21)

By taking the limits of the two sides of the equality ashi⟶ 0 we have

αi zα

zxαi

f(a) foralli 1 2 n (22)

Finally by substituting the values above αi in the formulaDαf(a)(h) 1113936

ni1 αihi the result is followed

Corollary 1 If a RVF f with n variables is α DF ata (a1 an) isin Rn each ai gt 0 then Dαf(a) is unique

Remark 2 If a RVF f with n variables is α DF ata (a1 an) isin Rn each ai gt 0 then the CoV of gradientof f of the order α isin (0 1] at a is

nablaαf(a) zα

zxα1

f(a) zα

zxαn

f(a)1113888 1113889 (23)

Also the matrix form (MF) of equation (19) is given asfollows

Dαf(a)(h) nablaαf(a) middot h

zα

zxα1

f(a) zα

zxαn

f(a)1113888 1113889 middot

h1

⋮

hn

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(24)

Theorem 10 Let α isin (0 1] f g X⟶ R be a RVF definedin an open set (OS) X sub Rn ni forall x (x1 xn) isin X eachxi gt 0 and a point a (a1 an) isin X If f g are α DF at athen we have

(i) Dα(λf + μg)(a) λDα(f)(a) + μDα(g)(a)forall λ μ isin R

(ii) Dα(fg)(a) Dα(f)(a) middot g(a) + f(a)Dα(g)(a)

Proof (i) follows fromDefinition 4 thus it follows the proofof (i)

For (ii) let A (a1 + h1a1minus α1 an + hna1minus α

n ) and thenwe have

limh⟶0

((fg)(A) minus (fg)(a)) minus Dαf(a) middot g(a) + f(a) middot D

αg(a)( 1113857(h)

h

limh⟶0

(f(A) minus f(a)) minus Dαf(a)(h)

hmiddot g(a) + f(a) middot

(g(A) minus g(a)) minus Dαg(a)(h)

h1113888 1113889 + lim

h⟶0

(f(A) minus f(a)) middot (g(A) minus g(a))

h

0 + 0 + limh⟶0

Dαf(a)(h)( 1113857 middot D

αg(a)(h)( 1113857

h lim

h⟶0D

αf(a)

hh

1113888 11138891113888 1113889 middot Dαg(a)

hh

1113888 11138891113888 1113889 middot h 0

(25)

Theorem 11 Let α isin (0 1] f X⟶ R be a RVF defined inan OS X sub Rn niforall x (x1 xn) isin X each xi gt 0 and apoint a (a1 an) isin X If the function f has all CoV ofPDrs of the order α at each point of a neighbourhood of thepoint a U(a) with U(a) sub X and they are continuous at athen f is α DF at a

Proof See eorem 21 proof in [27]

Remark 3 e above theorem allows defining the space ofRVFs with n variables by having continuous CoV of PDrs oforder α isin (0 1] in a domain X sub Rn which can be denotedby Cα(X R)

Finally we can easily extend all of the above results to theVVFs of SRVs

Theorem 12 Assume that α isin (0 1] f X⟶ Rm be a VVFdefined in an OS X sub Rn niforallx (x1 xn) isin X each

Journal of Mathematics 5

xi gt 0 and the point a (a1 an) isin X e function f isα DF at a iff its components are α DF at a and if these

components are f1 f2 fm then the components ofDαf(a) are the α-derivativesDαfj(a) for j 1 2 m ie

f f1 f2 fm( 1113857rArr Dαf(a) D

αf1(a) D

αf2(a) D

αfm(a)( 1113857 (26)

Proof It is similarly proven as same as traditional calculusby applying Dα instead of derivative

Remark 4 Assume that α isin (0 1] f X⟶ Rm is a VVFdefined in an OS X sub Rn niforall x (x1 xn) isin X eachxi gt 0 and a point a (a1 an) isin X If function f isα-differentiable at a then zαzxα

i fj(a) exists fori 1 2 n j 1 2 m and the CoV of α-Jacobian off of order α isin (0 1] at a is expressed as

Jαf(a)

zα

zxα1f1(a)

zα

zxαn

f1(a)

⋮ ⋮

zα

zxα1fm(a)

zα

zxαn

fm(a)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(27)

4 The Chain Rule

Let us prove the CR for the functions of SVs in 2 particularcases For the proofrsquos purpose the continuityrsquos hypothesis ofPDrs is given [28]

Theorem 13 (CR) Suppose that t isin R andx (x1 xn) isin Rn If f(t) (f1(t) fn(t)) is α DF

at agt 0 ni α isin (0 1] and a RVF g with n variablesx1 xn is α DF at f(a) isin Rn all fi(a)gt 0 α isin (0 1] en the composition g ∘f is α-differential at a and

Tαg ∘ f( 1113857(a) 1113944n

i1

zα

zxαi

g(f(a)) middot fi(a)( 1113857αminus 1

middot Tαfi( 1113857(a)

(28)

Proof Assumeg isin C1(U(f(a)) R) niU(f(a)) is the pointf(a) neighborhood Suppose thath(t) (g ∘ f)(t) g(f(t)) By setting u a + εa1minus α inDefinition 1 we see that

Tαh( 1113857(a) limt⟶a

h(t) minus h(a)

t minus aa1minus α

limt⟶a

g(f(t)) minus g(f(a))

t minus aa1minus α

(29)

Without loss of generality (WLOG) U(f(a)) is assumedto be an open ball (OB) represented by B(f(a) r) Since f isa CF then along with points (f1(a) fn(a)) and(f1(t) fn(t)) the points(f1(a) f2(t) fn(t)) (f1(a) f2(a) fn(t))

and lines connecting them must also be the ball B(f(a) r)In fact using the classical MVT for differentiable functions(DFs) of one variable is in the following computation [28]

h(t) minus h(a)

t minus aa1minus α

g(f(t)) minus g(f(a))

t minus aa1minus α

g f1(t) fn(t)( 1113857 minus g f1(a) f2(t) fn(t)( 1113857

t minus aa1minus α

+g f1(a) f2(t) fn(t)( 1113857 minus g f1(a) f2(a) f3(t) fn(t)( 1113857

t minus aa1minus α

+ middot middot middot

+g f1(a) f2(a) fn(t)( 1113857 minus g f1(a) fn(a)( 1113857

t minus aa1minus α

z

zx1g c1 f2(t) fn(t)( 1113857

f1(t) minus f1(a)

t minus aa1minus α

z

zx2g f1(a) c2 fn(t)( 1113857

f2(t) minus f2(a)

t minus aa1minus α

+ middot middot middot

+z

zx2g f1(a) c2 fn(t)( 1113857

f2(t) minus f2(a)

t minus aa1minus α

+ middot middot middot

+z

zxn

g f1(a) f2(a) cn( 1113857fn(t) minus fn(a)

t minus aa1minus α

(30)

6 Journal of Mathematics

where ci is between fi(a) and fi(t) forall i 1 2 n By taking limits as t⟶ a and using the continuity ofPDrs of g as well as the fact that ci⟶ fi(a) foralli 1 2 n equation (29) is expressed as

Tαh( 1113857(a) limt⟶a

h(t) minus h(a)

t minus aa1minus α

limt⟶a

g(f(t)) minus g(f(a))

t minus aa1minus α

limt⟶infin

z

zx1g c1 f2(t)( 1113857

f1(t) minus f1(a)

t minus aa1minus α

+ middot middot middot1113888 1113889

+z

zx2g f1(a) c2 fn(t)( 1113857

f2(t) minus f2(a)

t minus aa1minus α

+ middot middot middot

+z

zxn

g f1(a) f2(a)1minus α

cn1113872 1113873fn(t) minus fn(a)

t minus aa1minus α

1113889 +z

zx1g(f(a)) middot f

(1)1 (a) middot a

1minus α

+z

zx2g(f(a)) middot f

(1)2 (a) middot a

1minus αmiddot f1(a)

αminus 1middot f

(1)1 (a) middot a

1minus α

z

zx1g(f(a)) middot f1(a)

1minus αmiddot f1(a)

1minus αmiddot f1(a)

αminus 1middot f

(1)1 (a) middot a

1minus α+

z

zx2g(f(a)) middot f2(a)

1minus α

middot f2(a)αminus 1

middot f(1)2 (a) middot a

1minus α+ middot middot middot +

z

zxn

g(f(a)) middot f2(a)1minus α

middot f2(a)αminus 1

middot f(1)n (a) middot a

1minus α

zα

zxα1

g(f(a)) middot f1(a)αminus 1

middot Tαf1( 1113857(α) +zα

zxα2

g(f(a)) middot f2(a)αminus 1

middot Tαf2( 1113857(a) + middot middot middot

+zα

zxαn

g(f(a)) middot fn(a)αminus 1

middot Tαf1( 1113857(α)

(31)

Our proof is completely done Remark 5 e MF of equation (28) is expressed as

Dα(g middot f)(a)(h)

zα

zxα1

g(f(a)) zα

zxαn

g(f(a))1113888 1113889

middot

f1(a)αminus 1 0 0

0 ⋱ 0

0 0 fn(a)αminus 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠middot

Tαf1( 1113857(a)

⋮

Tαfn( 1113857(a)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠middot h

(32)

Theorem 14 (CR) Let x (x1 xn) isin Rn andy (y1 ym) isin Rm If f(x1 xn) (f1(x1

xn) fm(x1 xn)) is a α DF at a (a1 an) isin Rneach ai gt 0 ni α isin (0 1] and a RVF g with variables y1 ym

is α DF at f(a) isin Rn where allfi(a)gt 0 niα isin (0 1]en thecomposition g ∘ f is α-differentiable at a and we have

zα

zxαi

(g ∘ f)(a) 1113944m

j1

zα

zyαj

g(f(a)) middot fj(a)αminus 1

middotzα

zxαi

fj(a)

forall i 1 2 n

(33)

Journal of Mathematics 7

Proof Proof According to PDrrsquos definition and eorem14 the following is implied

Remark 6 e MF of equation (33) is expressed as

Dα(g ∘ f)(a)(h)

zα

zyα1

g(f(a)) zα

zyαm

g(f(a))1113888 1113889 middot

f1(a)αminus 1 0 0

0 ⋱ 0

0 0 fm(a)αminus 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠middot

zα

zxα1f1(a)

zα

zxαn

f1(a)

middot middot middot middot middot middot

zα

zxα1fn(a)

zα

zxαn

fn(a)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

middot

h1

⋮

hn

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (34)

5 Conformable Implicit Function Theorem

In this section the CoV of IFm for SVs is obtained Firstwe establish the conformable theorem of existence andregularity of the implicit function for the case of a singleequation

Theorem 15 (Conformable implicit function theorem forthe case of a single equation) Suppose that α isin (0 1]F X⟶ R is a RVF defined in an OS X sub Rn+1 ni forall(x1 xn y) isin X each xi ygt 0 and point(a1 an b) isin X Suppose that

(i) F(a1 an b) 0(ii) F isin Cα(X R)

(iii) zαzyαF(a1 an b)ne 0

en there is a neighbourhood U sub Rn of (a1 an)such that there is a unique function (UF) y g(x1 xn)

that satisfies the followingg(a1 an) b and F(x1 xn g(x1 xn))

0 forall(x1 xn) isin UFinally y g(x1 xn) is Cα in U and for every

i 1 2 n we have

zα

zxαi

g x1 xn( 1113857 minuszαzx

αi F x1 xn( 1113857 g x1 xn( 1113857( 1113857( 1113857

zαzy

αF x1 xn( 1113857 g x1 xn( 1113857( 1113857( 1113857 middot g x1 xn( 1113857( 1113857

αminus 1 (35)

Proof WLOG X is assumed to be an OB represented byB((a1 an b) ε0) Let ρ isin (0 ε0) If we call δ

ε20 minus ρ21113969

it is verified that

x1 xn( 1113857 minus a1 an( 1113857

lt δ and |y minus b|lt ρ1113960 1113961rArr x1 xn y( 1113857 isin B a1 an b( 1113857 ε0( 1113857 (36)

Note that in particular if |y minus b|lt ρ then(a1 an y) isin B((a1 an b) ε0)

Since zαzyαF(a1 an b)ne 0 it is assumed to bepositive (otherwise minus F is considered instead of F) From factthat F(a1 an b) 0 it follows that

F a1 an b minus ρ( 1113857gt 0

F a1 an b minus ρ( 1113857lt 0(37)

By the continuity of F at (a1 an b minus ρ) and(a1 an b + ρ) there exists δprime isin (0 δ) such that

x1 xn( 1113857 minus a1 an( 1113857

lt δprimerArr F x1 xn b minus ρ( 1113857gt 0 andF x1 xn b + ρ( 1113857lt 01113858 1113859 (38)

Since the function y↦F(x1 xn y) is CF on the in-terval [b minus ρ b + ρ ] forall (x1 xn) isin B((a1 an) δprime) andvia the classical Bolzanorsquos theorem (Bm) it implies thatexistyx isin ( b minus ρ b + ρ) niF(x1 xn yx) 0 for eachx (x1 xn) en yrsquos value is unique since a functionwhose derivative is positive has more than zero On the other

hand by having U B((a1 an) δprime) for each(x1 xn) isin U there exists a unique y

niF(x1 xn y) 0 we can write y g(x1 xn) andthen this function will be proven to be CF onB((a1 an) δprime)e continuity of the function g at the point(a1 an) is clear since for each ρgt 0 exist a value δprime gt 0 ni

8 Journal of Mathematics

x1 xn( 1113857 minus a1 an( 1113857

lt δprimerArr b minus yx

11138681113868111386811138681113868111386811138681113868

lt ρhArr b minus g x1 xn( 11138571113868111386811138681113868

1113868111386811138681113868lt ρ(39)

e function g continuity will be proven at any point(x1 xn) isin B((a1 an) δprime) by simply substitutingB((a1 an) δprime) for an OB B((x1 x)) is contained inB((a1 an) δprime)

Finally let us now show formula (35) By applyingeorem 14 to equation

F x1 xn y( 1113857 0 we have

zα

zxαi

F(x g(x)) +zα

zyα F(x g(x)) middot g(x)

αminus 1middot

zα

zxαi

g(x) 0

(40)

forall i 1 2 n nix (x1 xn) Solving zαzxαi g(x)

we get (35) In addition the formula (35) on the right side iscontinuous so the continuity of CoV of PDrs zαzxα

i g(x) foralli 1 2 n follows

eorem 15 will provide us a help in computing CoV ofPDrs of implicit function of SVs

Example 2 Consider

F(x y z) x3

+ 3y2

+ 4xz2

minus 3yz2

minus 5 0 (41)

is equationrsquos one solution is (1 1 1) F is obviously inCα which is OB represented by B((1 1 1) ε0) withx y zgt 0 for some α isin (0 1] since

zα

zzα F(1 1 1) 8xz

2minus αminus 6yz

2minus α1113872 11138731113961

(111) 2ne 0 (42)

eorem 15 implies that there is aneighbourhood U sub R2 of (1 1) niexist a UF z g(x y)

satisfies the followingg(1 1) 1 andF(x y g(x y)) 0 forall(x y) isin U

Moreover z g(x y) is Cα in U and we have

zα

zxα g(x y) minus

3x + 4z2

1113872 1113873x1minus α

(8x minus 6y)z

zα

zyα g(x y) minus

6y minus 3z2

1113872 1113873y1minus α

(8x minus 6y)z

(43)

Finally we obtain zαzxαg(1 1) minus (72) andzαzxαg(1 1) minus (32)

Finally CoV of IFm for a system of several equationsand SRVs is obtained

Theorem 16 (Conformable general implicit functiontheorem) Let α isin (0 1] F X⟶ Rm be a VVF defined inan OS X sub Rn+m ni forall (x y) (x1 xn y1 ym) isin Xeach xi yj gt 0 and point(a b) (a1 an b1 bm) isin X Assume that

(i) F(a b) 0(ii) F isin Cα(X Rm )

(iii) det[JαyF(a b)]ne 0

en there is a neighbourhood U sub Rn of a niexist a UFf U⟶ Rm x↦ y f(x) that satisfies f(a) b andF(x f(x) ) 0 forallx isin U

Finally y f(x) is class Cα in U and for everyi 1 2 n we have

zαfzxα

i

1113890 1113891

t

minus fαminus 11113872 1113873

minus 1middot J

αyF1113872 1113873

minus 1middot

zαFzxα

i

1113890 1113891

t

(44)

where

zαf

zxαi

1113890 1113891 zαf1

zxαi

zαfm

zxαi

1113888 1113889

fαminus 1

fαminus 11 0

⋮ ⋮

0 fαminus 1m

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

JαyF

zαF1

zyα1

zαF1

zyαm

zαFm

zyα1

zαFm

zyαm

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

zαF

zxαi

1113890 1113891 zαF1

zxαi

zαFm

zxαi

1113888 1113889

(45)

Proof e proof of existence and uniqueness of the implicitfunction is done similar to the traditional multivariablecalculus by applying mathematical induction on q and usingthe conformable implicit function theorem for severalvariables [28]

Let us now show formula (44) Assume that a systemwith several equations and SRVs is expressed as

F(x y) 0 or

F1 x1 xn y1 ym( 1113857 0

⋮

Fm x1 xn y1 ym( 1113857 0

⎧⎪⎪⎨

⎪⎪⎩ (46)

which satisfies hypotheses (i)ndash(iii) of the theorem then thissystem is defined in a neighbourhood U sub Rn of a theimplicit function y f(x) class Cα in U such that f(a) bwhich satisfies equation (1) ie

F(x f(x)) 0 or

F1 x f1(x) fm(x)( 1113857 0

Fm x f1(x) fm(x)( 1113857 0

⎧⎪⎪⎨

⎪⎪⎩(47)

By applying CoV of CR to the above equation we have

Journal of Mathematics 9

zαF1

zxαi

+zαF1

zyα1

middot fαminus 11 middot

zαf1

zxαi

+ middot middot middot +zαF1

zyαm

middot fαminus 1m middot

zαfm

zxαi

0

⋮

zαFm

zxαi

+zαFm

zyα1

middot fαminus 11 middot

zαf1

zxαi

+ middot middot middot +zαFm

zyαm

middot fαminus 1m middot

zαfm

zxαi

0

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

orzαF

zxαi

+ 1113944m

j1

zαF

zyαj

middot fαminus 1j middot

zαfj

zxαi

0

(48)

for all i 1 2 n In addition the MF of equation (48) is given as follows

zαF1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus

zαF1

zyα1

zαF1

zyαm

zαFm

zyα1

zαFm

zyαm

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

middot

fαminus 11 0

⋮ ⋮

0 fαminus 1m

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠middot

zαf1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

orzαFzxα

i

1113890 1113891

t

minus JαyF middot fαminus 1

middotzαfzxα

i

1113890 1113891

t

(49)

Since JαyF and fαminus 1 are regular matrices by hypothesis we

have

zαf1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus

fαminus 11 0

⋮ ⋮

0 fαminus 1m

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus 1

middot

zαF1

zyα1

zαF1

zyαm

zαFm

zyα1

zαFm

zyαm

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus 1

middot

zαF1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

orzαfzxα

i

1113890 1113891

t

minus fαminus 11113872 1113873

minus 1middot J

αyF1113872 1113873

minus 1middot

zαFzxα

i

1113890 1113891

t

(50)

which completes the proofWe will now show that eorem 16 can be used to

compute CoV of PDrs of systems with several equations andSRVs

Example 3 Consider a system of two equations and two realvariables

F1(x y z w) x2

+ y2

+ z2

+ w2

minus 6 0

F2(x y z w) x2

minus y2

+ z2

minus w2

0

⎧⎨

⎩ (51)

One solution of this equation is(x y z w) (1 1

2

radic

2

radic) Clearly F (F1 F2) is in Cα

which is OB B((1 12

radic

2

radic) ε0) with x y z wgt 0 for

some α isin (0 1] since

10 Journal of Mathematics

det JαzwF((

2

radic

2

radic 1 1))1113960 1113961 det

2z2minus α 2w2minus α

2z2minus α minus 2w2minus α⎛⎝ ⎞⎠⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

(112

radic

2

radic)

minus322propne 0 (52)

eorem 16 implies that there is aneighbourhood U sub R2 of (

2

radic

2

radic) niexist a UF f (f1 f2)

given by

z f1(x y)

w f2(x y)1113896 (53)

that satisfies

f1(1 1) 2

radic

f2(1 1) 2

radic

⎧⎨

⎩

F1 x y f1(x y) f2(x y)( 1113857 0

F2 x y f1(x y) f2(x y)( 1113857 0forall(x y) isin U1113896

(54)

Moreover f (f1 f2) is class Cα in U and we have

zαf1

zxα

zαf2

zxα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ minus

zαminus 1 0

0 wαminus 1⎛⎝ ⎞⎠

minus 1

middot2z2minus α 2w2minus α

2z2minus α minus 2w2minus α⎛⎝ ⎞⎠

minus 1

middot2x

2minus α

2x2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz1minus α 0

0 w1minus α

⎛⎜⎝ ⎞⎟⎠ middotzαminus 2

zαminus 2

wαminus 2

minus wαminus 2

⎛⎜⎝ ⎞⎟⎠

middot2x

2minus α

2x2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz

minus 1z

minus 1

wminus 1

minus wminus 1

⎛⎜⎝ ⎞⎟⎠ middot2x

2minus α

2x2minus α

⎛⎜⎝ ⎞⎟⎠ minus

x2minus α

z

0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

zαf1

zyα

zαf2

zyα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minuszαminus 1 0

0 wαminus 1⎛⎝ ⎞⎠

minus 1

middot2z2minus α 2w2minus α

2z2minus α minus 2w2minus α⎛⎝ ⎞⎠

minus 1

middot2y

2minus α

minus 2y2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz1minus α 0

0 w1minus α

⎛⎜⎝ ⎞⎟⎠ middotzαminus 2

zαminus 2

wαminus 2

minus wαminus 2

⎛⎜⎝ ⎞⎟⎠

middot2y

2minus α

minus 2y2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz

minus 1z

minus 1

wminus 1

minus wminus 1

⎛⎜⎝ ⎞⎟⎠ middot2y

2minus α

minus 2y2minus α

⎛⎜⎝ ⎞⎟⎠

0

minusy2minus α

w

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(55)

Finally we have

zαf1

zxα

zαf2

zxα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(112

radic

2

radic)

minus12

radic

0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

zαf1

zyα

zαf2

zyα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(112

radic

2

radic)

0

minus12

radic

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(56)

According to all of our numerical examples in this workwith the help of our investigation to all proposed theoremsand properties in this article the numerical examples showthat all our obtained results are valid and efficient As seen inall these examples the computations are simple and cost-

efficient which are important when scientific phenomenaare modelled in the sense of conformable calculus All in allthe simplicity of computations is always needed in modelingapplications in comparison to other complicated compu-tations that are needed using other approaches in otherfractional operators All our obtained results in this work areaccurate because our results coincide with the usual integer-order results

6 Conclusion

Anew investigation on the CoV ofmultivariable calculus hasbeen discussed in this work in detail e α-derivative of afunction of SVs and all related properties have been in-vestigatede CoV of CR for functions of SVs has also beenstudied e CoV of IFm has been presented in the lastpart of our results and numerical experiments have beenconducted to support our theoretical results e findings ofthis investigations show that all results formulated via CoD

Journal of Mathematics 11

coincide with the ones in the traditional integer case Allresults in this work can be potentially applied in modelingoceanographic phenomena such as Stommelrsquos box model ofthermohaline circulation and other related models wherethis studyrsquos analysis can be further extended or generalizedin 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

Authorsrsquo Contributions

MohammedK A Kaabar involved in actualization developedmethodology performed formal analysis validated and in-vestigated the study prepared the initial draft and supervisedand edited the original draft Francisco Martınez InmaculadaMartınez and Silvestre Paredes involved in actualizationdeveloped methodology performed formal analysis validatedand investigated the study and prepared the initial draftZailan Siri developed methodology performed formal anal-ysis validated and investigated the study and prepared theinitial draft All authors read and approved the final version

References

[1] M M A Khater M S Mohamed and R A M Attia ldquoOnsemi analytical and numerical simulations for a mathematicalbiological model the time-fractional nonlinear Kolmogor-ovndashPetrovskiindashPiskunov (KPP) equationrdquo Chaos Solitons ampFractals vol 144 Article ID 110676 2021

[2] M A Khater K S Nisar and M S Mohamed ldquoNumericalinvestigation for the fractional nonlinear space-time telegraphequation via the trigonometric Quintic B-spline schemerdquoMathematical Methods in the Applied Sciences vol 44 no 6pp 4598ndash4606 2021

[3] M M A Khater M S Mohamed and S K Elagan ldquoDiverseaccurate computational solutions of the nonlinearKleinndashFockndashGordon equationrdquo Results in Physics vol 23Article ID 104003 2021

[4] M F Li J R Ren and T Zhu ldquoFractional vector calculus andfractional special functionrdquo arXiv preprint10012889 2010

[5] D J Wollkind and B J Dichone Comprehensive AppliedMathematical Modeling in the Natural and Engineering Sci-ences Springer International Publishing Berlin Germany2017

[6] P J Olver Introduction to Partial Differential EquationsSpringer International Publishing Berlin Germany 2014

[7] J D Krehbiel and J B Freund ldquoStokes flow inside a sphere inan inviscid extensional flowrdquo Zeitschrift fur AngewandteMathematik und Physik vol 68 no 4 pp 68ndash81 2017

[8] A A Kilbas H M Srivastava and J J Trujillo ldquoeory andapplications of fractional differential equationsrdquo in North-Holland Mathematics StudiesVol 204 Elsevier AmsterdamNetherlands 2006

[9] K S Miller An Introduction to Fractional Calculus andFractional Differential Equations John Wiley and SonsHoboken NJ USA 1993

[10] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014

[11] R Khalil M Al Horani and M Abu Hammad ldquoGeometricmeaning of conformable derivative via fractional cordsrdquo eJournal of Mathematics and Computer Science vol 19 no 4pp 241ndash245 2019

[12] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journalof Computational and Applied Mathematics vol 279pp 57ndash66 2015

[13] O S Iyiola and E R Nwaeze ldquoSome new results on the newconformable fractional calculus with application usingDprimeAlambert approachrdquo Progress in Fractional Differentiationand Applications vol 2 no 2 pp 115ndash122 2016

[14] A Atangana D Baleanu and A Alsaedi ldquoNew properties ofconformable derivativerdquo Open Mathematics vol 13 pp 57ndash63 2015

[15] N Yazici and U Gozutok ldquoMultivariable conformablefractional calculusrdquo Filomat vol 32 no 2 pp 45ndash53 2018

[16] F Martınez I Martınez and S Paredes ldquoConformableEulerprimes theorem on homogeneous functionsrdquo ComputationalMathematical Methods vol 1 no 5 pp 1ndash11 2018

[17] A A Hyder and A H Soliman ldquoExact solutions of space-timelocal fractal nonlinear evolution equations a generalizedconformable derivative approachrdquo Results in Physics vol 17Article ID 103135 2020

[18] A A Hyder ldquoWhite noise theory and general improvedKudryashov method for stochastic nonlinear evolutionequations with conformable derivativesrdquo Advances in Dif-ference Equations vol 202019 pages 2020

[19] A H Soliman and A A Hyder ldquoClosed-form solutions ofstochastic KdV equation with generalized conformable deriv-ativesrdquo Physica Scripta vol 95 no 6 Article ID 065219 2020

[20] V E Tarasov ldquoNo violation of the Leibniz rule No fractionalderivativerdquo Communications in Nonlinear Science and Nu-merical Simulation vol 18 no 11 pp 2945ndash2948 2013

[21] R Khalil M Al Horani A Yousef and M SababhehldquoFractional analytic functionsrdquo Far East Journal of Mathe-matical Sciences vol 103 no 1 pp 113ndash123 2018

[22] S Uccedilar and N Y Ozgur ldquoComplex conformable derivativerdquoArabian Journal of Geosciences vol 12 no 201 2019

[23] F Martınez I Martınez I Martınez M K A Kaabar andS Paredes ldquoNew results on complex conformable integralrdquoAIMS Mathematics vol 5 no 6 pp 7695ndash7710 2020

[24] H Kiskinov M Petkova A Zahariev and M VeselinovaldquoSome results about conformable derivatives in Banach spacesand an application to the partial differential equationsrdquo inAIPConference Proceedingsvol 2333 no 1 AIP Publishing LLCArticle ID 120002 2021

[25] U Gozutok H A Ccediloban and Y Sagıroglu ldquoFrenet framewith respect to conformable derivativerdquo Filomat vol 33no 6 pp 1541ndash1550 2019

[26] J C Mayo-Maldonado G Fernandez-Anaya and O F Ruiz-Martinez ldquoStability of conformable linear differential systemsa behavioural framework with applications in fractional-ordercontrolrdquo IET Control eory amp Applications vol 14 no 18pp 2900ndash2913 2020

[27] M Al Horani and R Khalil ldquoTotal fractional differential withapplications to exact fractional differential equationsrdquo In-ternational Journal of Computer Mathematics vol 95 no 6-7pp 1444ndash1452 2018

[28] J E Marsden and M J Hoffman Vector Calculus WH Freeman and Company New York NY USA 4th edition1996

12 Journal of Mathematics

xi gt 0 and the point a (a1 an) isin X e function f isα DF at a iff its components are α DF at a and if these

components are f1 f2 fm then the components ofDαf(a) are the α-derivativesDαfj(a) for j 1 2 m ie

f f1 f2 fm( 1113857rArr Dαf(a) D

αf1(a) D

αf2(a) D

αfm(a)( 1113857 (26)

Proof It is similarly proven as same as traditional calculusby applying Dα instead of derivative

Remark 4 Assume that α isin (0 1] f X⟶ Rm is a VVFdefined in an OS X sub Rn niforall x (x1 xn) isin X eachxi gt 0 and a point a (a1 an) isin X If function f isα-differentiable at a then zαzxα

i fj(a) exists fori 1 2 n j 1 2 m and the CoV of α-Jacobian off of order α isin (0 1] at a is expressed as

Jαf(a)

zα

zxα1f1(a)

zα

zxαn

f1(a)

⋮ ⋮

zα

zxα1fm(a)

zα

zxαn

fm(a)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(27)

4 The Chain Rule

Let us prove the CR for the functions of SVs in 2 particularcases For the proofrsquos purpose the continuityrsquos hypothesis ofPDrs is given [28]

Theorem 13 (CR) Suppose that t isin R andx (x1 xn) isin Rn If f(t) (f1(t) fn(t)) is α DF

at agt 0 ni α isin (0 1] and a RVF g with n variablesx1 xn is α DF at f(a) isin Rn all fi(a)gt 0 α isin (0 1] en the composition g ∘f is α-differential at a and

Tαg ∘ f( 1113857(a) 1113944n

i1

zα

zxαi

g(f(a)) middot fi(a)( 1113857αminus 1

middot Tαfi( 1113857(a)

(28)

Proof Assumeg isin C1(U(f(a)) R) niU(f(a)) is the pointf(a) neighborhood Suppose thath(t) (g ∘ f)(t) g(f(t)) By setting u a + εa1minus α inDefinition 1 we see that

Tαh( 1113857(a) limt⟶a

h(t) minus h(a)

t minus aa1minus α

limt⟶a

g(f(t)) minus g(f(a))

t minus aa1minus α

(29)

Without loss of generality (WLOG) U(f(a)) is assumedto be an open ball (OB) represented by B(f(a) r) Since f isa CF then along with points (f1(a) fn(a)) and(f1(t) fn(t)) the points(f1(a) f2(t) fn(t)) (f1(a) f2(a) fn(t))

and lines connecting them must also be the ball B(f(a) r)In fact using the classical MVT for differentiable functions(DFs) of one variable is in the following computation [28]

h(t) minus h(a)

t minus aa1minus α

g(f(t)) minus g(f(a))

t minus aa1minus α

g f1(t) fn(t)( 1113857 minus g f1(a) f2(t) fn(t)( 1113857

t minus aa1minus α

+g f1(a) f2(t) fn(t)( 1113857 minus g f1(a) f2(a) f3(t) fn(t)( 1113857

t minus aa1minus α

+ middot middot middot

+g f1(a) f2(a) fn(t)( 1113857 minus g f1(a) fn(a)( 1113857

t minus aa1minus α

z

zx1g c1 f2(t) fn(t)( 1113857

f1(t) minus f1(a)

t minus aa1minus α

z

zx2g f1(a) c2 fn(t)( 1113857

f2(t) minus f2(a)

t minus aa1minus α

+ middot middot middot

+z

zx2g f1(a) c2 fn(t)( 1113857

f2(t) minus f2(a)

t minus aa1minus α

+ middot middot middot

+z

zxn

g f1(a) f2(a) cn( 1113857fn(t) minus fn(a)

t minus aa1minus α

(30)

6 Journal of Mathematics

where ci is between fi(a) and fi(t) forall i 1 2 n By taking limits as t⟶ a and using the continuity ofPDrs of g as well as the fact that ci⟶ fi(a) foralli 1 2 n equation (29) is expressed as

Tαh( 1113857(a) limt⟶a

h(t) minus h(a)

t minus aa1minus α

limt⟶a

g(f(t)) minus g(f(a))

t minus aa1minus α

limt⟶infin

z

zx1g c1 f2(t)( 1113857

f1(t) minus f1(a)

t minus aa1minus α

+ middot middot middot1113888 1113889

+z

zx2g f1(a) c2 fn(t)( 1113857

f2(t) minus f2(a)

t minus aa1minus α

+ middot middot middot

+z

zxn

g f1(a) f2(a)1minus α

cn1113872 1113873fn(t) minus fn(a)

t minus aa1minus α

1113889 +z

zx1g(f(a)) middot f

(1)1 (a) middot a

1minus α

+z

zx2g(f(a)) middot f

(1)2 (a) middot a

1minus αmiddot f1(a)

αminus 1middot f

(1)1 (a) middot a

1minus α

z

zx1g(f(a)) middot f1(a)

1minus αmiddot f1(a)

1minus αmiddot f1(a)

αminus 1middot f

(1)1 (a) middot a

1minus α+

z

zx2g(f(a)) middot f2(a)

1minus α

middot f2(a)αminus 1

middot f(1)2 (a) middot a

1minus α+ middot middot middot +

z

zxn

g(f(a)) middot f2(a)1minus α

middot f2(a)αminus 1

middot f(1)n (a) middot a

1minus α

zα

zxα1

g(f(a)) middot f1(a)αminus 1

middot Tαf1( 1113857(α) +zα

zxα2

g(f(a)) middot f2(a)αminus 1

middot Tαf2( 1113857(a) + middot middot middot

+zα

zxαn

g(f(a)) middot fn(a)αminus 1

middot Tαf1( 1113857(α)

(31)

Our proof is completely done Remark 5 e MF of equation (28) is expressed as

Dα(g middot f)(a)(h)

zα

zxα1

g(f(a)) zα

zxαn

g(f(a))1113888 1113889

middot

f1(a)αminus 1 0 0

0 ⋱ 0

0 0 fn(a)αminus 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠middot

Tαf1( 1113857(a)

⋮

Tαfn( 1113857(a)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠middot h

(32)

Theorem 14 (CR) Let x (x1 xn) isin Rn andy (y1 ym) isin Rm If f(x1 xn) (f1(x1

xn) fm(x1 xn)) is a α DF at a (a1 an) isin Rneach ai gt 0 ni α isin (0 1] and a RVF g with variables y1 ym

is α DF at f(a) isin Rn where allfi(a)gt 0 niα isin (0 1]en thecomposition g ∘ f is α-differentiable at a and we have

zα

zxαi

(g ∘ f)(a) 1113944m

j1

zα

zyαj

g(f(a)) middot fj(a)αminus 1

middotzα

zxαi

fj(a)

forall i 1 2 n

(33)

Journal of Mathematics 7

Proof Proof According to PDrrsquos definition and eorem14 the following is implied

Remark 6 e MF of equation (33) is expressed as

Dα(g ∘ f)(a)(h)

zα

zyα1

g(f(a)) zα

zyαm

g(f(a))1113888 1113889 middot

f1(a)αminus 1 0 0

0 ⋱ 0

0 0 fm(a)αminus 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠middot

zα

zxα1f1(a)

zα

zxαn

f1(a)

middot middot middot middot middot middot

zα

zxα1fn(a)

zα

zxαn

fn(a)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

middot

h1

⋮

hn

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (34)

5 Conformable Implicit Function Theorem

In this section the CoV of IFm for SVs is obtained Firstwe establish the conformable theorem of existence andregularity of the implicit function for the case of a singleequation

Theorem 15 (Conformable implicit function theorem forthe case of a single equation) Suppose that α isin (0 1]F X⟶ R is a RVF defined in an OS X sub Rn+1 ni forall(x1 xn y) isin X each xi ygt 0 and point(a1 an b) isin X Suppose that

(i) F(a1 an b) 0(ii) F isin Cα(X R)

(iii) zαzyαF(a1 an b)ne 0

en there is a neighbourhood U sub Rn of (a1 an)such that there is a unique function (UF) y g(x1 xn)

that satisfies the followingg(a1 an) b and F(x1 xn g(x1 xn))

0 forall(x1 xn) isin UFinally y g(x1 xn) is Cα in U and for every

i 1 2 n we have

zα

zxαi

g x1 xn( 1113857 minuszαzx

αi F x1 xn( 1113857 g x1 xn( 1113857( 1113857( 1113857

zαzy

αF x1 xn( 1113857 g x1 xn( 1113857( 1113857( 1113857 middot g x1 xn( 1113857( 1113857

αminus 1 (35)

Proof WLOG X is assumed to be an OB represented byB((a1 an b) ε0) Let ρ isin (0 ε0) If we call δ

ε20 minus ρ21113969

it is verified that

x1 xn( 1113857 minus a1 an( 1113857

lt δ and |y minus b|lt ρ1113960 1113961rArr x1 xn y( 1113857 isin B a1 an b( 1113857 ε0( 1113857 (36)

Note that in particular if |y minus b|lt ρ then(a1 an y) isin B((a1 an b) ε0)

Since zαzyαF(a1 an b)ne 0 it is assumed to bepositive (otherwise minus F is considered instead of F) From factthat F(a1 an b) 0 it follows that

F a1 an b minus ρ( 1113857gt 0

F a1 an b minus ρ( 1113857lt 0(37)

By the continuity of F at (a1 an b minus ρ) and(a1 an b + ρ) there exists δprime isin (0 δ) such that

x1 xn( 1113857 minus a1 an( 1113857

lt δprimerArr F x1 xn b minus ρ( 1113857gt 0 andF x1 xn b + ρ( 1113857lt 01113858 1113859 (38)

Since the function y↦F(x1 xn y) is CF on the in-terval [b minus ρ b + ρ ] forall (x1 xn) isin B((a1 an) δprime) andvia the classical Bolzanorsquos theorem (Bm) it implies thatexistyx isin ( b minus ρ b + ρ) niF(x1 xn yx) 0 for eachx (x1 xn) en yrsquos value is unique since a functionwhose derivative is positive has more than zero On the other

hand by having U B((a1 an) δprime) for each(x1 xn) isin U there exists a unique y

niF(x1 xn y) 0 we can write y g(x1 xn) andthen this function will be proven to be CF onB((a1 an) δprime)e continuity of the function g at the point(a1 an) is clear since for each ρgt 0 exist a value δprime gt 0 ni

8 Journal of Mathematics

x1 xn( 1113857 minus a1 an( 1113857

lt δprimerArr b minus yx

11138681113868111386811138681113868111386811138681113868

lt ρhArr b minus g x1 xn( 11138571113868111386811138681113868

1113868111386811138681113868lt ρ(39)

e function g continuity will be proven at any point(x1 xn) isin B((a1 an) δprime) by simply substitutingB((a1 an) δprime) for an OB B((x1 x)) is contained inB((a1 an) δprime)

Finally let us now show formula (35) By applyingeorem 14 to equation

F x1 xn y( 1113857 0 we have

zα

zxαi

F(x g(x)) +zα

zyα F(x g(x)) middot g(x)

αminus 1middot

zα

zxαi

g(x) 0

(40)

forall i 1 2 n nix (x1 xn) Solving zαzxαi g(x)

we get (35) In addition the formula (35) on the right side iscontinuous so the continuity of CoV of PDrs zαzxα

i g(x) foralli 1 2 n follows

eorem 15 will provide us a help in computing CoV ofPDrs of implicit function of SVs

Example 2 Consider

F(x y z) x3

+ 3y2

+ 4xz2

minus 3yz2

minus 5 0 (41)

is equationrsquos one solution is (1 1 1) F is obviously inCα which is OB represented by B((1 1 1) ε0) withx y zgt 0 for some α isin (0 1] since

zα

zzα F(1 1 1) 8xz

2minus αminus 6yz

2minus α1113872 11138731113961

(111) 2ne 0 (42)

eorem 15 implies that there is aneighbourhood U sub R2 of (1 1) niexist a UF z g(x y)

satisfies the followingg(1 1) 1 andF(x y g(x y)) 0 forall(x y) isin U

Moreover z g(x y) is Cα in U and we have

zα

zxα g(x y) minus

3x + 4z2

1113872 1113873x1minus α

(8x minus 6y)z

zα

zyα g(x y) minus

6y minus 3z2

1113872 1113873y1minus α

(8x minus 6y)z

(43)

Finally we obtain zαzxαg(1 1) minus (72) andzαzxαg(1 1) minus (32)

Finally CoV of IFm for a system of several equationsand SRVs is obtained

Theorem 16 (Conformable general implicit functiontheorem) Let α isin (0 1] F X⟶ Rm be a VVF defined inan OS X sub Rn+m ni forall (x y) (x1 xn y1 ym) isin Xeach xi yj gt 0 and point(a b) (a1 an b1 bm) isin X Assume that

(i) F(a b) 0(ii) F isin Cα(X Rm )

(iii) det[JαyF(a b)]ne 0

en there is a neighbourhood U sub Rn of a niexist a UFf U⟶ Rm x↦ y f(x) that satisfies f(a) b andF(x f(x) ) 0 forallx isin U

Finally y f(x) is class Cα in U and for everyi 1 2 n we have

zαfzxα

i

1113890 1113891

t

minus fαminus 11113872 1113873

minus 1middot J

αyF1113872 1113873

minus 1middot

zαFzxα

i

1113890 1113891

t

(44)

where

zαf

zxαi

1113890 1113891 zαf1

zxαi

zαfm

zxαi

1113888 1113889

fαminus 1

fαminus 11 0

⋮ ⋮

0 fαminus 1m

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

JαyF

zαF1

zyα1

zαF1

zyαm

zαFm

zyα1

zαFm

zyαm

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

zαF

zxαi

1113890 1113891 zαF1

zxαi

zαFm

zxαi

1113888 1113889

(45)

Proof e proof of existence and uniqueness of the implicitfunction is done similar to the traditional multivariablecalculus by applying mathematical induction on q and usingthe conformable implicit function theorem for severalvariables [28]

Let us now show formula (44) Assume that a systemwith several equations and SRVs is expressed as

F(x y) 0 or

F1 x1 xn y1 ym( 1113857 0

⋮

Fm x1 xn y1 ym( 1113857 0

⎧⎪⎪⎨

⎪⎪⎩ (46)

which satisfies hypotheses (i)ndash(iii) of the theorem then thissystem is defined in a neighbourhood U sub Rn of a theimplicit function y f(x) class Cα in U such that f(a) bwhich satisfies equation (1) ie

F(x f(x)) 0 or

F1 x f1(x) fm(x)( 1113857 0

Fm x f1(x) fm(x)( 1113857 0

⎧⎪⎪⎨

⎪⎪⎩(47)

By applying CoV of CR to the above equation we have

Journal of Mathematics 9

zαF1

zxαi

+zαF1

zyα1

middot fαminus 11 middot

zαf1

zxαi

+ middot middot middot +zαF1

zyαm

middot fαminus 1m middot

zαfm

zxαi

0

⋮

zαFm

zxαi

+zαFm

zyα1

middot fαminus 11 middot

zαf1

zxαi

+ middot middot middot +zαFm

zyαm

middot fαminus 1m middot

zαfm

zxαi

0

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

orzαF

zxαi

+ 1113944m

j1

zαF

zyαj

middot fαminus 1j middot

zαfj

zxαi

0

(48)

for all i 1 2 n In addition the MF of equation (48) is given as follows

zαF1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus

zαF1

zyα1

zαF1

zyαm

zαFm

zyα1

zαFm

zyαm

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

middot

fαminus 11 0

⋮ ⋮

0 fαminus 1m

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠middot

zαf1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

orzαFzxα

i

1113890 1113891

t

minus JαyF middot fαminus 1

middotzαfzxα

i

1113890 1113891

t

(49)

Since JαyF and fαminus 1 are regular matrices by hypothesis we

have

zαf1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus

fαminus 11 0

⋮ ⋮

0 fαminus 1m

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus 1

middot

zαF1

zyα1

zαF1

zyαm

zαFm

zyα1

zαFm

zyαm

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus 1

middot

zαF1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

orzαfzxα

i

1113890 1113891

t

minus fαminus 11113872 1113873

minus 1middot J

αyF1113872 1113873

minus 1middot

zαFzxα

i

1113890 1113891

t

(50)

which completes the proofWe will now show that eorem 16 can be used to

compute CoV of PDrs of systems with several equations andSRVs

Example 3 Consider a system of two equations and two realvariables

F1(x y z w) x2

+ y2

+ z2

+ w2

minus 6 0

F2(x y z w) x2

minus y2

+ z2

minus w2

0

⎧⎨

⎩ (51)

One solution of this equation is(x y z w) (1 1

2

radic

2

radic) Clearly F (F1 F2) is in Cα

which is OB B((1 12

radic

2

radic) ε0) with x y z wgt 0 for

some α isin (0 1] since

10 Journal of Mathematics

det JαzwF((

2

radic

2

radic 1 1))1113960 1113961 det

2z2minus α 2w2minus α

2z2minus α minus 2w2minus α⎛⎝ ⎞⎠⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

(112

radic

2

radic)

minus322propne 0 (52)

eorem 16 implies that there is aneighbourhood U sub R2 of (

2

radic

2

radic) niexist a UF f (f1 f2)

given by

z f1(x y)

w f2(x y)1113896 (53)

that satisfies

f1(1 1) 2

radic

f2(1 1) 2

radic

⎧⎨

⎩

F1 x y f1(x y) f2(x y)( 1113857 0

F2 x y f1(x y) f2(x y)( 1113857 0forall(x y) isin U1113896

(54)

Moreover f (f1 f2) is class Cα in U and we have

zαf1

zxα

zαf2

zxα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ minus

zαminus 1 0

0 wαminus 1⎛⎝ ⎞⎠

minus 1

middot2z2minus α 2w2minus α

2z2minus α minus 2w2minus α⎛⎝ ⎞⎠

minus 1

middot2x

2minus α

2x2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz1minus α 0

0 w1minus α

⎛⎜⎝ ⎞⎟⎠ middotzαminus 2

zαminus 2

wαminus 2

minus wαminus 2

⎛⎜⎝ ⎞⎟⎠

middot2x

2minus α

2x2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz

minus 1z

minus 1

wminus 1

minus wminus 1

⎛⎜⎝ ⎞⎟⎠ middot2x

2minus α

2x2minus α

⎛⎜⎝ ⎞⎟⎠ minus

x2minus α

z

0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

zαf1

zyα

zαf2

zyα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minuszαminus 1 0

0 wαminus 1⎛⎝ ⎞⎠

minus 1

middot2z2minus α 2w2minus α

2z2minus α minus 2w2minus α⎛⎝ ⎞⎠

minus 1

middot2y

2minus α

minus 2y2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz1minus α 0

0 w1minus α

⎛⎜⎝ ⎞⎟⎠ middotzαminus 2

zαminus 2

wαminus 2

minus wαminus 2

⎛⎜⎝ ⎞⎟⎠

middot2y

2minus α

minus 2y2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz

minus 1z

minus 1

wminus 1

minus wminus 1

⎛⎜⎝ ⎞⎟⎠ middot2y

2minus α

minus 2y2minus α

⎛⎜⎝ ⎞⎟⎠

0

minusy2minus α

w

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(55)

Finally we have

zαf1

zxα

zαf2

zxα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(112

radic

2

radic)

minus12

radic

0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

zαf1

zyα

zαf2

zyα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(112

radic

2

radic)

0

minus12

radic

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(56)

According to all of our numerical examples in this workwith the help of our investigation to all proposed theoremsand properties in this article the numerical examples showthat all our obtained results are valid and efficient As seen inall these examples the computations are simple and cost-

efficient which are important when scientific phenomenaare modelled in the sense of conformable calculus All in allthe simplicity of computations is always needed in modelingapplications in comparison to other complicated compu-tations that are needed using other approaches in otherfractional operators All our obtained results in this work areaccurate because our results coincide with the usual integer-order results

6 Conclusion

Anew investigation on the CoV ofmultivariable calculus hasbeen discussed in this work in detail e α-derivative of afunction of SVs and all related properties have been in-vestigatede CoV of CR for functions of SVs has also beenstudied e CoV of IFm has been presented in the lastpart of our results and numerical experiments have beenconducted to support our theoretical results e findings ofthis investigations show that all results formulated via CoD

Journal of Mathematics 11

coincide with the ones in the traditional integer case Allresults in this work can be potentially applied in modelingoceanographic phenomena such as Stommelrsquos box model ofthermohaline circulation and other related models wherethis studyrsquos analysis can be further extended or generalizedin 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

Authorsrsquo Contributions

MohammedK A Kaabar involved in actualization developedmethodology performed formal analysis validated and in-vestigated the study prepared the initial draft and supervisedand edited the original draft Francisco Martınez InmaculadaMartınez and Silvestre Paredes involved in actualizationdeveloped methodology performed formal analysis validatedand investigated the study and prepared the initial draftZailan Siri developed methodology performed formal anal-ysis validated and investigated the study and prepared theinitial draft All authors read and approved the final version

References

[1] M M A Khater M S Mohamed and R A M Attia ldquoOnsemi analytical and numerical simulations for a mathematicalbiological model the time-fractional nonlinear Kolmogor-ovndashPetrovskiindashPiskunov (KPP) equationrdquo Chaos Solitons ampFractals vol 144 Article ID 110676 2021

[2] M A Khater K S Nisar and M S Mohamed ldquoNumericalinvestigation for the fractional nonlinear space-time telegraphequation via the trigonometric Quintic B-spline schemerdquoMathematical Methods in the Applied Sciences vol 44 no 6pp 4598ndash4606 2021

[3] M M A Khater M S Mohamed and S K Elagan ldquoDiverseaccurate computational solutions of the nonlinearKleinndashFockndashGordon equationrdquo Results in Physics vol 23Article ID 104003 2021

[4] M F Li J R Ren and T Zhu ldquoFractional vector calculus andfractional special functionrdquo arXiv preprint10012889 2010

[5] D J Wollkind and B J Dichone Comprehensive AppliedMathematical Modeling in the Natural and Engineering Sci-ences Springer International Publishing Berlin Germany2017

[6] P J Olver Introduction to Partial Differential EquationsSpringer International Publishing Berlin Germany 2014

[7] J D Krehbiel and J B Freund ldquoStokes flow inside a sphere inan inviscid extensional flowrdquo Zeitschrift fur AngewandteMathematik und Physik vol 68 no 4 pp 68ndash81 2017

[8] A A Kilbas H M Srivastava and J J Trujillo ldquoeory andapplications of fractional differential equationsrdquo in North-Holland Mathematics StudiesVol 204 Elsevier AmsterdamNetherlands 2006

[9] K S Miller An Introduction to Fractional Calculus andFractional Differential Equations John Wiley and SonsHoboken NJ USA 1993

[10] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014

[11] R Khalil M Al Horani and M Abu Hammad ldquoGeometricmeaning of conformable derivative via fractional cordsrdquo eJournal of Mathematics and Computer Science vol 19 no 4pp 241ndash245 2019

[12] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journalof Computational and Applied Mathematics vol 279pp 57ndash66 2015

[13] O S Iyiola and E R Nwaeze ldquoSome new results on the newconformable fractional calculus with application usingDprimeAlambert approachrdquo Progress in Fractional Differentiationand Applications vol 2 no 2 pp 115ndash122 2016

[14] A Atangana D Baleanu and A Alsaedi ldquoNew properties ofconformable derivativerdquo Open Mathematics vol 13 pp 57ndash63 2015

[15] N Yazici and U Gozutok ldquoMultivariable conformablefractional calculusrdquo Filomat vol 32 no 2 pp 45ndash53 2018

[16] F Martınez I Martınez and S Paredes ldquoConformableEulerprimes theorem on homogeneous functionsrdquo ComputationalMathematical Methods vol 1 no 5 pp 1ndash11 2018

[17] A A Hyder and A H Soliman ldquoExact solutions of space-timelocal fractal nonlinear evolution equations a generalizedconformable derivative approachrdquo Results in Physics vol 17Article ID 103135 2020

[18] A A Hyder ldquoWhite noise theory and general improvedKudryashov method for stochastic nonlinear evolutionequations with conformable derivativesrdquo Advances in Dif-ference Equations vol 202019 pages 2020

[19] A H Soliman and A A Hyder ldquoClosed-form solutions ofstochastic KdV equation with generalized conformable deriv-ativesrdquo Physica Scripta vol 95 no 6 Article ID 065219 2020

[20] V E Tarasov ldquoNo violation of the Leibniz rule No fractionalderivativerdquo Communications in Nonlinear Science and Nu-merical Simulation vol 18 no 11 pp 2945ndash2948 2013

[21] R Khalil M Al Horani A Yousef and M SababhehldquoFractional analytic functionsrdquo Far East Journal of Mathe-matical Sciences vol 103 no 1 pp 113ndash123 2018

[22] S Uccedilar and N Y Ozgur ldquoComplex conformable derivativerdquoArabian Journal of Geosciences vol 12 no 201 2019

[23] F Martınez I Martınez I Martınez M K A Kaabar andS Paredes ldquoNew results on complex conformable integralrdquoAIMS Mathematics vol 5 no 6 pp 7695ndash7710 2020

[24] H Kiskinov M Petkova A Zahariev and M VeselinovaldquoSome results about conformable derivatives in Banach spacesand an application to the partial differential equationsrdquo inAIPConference Proceedingsvol 2333 no 1 AIP Publishing LLCArticle ID 120002 2021

[25] U Gozutok H A Ccediloban and Y Sagıroglu ldquoFrenet framewith respect to conformable derivativerdquo Filomat vol 33no 6 pp 1541ndash1550 2019

[26] J C Mayo-Maldonado G Fernandez-Anaya and O F Ruiz-Martinez ldquoStability of conformable linear differential systemsa behavioural framework with applications in fractional-ordercontrolrdquo IET Control eory amp Applications vol 14 no 18pp 2900ndash2913 2020

[27] M Al Horani and R Khalil ldquoTotal fractional differential withapplications to exact fractional differential equationsrdquo In-ternational Journal of Computer Mathematics vol 95 no 6-7pp 1444ndash1452 2018

[28] J E Marsden and M J Hoffman Vector Calculus WH Freeman and Company New York NY USA 4th edition1996

12 Journal of Mathematics

where ci is between fi(a) and fi(t) forall i 1 2 n By taking limits as t⟶ a and using the continuity ofPDrs of g as well as the fact that ci⟶ fi(a) foralli 1 2 n equation (29) is expressed as

Tαh( 1113857(a) limt⟶a

h(t) minus h(a)

t minus aa1minus α

limt⟶a

g(f(t)) minus g(f(a))

t minus aa1minus α

limt⟶infin

z

zx1g c1 f2(t)( 1113857

f1(t) minus f1(a)

t minus aa1minus α

+ middot middot middot1113888 1113889

+z

zx2g f1(a) c2 fn(t)( 1113857

f2(t) minus f2(a)

t minus aa1minus α

+ middot middot middot

+z

zxn

g f1(a) f2(a)1minus α

cn1113872 1113873fn(t) minus fn(a)

t minus aa1minus α

1113889 +z

zx1g(f(a)) middot f

(1)1 (a) middot a

1minus α

+z

zx2g(f(a)) middot f

(1)2 (a) middot a

1minus αmiddot f1(a)

αminus 1middot f

(1)1 (a) middot a

1minus α

z

zx1g(f(a)) middot f1(a)

1minus αmiddot f1(a)

1minus αmiddot f1(a)

αminus 1middot f

(1)1 (a) middot a

1minus α+

z

zx2g(f(a)) middot f2(a)

1minus α

middot f2(a)αminus 1

middot f(1)2 (a) middot a

1minus α+ middot middot middot +

z

zxn

g(f(a)) middot f2(a)1minus α

middot f2(a)αminus 1

middot f(1)n (a) middot a

1minus α

zα

zxα1

g(f(a)) middot f1(a)αminus 1

middot Tαf1( 1113857(α) +zα

zxα2

g(f(a)) middot f2(a)αminus 1

middot Tαf2( 1113857(a) + middot middot middot

+zα

zxαn

g(f(a)) middot fn(a)αminus 1

middot Tαf1( 1113857(α)

(31)

Our proof is completely done Remark 5 e MF of equation (28) is expressed as

Dα(g middot f)(a)(h)

zα

zxα1

g(f(a)) zα

zxαn

g(f(a))1113888 1113889

middot

f1(a)αminus 1 0 0

0 ⋱ 0

0 0 fn(a)αminus 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠middot

Tαf1( 1113857(a)

⋮

Tαfn( 1113857(a)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠middot h

(32)

Theorem 14 (CR) Let x (x1 xn) isin Rn andy (y1 ym) isin Rm If f(x1 xn) (f1(x1

xn) fm(x1 xn)) is a α DF at a (a1 an) isin Rneach ai gt 0 ni α isin (0 1] and a RVF g with variables y1 ym

is α DF at f(a) isin Rn where allfi(a)gt 0 niα isin (0 1]en thecomposition g ∘ f is α-differentiable at a and we have

zα

zxαi

(g ∘ f)(a) 1113944m

j1

zα

zyαj

g(f(a)) middot fj(a)αminus 1

middotzα

zxαi

fj(a)

forall i 1 2 n

(33)

Journal of Mathematics 7

Proof Proof According to PDrrsquos definition and eorem14 the following is implied

Remark 6 e MF of equation (33) is expressed as

Dα(g ∘ f)(a)(h)

zα

zyα1

g(f(a)) zα

zyαm

g(f(a))1113888 1113889 middot

f1(a)αminus 1 0 0

0 ⋱ 0

0 0 fm(a)αminus 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠middot

zα

zxα1f1(a)

zα

zxαn

f1(a)

middot middot middot middot middot middot

zα

zxα1fn(a)

zα

zxαn

fn(a)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

middot

h1

⋮

hn

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (34)

5 Conformable Implicit Function Theorem

In this section the CoV of IFm for SVs is obtained Firstwe establish the conformable theorem of existence andregularity of the implicit function for the case of a singleequation

Theorem 15 (Conformable implicit function theorem forthe case of a single equation) Suppose that α isin (0 1]F X⟶ R is a RVF defined in an OS X sub Rn+1 ni forall(x1 xn y) isin X each xi ygt 0 and point(a1 an b) isin X Suppose that

(i) F(a1 an b) 0(ii) F isin Cα(X R)

(iii) zαzyαF(a1 an b)ne 0

en there is a neighbourhood U sub Rn of (a1 an)such that there is a unique function (UF) y g(x1 xn)

that satisfies the followingg(a1 an) b and F(x1 xn g(x1 xn))

0 forall(x1 xn) isin UFinally y g(x1 xn) is Cα in U and for every

i 1 2 n we have

zα

zxαi

g x1 xn( 1113857 minuszαzx

αi F x1 xn( 1113857 g x1 xn( 1113857( 1113857( 1113857

zαzy

αF x1 xn( 1113857 g x1 xn( 1113857( 1113857( 1113857 middot g x1 xn( 1113857( 1113857

αminus 1 (35)

Proof WLOG X is assumed to be an OB represented byB((a1 an b) ε0) Let ρ isin (0 ε0) If we call δ

ε20 minus ρ21113969

it is verified that

x1 xn( 1113857 minus a1 an( 1113857

lt δ and |y minus b|lt ρ1113960 1113961rArr x1 xn y( 1113857 isin B a1 an b( 1113857 ε0( 1113857 (36)

Note that in particular if |y minus b|lt ρ then(a1 an y) isin B((a1 an b) ε0)

Since zαzyαF(a1 an b)ne 0 it is assumed to bepositive (otherwise minus F is considered instead of F) From factthat F(a1 an b) 0 it follows that

F a1 an b minus ρ( 1113857gt 0

F a1 an b minus ρ( 1113857lt 0(37)

By the continuity of F at (a1 an b minus ρ) and(a1 an b + ρ) there exists δprime isin (0 δ) such that

x1 xn( 1113857 minus a1 an( 1113857

lt δprimerArr F x1 xn b minus ρ( 1113857gt 0 andF x1 xn b + ρ( 1113857lt 01113858 1113859 (38)

Since the function y↦F(x1 xn y) is CF on the in-terval [b minus ρ b + ρ ] forall (x1 xn) isin B((a1 an) δprime) andvia the classical Bolzanorsquos theorem (Bm) it implies thatexistyx isin ( b minus ρ b + ρ) niF(x1 xn yx) 0 for eachx (x1 xn) en yrsquos value is unique since a functionwhose derivative is positive has more than zero On the other

hand by having U B((a1 an) δprime) for each(x1 xn) isin U there exists a unique y

niF(x1 xn y) 0 we can write y g(x1 xn) andthen this function will be proven to be CF onB((a1 an) δprime)e continuity of the function g at the point(a1 an) is clear since for each ρgt 0 exist a value δprime gt 0 ni

8 Journal of Mathematics

x1 xn( 1113857 minus a1 an( 1113857

lt δprimerArr b minus yx

11138681113868111386811138681113868111386811138681113868

lt ρhArr b minus g x1 xn( 11138571113868111386811138681113868

1113868111386811138681113868lt ρ(39)

e function g continuity will be proven at any point(x1 xn) isin B((a1 an) δprime) by simply substitutingB((a1 an) δprime) for an OB B((x1 x)) is contained inB((a1 an) δprime)

Finally let us now show formula (35) By applyingeorem 14 to equation

F x1 xn y( 1113857 0 we have

zα

zxαi

F(x g(x)) +zα

zyα F(x g(x)) middot g(x)

αminus 1middot

zα

zxαi

g(x) 0

(40)

forall i 1 2 n nix (x1 xn) Solving zαzxαi g(x)

we get (35) In addition the formula (35) on the right side iscontinuous so the continuity of CoV of PDrs zαzxα

i g(x) foralli 1 2 n follows

eorem 15 will provide us a help in computing CoV ofPDrs of implicit function of SVs

Example 2 Consider

F(x y z) x3

+ 3y2

+ 4xz2

minus 3yz2

minus 5 0 (41)

is equationrsquos one solution is (1 1 1) F is obviously inCα which is OB represented by B((1 1 1) ε0) withx y zgt 0 for some α isin (0 1] since

zα

zzα F(1 1 1) 8xz

2minus αminus 6yz

2minus α1113872 11138731113961

(111) 2ne 0 (42)

eorem 15 implies that there is aneighbourhood U sub R2 of (1 1) niexist a UF z g(x y)

satisfies the followingg(1 1) 1 andF(x y g(x y)) 0 forall(x y) isin U

Moreover z g(x y) is Cα in U and we have

zα

zxα g(x y) minus

3x + 4z2

1113872 1113873x1minus α

(8x minus 6y)z

zα

zyα g(x y) minus

6y minus 3z2

1113872 1113873y1minus α

(8x minus 6y)z

(43)

Finally we obtain zαzxαg(1 1) minus (72) andzαzxαg(1 1) minus (32)

Finally CoV of IFm for a system of several equationsand SRVs is obtained

Theorem 16 (Conformable general implicit functiontheorem) Let α isin (0 1] F X⟶ Rm be a VVF defined inan OS X sub Rn+m ni forall (x y) (x1 xn y1 ym) isin Xeach xi yj gt 0 and point(a b) (a1 an b1 bm) isin X Assume that

(i) F(a b) 0(ii) F isin Cα(X Rm )

(iii) det[JαyF(a b)]ne 0

en there is a neighbourhood U sub Rn of a niexist a UFf U⟶ Rm x↦ y f(x) that satisfies f(a) b andF(x f(x) ) 0 forallx isin U

Finally y f(x) is class Cα in U and for everyi 1 2 n we have

zαfzxα

i

1113890 1113891

t

minus fαminus 11113872 1113873

minus 1middot J

αyF1113872 1113873

minus 1middot

zαFzxα

i

1113890 1113891

t

(44)

where

zαf

zxαi

1113890 1113891 zαf1

zxαi

zαfm

zxαi

1113888 1113889

fαminus 1

fαminus 11 0

⋮ ⋮

0 fαminus 1m

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

JαyF

zαF1

zyα1

zαF1

zyαm

zαFm

zyα1

zαFm

zyαm

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

zαF

zxαi

1113890 1113891 zαF1

zxαi

zαFm

zxαi

1113888 1113889

(45)

Proof e proof of existence and uniqueness of the implicitfunction is done similar to the traditional multivariablecalculus by applying mathematical induction on q and usingthe conformable implicit function theorem for severalvariables [28]

Let us now show formula (44) Assume that a systemwith several equations and SRVs is expressed as

F(x y) 0 or

F1 x1 xn y1 ym( 1113857 0

⋮

Fm x1 xn y1 ym( 1113857 0

⎧⎪⎪⎨

⎪⎪⎩ (46)

which satisfies hypotheses (i)ndash(iii) of the theorem then thissystem is defined in a neighbourhood U sub Rn of a theimplicit function y f(x) class Cα in U such that f(a) bwhich satisfies equation (1) ie

F(x f(x)) 0 or

F1 x f1(x) fm(x)( 1113857 0

Fm x f1(x) fm(x)( 1113857 0

⎧⎪⎪⎨

⎪⎪⎩(47)

By applying CoV of CR to the above equation we have

Journal of Mathematics 9

zαF1

zxαi

+zαF1

zyα1

middot fαminus 11 middot

zαf1

zxαi

+ middot middot middot +zαF1

zyαm

middot fαminus 1m middot

zαfm

zxαi

0

⋮

zαFm

zxαi

+zαFm

zyα1

middot fαminus 11 middot

zαf1

zxαi

+ middot middot middot +zαFm

zyαm

middot fαminus 1m middot

zαfm

zxαi

0

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

orzαF

zxαi

+ 1113944m

j1

zαF

zyαj

middot fαminus 1j middot

zαfj

zxαi

0

(48)

for all i 1 2 n In addition the MF of equation (48) is given as follows

zαF1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus

zαF1

zyα1

zαF1

zyαm

zαFm

zyα1

zαFm

zyαm

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

middot

fαminus 11 0

⋮ ⋮

0 fαminus 1m

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠middot

zαf1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

orzαFzxα

i

1113890 1113891

t

minus JαyF middot fαminus 1

middotzαfzxα

i

1113890 1113891

t

(49)

Since JαyF and fαminus 1 are regular matrices by hypothesis we

have

zαf1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus

fαminus 11 0

⋮ ⋮

0 fαminus 1m

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus 1

middot

zαF1

zyα1

zαF1

zyαm

zαFm

zyα1

zαFm

zyαm

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus 1

middot

zαF1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

orzαfzxα

i

1113890 1113891

t

minus fαminus 11113872 1113873

minus 1middot J

αyF1113872 1113873

minus 1middot

zαFzxα

i

1113890 1113891

t

(50)

which completes the proofWe will now show that eorem 16 can be used to

compute CoV of PDrs of systems with several equations andSRVs

Example 3 Consider a system of two equations and two realvariables

F1(x y z w) x2

+ y2

+ z2

+ w2

minus 6 0

F2(x y z w) x2

minus y2

+ z2

minus w2

0

⎧⎨

⎩ (51)

One solution of this equation is(x y z w) (1 1

2

radic

2

radic) Clearly F (F1 F2) is in Cα

which is OB B((1 12

radic

2

radic) ε0) with x y z wgt 0 for

some α isin (0 1] since

10 Journal of Mathematics

det JαzwF((

2

radic

2

radic 1 1))1113960 1113961 det

2z2minus α 2w2minus α

2z2minus α minus 2w2minus α⎛⎝ ⎞⎠⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

(112

radic

2

radic)

minus322propne 0 (52)

eorem 16 implies that there is aneighbourhood U sub R2 of (

2

radic

2

radic) niexist a UF f (f1 f2)

given by

z f1(x y)

w f2(x y)1113896 (53)

that satisfies

f1(1 1) 2

radic

f2(1 1) 2

radic

⎧⎨

⎩

F1 x y f1(x y) f2(x y)( 1113857 0

F2 x y f1(x y) f2(x y)( 1113857 0forall(x y) isin U1113896

(54)

Moreover f (f1 f2) is class Cα in U and we have

zαf1

zxα

zαf2

zxα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ minus

zαminus 1 0

0 wαminus 1⎛⎝ ⎞⎠

minus 1

middot2z2minus α 2w2minus α

2z2minus α minus 2w2minus α⎛⎝ ⎞⎠

minus 1

middot2x

2minus α

2x2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz1minus α 0

0 w1minus α

⎛⎜⎝ ⎞⎟⎠ middotzαminus 2

zαminus 2

wαminus 2

minus wαminus 2

⎛⎜⎝ ⎞⎟⎠

middot2x

2minus α

2x2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz

minus 1z

minus 1

wminus 1

minus wminus 1

⎛⎜⎝ ⎞⎟⎠ middot2x

2minus α

2x2minus α

⎛⎜⎝ ⎞⎟⎠ minus

x2minus α

z

0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

zαf1

zyα

zαf2

zyα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minuszαminus 1 0

0 wαminus 1⎛⎝ ⎞⎠

minus 1

middot2z2minus α 2w2minus α

2z2minus α minus 2w2minus α⎛⎝ ⎞⎠

minus 1

middot2y

2minus α

minus 2y2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz1minus α 0

0 w1minus α

⎛⎜⎝ ⎞⎟⎠ middotzαminus 2

zαminus 2

wαminus 2

minus wαminus 2

⎛⎜⎝ ⎞⎟⎠

middot2y

2minus α

minus 2y2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz

minus 1z

minus 1

wminus 1

minus wminus 1

⎛⎜⎝ ⎞⎟⎠ middot2y

2minus α

minus 2y2minus α

⎛⎜⎝ ⎞⎟⎠

0

minusy2minus α

w

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(55)

Finally we have

zαf1

zxα

zαf2

zxα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(112

radic

2

radic)

minus12

radic

0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

zαf1

zyα

zαf2

zyα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(112

radic

2

radic)

0

minus12

radic

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(56)

According to all of our numerical examples in this workwith the help of our investigation to all proposed theoremsand properties in this article the numerical examples showthat all our obtained results are valid and efficient As seen inall these examples the computations are simple and cost-

efficient which are important when scientific phenomenaare modelled in the sense of conformable calculus All in allthe simplicity of computations is always needed in modelingapplications in comparison to other complicated compu-tations that are needed using other approaches in otherfractional operators All our obtained results in this work areaccurate because our results coincide with the usual integer-order results

6 Conclusion

Anew investigation on the CoV ofmultivariable calculus hasbeen discussed in this work in detail e α-derivative of afunction of SVs and all related properties have been in-vestigatede CoV of CR for functions of SVs has also beenstudied e CoV of IFm has been presented in the lastpart of our results and numerical experiments have beenconducted to support our theoretical results e findings ofthis investigations show that all results formulated via CoD

Journal of Mathematics 11

coincide with the ones in the traditional integer case Allresults in this work can be potentially applied in modelingoceanographic phenomena such as Stommelrsquos box model ofthermohaline circulation and other related models wherethis studyrsquos analysis can be further extended or generalizedin 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

Authorsrsquo Contributions

MohammedK A Kaabar involved in actualization developedmethodology performed formal analysis validated and in-vestigated the study prepared the initial draft and supervisedand edited the original draft Francisco Martınez InmaculadaMartınez and Silvestre Paredes involved in actualizationdeveloped methodology performed formal analysis validatedand investigated the study and prepared the initial draftZailan Siri developed methodology performed formal anal-ysis validated and investigated the study and prepared theinitial draft All authors read and approved the final version

References

[1] M M A Khater M S Mohamed and R A M Attia ldquoOnsemi analytical and numerical simulations for a mathematicalbiological model the time-fractional nonlinear Kolmogor-ovndashPetrovskiindashPiskunov (KPP) equationrdquo Chaos Solitons ampFractals vol 144 Article ID 110676 2021

[2] M A Khater K S Nisar and M S Mohamed ldquoNumericalinvestigation for the fractional nonlinear space-time telegraphequation via the trigonometric Quintic B-spline schemerdquoMathematical Methods in the Applied Sciences vol 44 no 6pp 4598ndash4606 2021

[3] M M A Khater M S Mohamed and S K Elagan ldquoDiverseaccurate computational solutions of the nonlinearKleinndashFockndashGordon equationrdquo Results in Physics vol 23Article ID 104003 2021

[4] M F Li J R Ren and T Zhu ldquoFractional vector calculus andfractional special functionrdquo arXiv preprint10012889 2010

[5] D J Wollkind and B J Dichone Comprehensive AppliedMathematical Modeling in the Natural and Engineering Sci-ences Springer International Publishing Berlin Germany2017

[6] P J Olver Introduction to Partial Differential EquationsSpringer International Publishing Berlin Germany 2014

[7] J D Krehbiel and J B Freund ldquoStokes flow inside a sphere inan inviscid extensional flowrdquo Zeitschrift fur AngewandteMathematik und Physik vol 68 no 4 pp 68ndash81 2017

[8] A A Kilbas H M Srivastava and J J Trujillo ldquoeory andapplications of fractional differential equationsrdquo in North-Holland Mathematics StudiesVol 204 Elsevier AmsterdamNetherlands 2006

[9] K S Miller An Introduction to Fractional Calculus andFractional Differential Equations John Wiley and SonsHoboken NJ USA 1993

[10] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014

[11] R Khalil M Al Horani and M Abu Hammad ldquoGeometricmeaning of conformable derivative via fractional cordsrdquo eJournal of Mathematics and Computer Science vol 19 no 4pp 241ndash245 2019

[12] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journalof Computational and Applied Mathematics vol 279pp 57ndash66 2015

[13] O S Iyiola and E R Nwaeze ldquoSome new results on the newconformable fractional calculus with application usingDprimeAlambert approachrdquo Progress in Fractional Differentiationand Applications vol 2 no 2 pp 115ndash122 2016

[14] A Atangana D Baleanu and A Alsaedi ldquoNew properties ofconformable derivativerdquo Open Mathematics vol 13 pp 57ndash63 2015

[15] N Yazici and U Gozutok ldquoMultivariable conformablefractional calculusrdquo Filomat vol 32 no 2 pp 45ndash53 2018

[16] F Martınez I Martınez and S Paredes ldquoConformableEulerprimes theorem on homogeneous functionsrdquo ComputationalMathematical Methods vol 1 no 5 pp 1ndash11 2018

[17] A A Hyder and A H Soliman ldquoExact solutions of space-timelocal fractal nonlinear evolution equations a generalizedconformable derivative approachrdquo Results in Physics vol 17Article ID 103135 2020

[18] A A Hyder ldquoWhite noise theory and general improvedKudryashov method for stochastic nonlinear evolutionequations with conformable derivativesrdquo Advances in Dif-ference Equations vol 202019 pages 2020

[19] A H Soliman and A A Hyder ldquoClosed-form solutions ofstochastic KdV equation with generalized conformable deriv-ativesrdquo Physica Scripta vol 95 no 6 Article ID 065219 2020

[20] V E Tarasov ldquoNo violation of the Leibniz rule No fractionalderivativerdquo Communications in Nonlinear Science and Nu-merical Simulation vol 18 no 11 pp 2945ndash2948 2013

[21] R Khalil M Al Horani A Yousef and M SababhehldquoFractional analytic functionsrdquo Far East Journal of Mathe-matical Sciences vol 103 no 1 pp 113ndash123 2018

[22] S Uccedilar and N Y Ozgur ldquoComplex conformable derivativerdquoArabian Journal of Geosciences vol 12 no 201 2019

[23] F Martınez I Martınez I Martınez M K A Kaabar andS Paredes ldquoNew results on complex conformable integralrdquoAIMS Mathematics vol 5 no 6 pp 7695ndash7710 2020

[24] H Kiskinov M Petkova A Zahariev and M VeselinovaldquoSome results about conformable derivatives in Banach spacesand an application to the partial differential equationsrdquo inAIPConference Proceedingsvol 2333 no 1 AIP Publishing LLCArticle ID 120002 2021

[25] U Gozutok H A Ccediloban and Y Sagıroglu ldquoFrenet framewith respect to conformable derivativerdquo Filomat vol 33no 6 pp 1541ndash1550 2019

[26] J C Mayo-Maldonado G Fernandez-Anaya and O F Ruiz-Martinez ldquoStability of conformable linear differential systemsa behavioural framework with applications in fractional-ordercontrolrdquo IET Control eory amp Applications vol 14 no 18pp 2900ndash2913 2020

[27] M Al Horani and R Khalil ldquoTotal fractional differential withapplications to exact fractional differential equationsrdquo In-ternational Journal of Computer Mathematics vol 95 no 6-7pp 1444ndash1452 2018

[28] J E Marsden and M J Hoffman Vector Calculus WH Freeman and Company New York NY USA 4th edition1996

12 Journal of Mathematics

Proof Proof According to PDrrsquos definition and eorem14 the following is implied

Remark 6 e MF of equation (33) is expressed as

Dα(g ∘ f)(a)(h)

zα

zyα1

g(f(a)) zα

zyαm

g(f(a))1113888 1113889 middot

f1(a)αminus 1 0 0

0 ⋱ 0

0 0 fm(a)αminus 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠middot

zα

zxα1f1(a)

zα

zxαn

f1(a)

middot middot middot middot middot middot

zα

zxα1fn(a)

zα

zxαn

fn(a)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

middot

h1

⋮

hn

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (34)

5 Conformable Implicit Function Theorem

In this section the CoV of IFm for SVs is obtained Firstwe establish the conformable theorem of existence andregularity of the implicit function for the case of a singleequation

Theorem 15 (Conformable implicit function theorem forthe case of a single equation) Suppose that α isin (0 1]F X⟶ R is a RVF defined in an OS X sub Rn+1 ni forall(x1 xn y) isin X each xi ygt 0 and point(a1 an b) isin X Suppose that

(i) F(a1 an b) 0(ii) F isin Cα(X R)

(iii) zαzyαF(a1 an b)ne 0

en there is a neighbourhood U sub Rn of (a1 an)such that there is a unique function (UF) y g(x1 xn)

that satisfies the followingg(a1 an) b and F(x1 xn g(x1 xn))

0 forall(x1 xn) isin UFinally y g(x1 xn) is Cα in U and for every

i 1 2 n we have

zα

zxαi

g x1 xn( 1113857 minuszαzx

αi F x1 xn( 1113857 g x1 xn( 1113857( 1113857( 1113857

zαzy

αF x1 xn( 1113857 g x1 xn( 1113857( 1113857( 1113857 middot g x1 xn( 1113857( 1113857

αminus 1 (35)

Proof WLOG X is assumed to be an OB represented byB((a1 an b) ε0) Let ρ isin (0 ε0) If we call δ

ε20 minus ρ21113969

it is verified that

x1 xn( 1113857 minus a1 an( 1113857

lt δ and |y minus b|lt ρ1113960 1113961rArr x1 xn y( 1113857 isin B a1 an b( 1113857 ε0( 1113857 (36)

Note that in particular if |y minus b|lt ρ then(a1 an y) isin B((a1 an b) ε0)

Since zαzyαF(a1 an b)ne 0 it is assumed to bepositive (otherwise minus F is considered instead of F) From factthat F(a1 an b) 0 it follows that

F a1 an b minus ρ( 1113857gt 0

F a1 an b minus ρ( 1113857lt 0(37)

By the continuity of F at (a1 an b minus ρ) and(a1 an b + ρ) there exists δprime isin (0 δ) such that

x1 xn( 1113857 minus a1 an( 1113857

lt δprimerArr F x1 xn b minus ρ( 1113857gt 0 andF x1 xn b + ρ( 1113857lt 01113858 1113859 (38)

Since the function y↦F(x1 xn y) is CF on the in-terval [b minus ρ b + ρ ] forall (x1 xn) isin B((a1 an) δprime) andvia the classical Bolzanorsquos theorem (Bm) it implies thatexistyx isin ( b minus ρ b + ρ) niF(x1 xn yx) 0 for eachx (x1 xn) en yrsquos value is unique since a functionwhose derivative is positive has more than zero On the other

hand by having U B((a1 an) δprime) for each(x1 xn) isin U there exists a unique y

niF(x1 xn y) 0 we can write y g(x1 xn) andthen this function will be proven to be CF onB((a1 an) δprime)e continuity of the function g at the point(a1 an) is clear since for each ρgt 0 exist a value δprime gt 0 ni

8 Journal of Mathematics

x1 xn( 1113857 minus a1 an( 1113857

lt δprimerArr b minus yx

11138681113868111386811138681113868111386811138681113868

lt ρhArr b minus g x1 xn( 11138571113868111386811138681113868

1113868111386811138681113868lt ρ(39)

e function g continuity will be proven at any point(x1 xn) isin B((a1 an) δprime) by simply substitutingB((a1 an) δprime) for an OB B((x1 x)) is contained inB((a1 an) δprime)

Finally let us now show formula (35) By applyingeorem 14 to equation

F x1 xn y( 1113857 0 we have

zα

zxαi

F(x g(x)) +zα

zyα F(x g(x)) middot g(x)

αminus 1middot

zα

zxαi

g(x) 0

(40)

forall i 1 2 n nix (x1 xn) Solving zαzxαi g(x)

we get (35) In addition the formula (35) on the right side iscontinuous so the continuity of CoV of PDrs zαzxα

i g(x) foralli 1 2 n follows

eorem 15 will provide us a help in computing CoV ofPDrs of implicit function of SVs

Example 2 Consider

F(x y z) x3

+ 3y2

+ 4xz2

minus 3yz2

minus 5 0 (41)

is equationrsquos one solution is (1 1 1) F is obviously inCα which is OB represented by B((1 1 1) ε0) withx y zgt 0 for some α isin (0 1] since

zα

zzα F(1 1 1) 8xz

2minus αminus 6yz

2minus α1113872 11138731113961

(111) 2ne 0 (42)

eorem 15 implies that there is aneighbourhood U sub R2 of (1 1) niexist a UF z g(x y)

satisfies the followingg(1 1) 1 andF(x y g(x y)) 0 forall(x y) isin U

Moreover z g(x y) is Cα in U and we have

zα

zxα g(x y) minus

3x + 4z2

1113872 1113873x1minus α

(8x minus 6y)z

zα

zyα g(x y) minus

6y minus 3z2

1113872 1113873y1minus α

(8x minus 6y)z

(43)

Finally we obtain zαzxαg(1 1) minus (72) andzαzxαg(1 1) minus (32)

Finally CoV of IFm for a system of several equationsand SRVs is obtained

Theorem 16 (Conformable general implicit functiontheorem) Let α isin (0 1] F X⟶ Rm be a VVF defined inan OS X sub Rn+m ni forall (x y) (x1 xn y1 ym) isin Xeach xi yj gt 0 and point(a b) (a1 an b1 bm) isin X Assume that

(i) F(a b) 0(ii) F isin Cα(X Rm )

(iii) det[JαyF(a b)]ne 0

en there is a neighbourhood U sub Rn of a niexist a UFf U⟶ Rm x↦ y f(x) that satisfies f(a) b andF(x f(x) ) 0 forallx isin U

Finally y f(x) is class Cα in U and for everyi 1 2 n we have

zαfzxα

i

1113890 1113891

t

minus fαminus 11113872 1113873

minus 1middot J

αyF1113872 1113873

minus 1middot

zαFzxα

i

1113890 1113891

t

(44)

where

zαf

zxαi

1113890 1113891 zαf1

zxαi

zαfm

zxαi

1113888 1113889

fαminus 1

fαminus 11 0

⋮ ⋮

0 fαminus 1m

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

JαyF

zαF1

zyα1

zαF1

zyαm

zαFm

zyα1

zαFm

zyαm

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

zαF

zxαi

1113890 1113891 zαF1

zxαi

zαFm

zxαi

1113888 1113889

(45)

Proof e proof of existence and uniqueness of the implicitfunction is done similar to the traditional multivariablecalculus by applying mathematical induction on q and usingthe conformable implicit function theorem for severalvariables [28]

Let us now show formula (44) Assume that a systemwith several equations and SRVs is expressed as

F(x y) 0 or

F1 x1 xn y1 ym( 1113857 0

⋮

Fm x1 xn y1 ym( 1113857 0

⎧⎪⎪⎨

⎪⎪⎩ (46)

which satisfies hypotheses (i)ndash(iii) of the theorem then thissystem is defined in a neighbourhood U sub Rn of a theimplicit function y f(x) class Cα in U such that f(a) bwhich satisfies equation (1) ie

F(x f(x)) 0 or

F1 x f1(x) fm(x)( 1113857 0

Fm x f1(x) fm(x)( 1113857 0

⎧⎪⎪⎨

⎪⎪⎩(47)

By applying CoV of CR to the above equation we have

Journal of Mathematics 9

zαF1

zxαi

+zαF1

zyα1

middot fαminus 11 middot

zαf1

zxαi

+ middot middot middot +zαF1

zyαm

middot fαminus 1m middot

zαfm

zxαi

0

⋮

zαFm

zxαi

+zαFm

zyα1

middot fαminus 11 middot

zαf1

zxαi

+ middot middot middot +zαFm

zyαm

middot fαminus 1m middot

zαfm

zxαi

0

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

orzαF

zxαi

+ 1113944m

j1

zαF

zyαj

middot fαminus 1j middot

zαfj

zxαi

0

(48)

for all i 1 2 n In addition the MF of equation (48) is given as follows

zαF1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus

zαF1

zyα1

zαF1

zyαm

zαFm

zyα1

zαFm

zyαm

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

middot

fαminus 11 0

⋮ ⋮

0 fαminus 1m

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠middot

zαf1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

orzαFzxα

i

1113890 1113891

t

minus JαyF middot fαminus 1

middotzαfzxα

i

1113890 1113891

t

(49)

Since JαyF and fαminus 1 are regular matrices by hypothesis we

have

zαf1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus

fαminus 11 0

⋮ ⋮

0 fαminus 1m

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus 1

middot

zαF1

zyα1

zαF1

zyαm

zαFm

zyα1

zαFm

zyαm

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus 1

middot

zαF1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

orzαfzxα

i

1113890 1113891

t

minus fαminus 11113872 1113873

minus 1middot J

αyF1113872 1113873

minus 1middot

zαFzxα

i

1113890 1113891

t

(50)

which completes the proofWe will now show that eorem 16 can be used to

compute CoV of PDrs of systems with several equations andSRVs

Example 3 Consider a system of two equations and two realvariables

F1(x y z w) x2

+ y2

+ z2

+ w2

minus 6 0

F2(x y z w) x2

minus y2

+ z2

minus w2

0

⎧⎨

⎩ (51)

One solution of this equation is(x y z w) (1 1

2

radic

2

radic) Clearly F (F1 F2) is in Cα

which is OB B((1 12

radic

2

radic) ε0) with x y z wgt 0 for

some α isin (0 1] since

10 Journal of Mathematics

det JαzwF((

2

radic

2

radic 1 1))1113960 1113961 det

2z2minus α 2w2minus α

2z2minus α minus 2w2minus α⎛⎝ ⎞⎠⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

(112

radic

2

radic)

minus322propne 0 (52)

eorem 16 implies that there is aneighbourhood U sub R2 of (

2

radic

2

radic) niexist a UF f (f1 f2)

given by

z f1(x y)

w f2(x y)1113896 (53)

that satisfies

f1(1 1) 2

radic

f2(1 1) 2

radic

⎧⎨

⎩

F1 x y f1(x y) f2(x y)( 1113857 0

F2 x y f1(x y) f2(x y)( 1113857 0forall(x y) isin U1113896

(54)

Moreover f (f1 f2) is class Cα in U and we have

zαf1

zxα

zαf2

zxα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ minus

zαminus 1 0

0 wαminus 1⎛⎝ ⎞⎠

minus 1

middot2z2minus α 2w2minus α

2z2minus α minus 2w2minus α⎛⎝ ⎞⎠

minus 1

middot2x

2minus α

2x2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz1minus α 0

0 w1minus α

⎛⎜⎝ ⎞⎟⎠ middotzαminus 2

zαminus 2

wαminus 2

minus wαminus 2

⎛⎜⎝ ⎞⎟⎠

middot2x

2minus α

2x2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz

minus 1z

minus 1

wminus 1

minus wminus 1

⎛⎜⎝ ⎞⎟⎠ middot2x

2minus α

2x2minus α

⎛⎜⎝ ⎞⎟⎠ minus

x2minus α

z

0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

zαf1

zyα

zαf2

zyα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minuszαminus 1 0

0 wαminus 1⎛⎝ ⎞⎠

minus 1

middot2z2minus α 2w2minus α

2z2minus α minus 2w2minus α⎛⎝ ⎞⎠

minus 1

middot2y

2minus α

minus 2y2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz1minus α 0

0 w1minus α

⎛⎜⎝ ⎞⎟⎠ middotzαminus 2

zαminus 2

wαminus 2

minus wαminus 2

⎛⎜⎝ ⎞⎟⎠

middot2y

2minus α

minus 2y2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz

minus 1z

minus 1

wminus 1

minus wminus 1

⎛⎜⎝ ⎞⎟⎠ middot2y

2minus α

minus 2y2minus α

⎛⎜⎝ ⎞⎟⎠

0

minusy2minus α

w

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(55)

Finally we have

zαf1

zxα

zαf2

zxα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(112

radic

2

radic)

minus12

radic

0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

zαf1

zyα

zαf2

zyα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(112

radic

2

radic)

0

minus12

radic

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(56)

According to all of our numerical examples in this workwith the help of our investigation to all proposed theoremsand properties in this article the numerical examples showthat all our obtained results are valid and efficient As seen inall these examples the computations are simple and cost-

efficient which are important when scientific phenomenaare modelled in the sense of conformable calculus All in allthe simplicity of computations is always needed in modelingapplications in comparison to other complicated compu-tations that are needed using other approaches in otherfractional operators All our obtained results in this work areaccurate because our results coincide with the usual integer-order results

6 Conclusion

Anew investigation on the CoV ofmultivariable calculus hasbeen discussed in this work in detail e α-derivative of afunction of SVs and all related properties have been in-vestigatede CoV of CR for functions of SVs has also beenstudied e CoV of IFm has been presented in the lastpart of our results and numerical experiments have beenconducted to support our theoretical results e findings ofthis investigations show that all results formulated via CoD

Journal of Mathematics 11

coincide with the ones in the traditional integer case Allresults in this work can be potentially applied in modelingoceanographic phenomena such as Stommelrsquos box model ofthermohaline circulation and other related models wherethis studyrsquos analysis can be further extended or generalizedin 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

Authorsrsquo Contributions

MohammedK A Kaabar involved in actualization developedmethodology performed formal analysis validated and in-vestigated the study prepared the initial draft and supervisedand edited the original draft Francisco Martınez InmaculadaMartınez and Silvestre Paredes involved in actualizationdeveloped methodology performed formal analysis validatedand investigated the study and prepared the initial draftZailan Siri developed methodology performed formal anal-ysis validated and investigated the study and prepared theinitial draft All authors read and approved the final version

References

[1] M M A Khater M S Mohamed and R A M Attia ldquoOnsemi analytical and numerical simulations for a mathematicalbiological model the time-fractional nonlinear Kolmogor-ovndashPetrovskiindashPiskunov (KPP) equationrdquo Chaos Solitons ampFractals vol 144 Article ID 110676 2021

[2] M A Khater K S Nisar and M S Mohamed ldquoNumericalinvestigation for the fractional nonlinear space-time telegraphequation via the trigonometric Quintic B-spline schemerdquoMathematical Methods in the Applied Sciences vol 44 no 6pp 4598ndash4606 2021

[3] M M A Khater M S Mohamed and S K Elagan ldquoDiverseaccurate computational solutions of the nonlinearKleinndashFockndashGordon equationrdquo Results in Physics vol 23Article ID 104003 2021

[4] M F Li J R Ren and T Zhu ldquoFractional vector calculus andfractional special functionrdquo arXiv preprint10012889 2010

[5] D J Wollkind and B J Dichone Comprehensive AppliedMathematical Modeling in the Natural and Engineering Sci-ences Springer International Publishing Berlin Germany2017

[6] P J Olver Introduction to Partial Differential EquationsSpringer International Publishing Berlin Germany 2014

[7] J D Krehbiel and J B Freund ldquoStokes flow inside a sphere inan inviscid extensional flowrdquo Zeitschrift fur AngewandteMathematik und Physik vol 68 no 4 pp 68ndash81 2017

[8] A A Kilbas H M Srivastava and J J Trujillo ldquoeory andapplications of fractional differential equationsrdquo in North-Holland Mathematics StudiesVol 204 Elsevier AmsterdamNetherlands 2006

[9] K S Miller An Introduction to Fractional Calculus andFractional Differential Equations John Wiley and SonsHoboken NJ USA 1993

[10] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014

[11] R Khalil M Al Horani and M Abu Hammad ldquoGeometricmeaning of conformable derivative via fractional cordsrdquo eJournal of Mathematics and Computer Science vol 19 no 4pp 241ndash245 2019

[12] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journalof Computational and Applied Mathematics vol 279pp 57ndash66 2015

[13] O S Iyiola and E R Nwaeze ldquoSome new results on the newconformable fractional calculus with application usingDprimeAlambert approachrdquo Progress in Fractional Differentiationand Applications vol 2 no 2 pp 115ndash122 2016

[14] A Atangana D Baleanu and A Alsaedi ldquoNew properties ofconformable derivativerdquo Open Mathematics vol 13 pp 57ndash63 2015

[15] N Yazici and U Gozutok ldquoMultivariable conformablefractional calculusrdquo Filomat vol 32 no 2 pp 45ndash53 2018

[16] F Martınez I Martınez and S Paredes ldquoConformableEulerprimes theorem on homogeneous functionsrdquo ComputationalMathematical Methods vol 1 no 5 pp 1ndash11 2018

[17] A A Hyder and A H Soliman ldquoExact solutions of space-timelocal fractal nonlinear evolution equations a generalizedconformable derivative approachrdquo Results in Physics vol 17Article ID 103135 2020

[18] A A Hyder ldquoWhite noise theory and general improvedKudryashov method for stochastic nonlinear evolutionequations with conformable derivativesrdquo Advances in Dif-ference Equations vol 202019 pages 2020

[19] A H Soliman and A A Hyder ldquoClosed-form solutions ofstochastic KdV equation with generalized conformable deriv-ativesrdquo Physica Scripta vol 95 no 6 Article ID 065219 2020

[20] V E Tarasov ldquoNo violation of the Leibniz rule No fractionalderivativerdquo Communications in Nonlinear Science and Nu-merical Simulation vol 18 no 11 pp 2945ndash2948 2013

[21] R Khalil M Al Horani A Yousef and M SababhehldquoFractional analytic functionsrdquo Far East Journal of Mathe-matical Sciences vol 103 no 1 pp 113ndash123 2018

[22] S Uccedilar and N Y Ozgur ldquoComplex conformable derivativerdquoArabian Journal of Geosciences vol 12 no 201 2019

[23] F Martınez I Martınez I Martınez M K A Kaabar andS Paredes ldquoNew results on complex conformable integralrdquoAIMS Mathematics vol 5 no 6 pp 7695ndash7710 2020

[24] H Kiskinov M Petkova A Zahariev and M VeselinovaldquoSome results about conformable derivatives in Banach spacesand an application to the partial differential equationsrdquo inAIPConference Proceedingsvol 2333 no 1 AIP Publishing LLCArticle ID 120002 2021

[25] U Gozutok H A Ccediloban and Y Sagıroglu ldquoFrenet framewith respect to conformable derivativerdquo Filomat vol 33no 6 pp 1541ndash1550 2019

[26] J C Mayo-Maldonado G Fernandez-Anaya and O F Ruiz-Martinez ldquoStability of conformable linear differential systemsa behavioural framework with applications in fractional-ordercontrolrdquo IET Control eory amp Applications vol 14 no 18pp 2900ndash2913 2020

[27] M Al Horani and R Khalil ldquoTotal fractional differential withapplications to exact fractional differential equationsrdquo In-ternational Journal of Computer Mathematics vol 95 no 6-7pp 1444ndash1452 2018

[28] J E Marsden and M J Hoffman Vector Calculus WH Freeman and Company New York NY USA 4th edition1996

12 Journal of Mathematics

x1 xn( 1113857 minus a1 an( 1113857

lt δprimerArr b minus yx

11138681113868111386811138681113868111386811138681113868

lt ρhArr b minus g x1 xn( 11138571113868111386811138681113868

1113868111386811138681113868lt ρ(39)

e function g continuity will be proven at any point(x1 xn) isin B((a1 an) δprime) by simply substitutingB((a1 an) δprime) for an OB B((x1 x)) is contained inB((a1 an) δprime)

Finally let us now show formula (35) By applyingeorem 14 to equation

F x1 xn y( 1113857 0 we have

zα

zxαi

F(x g(x)) +zα

zyα F(x g(x)) middot g(x)

αminus 1middot

zα

zxαi

g(x) 0

(40)

forall i 1 2 n nix (x1 xn) Solving zαzxαi g(x)

we get (35) In addition the formula (35) on the right side iscontinuous so the continuity of CoV of PDrs zαzxα

i g(x) foralli 1 2 n follows

eorem 15 will provide us a help in computing CoV ofPDrs of implicit function of SVs

Example 2 Consider

F(x y z) x3

+ 3y2

+ 4xz2

minus 3yz2

minus 5 0 (41)

is equationrsquos one solution is (1 1 1) F is obviously inCα which is OB represented by B((1 1 1) ε0) withx y zgt 0 for some α isin (0 1] since

zα

zzα F(1 1 1) 8xz

2minus αminus 6yz

2minus α1113872 11138731113961

(111) 2ne 0 (42)

eorem 15 implies that there is aneighbourhood U sub R2 of (1 1) niexist a UF z g(x y)

satisfies the followingg(1 1) 1 andF(x y g(x y)) 0 forall(x y) isin U

Moreover z g(x y) is Cα in U and we have

zα

zxα g(x y) minus

3x + 4z2

1113872 1113873x1minus α

(8x minus 6y)z

zα

zyα g(x y) minus

6y minus 3z2

1113872 1113873y1minus α

(8x minus 6y)z

(43)

Finally we obtain zαzxαg(1 1) minus (72) andzαzxαg(1 1) minus (32)

Finally CoV of IFm for a system of several equationsand SRVs is obtained

Theorem 16 (Conformable general implicit functiontheorem) Let α isin (0 1] F X⟶ Rm be a VVF defined inan OS X sub Rn+m ni forall (x y) (x1 xn y1 ym) isin Xeach xi yj gt 0 and point(a b) (a1 an b1 bm) isin X Assume that

(i) F(a b) 0(ii) F isin Cα(X Rm )

(iii) det[JαyF(a b)]ne 0

en there is a neighbourhood U sub Rn of a niexist a UFf U⟶ Rm x↦ y f(x) that satisfies f(a) b andF(x f(x) ) 0 forallx isin U

Finally y f(x) is class Cα in U and for everyi 1 2 n we have

zαfzxα

i

1113890 1113891

t

minus fαminus 11113872 1113873

minus 1middot J

αyF1113872 1113873

minus 1middot

zαFzxα

i

1113890 1113891

t

(44)

where

zαf

zxαi

1113890 1113891 zαf1

zxαi

zαfm

zxαi

1113888 1113889

fαminus 1

fαminus 11 0

⋮ ⋮

0 fαminus 1m

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

JαyF

zαF1

zyα1

zαF1

zyαm

zαFm

zyα1

zαFm

zyαm

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

zαF

zxαi

1113890 1113891 zαF1

zxαi

zαFm

zxαi

1113888 1113889

(45)

Proof e proof of existence and uniqueness of the implicitfunction is done similar to the traditional multivariablecalculus by applying mathematical induction on q and usingthe conformable implicit function theorem for severalvariables [28]

Let us now show formula (44) Assume that a systemwith several equations and SRVs is expressed as

F(x y) 0 or

F1 x1 xn y1 ym( 1113857 0

⋮

Fm x1 xn y1 ym( 1113857 0

⎧⎪⎪⎨

⎪⎪⎩ (46)

which satisfies hypotheses (i)ndash(iii) of the theorem then thissystem is defined in a neighbourhood U sub Rn of a theimplicit function y f(x) class Cα in U such that f(a) bwhich satisfies equation (1) ie

F(x f(x)) 0 or

F1 x f1(x) fm(x)( 1113857 0

Fm x f1(x) fm(x)( 1113857 0

⎧⎪⎪⎨

⎪⎪⎩(47)

By applying CoV of CR to the above equation we have

Journal of Mathematics 9

zαF1

zxαi

+zαF1

zyα1

middot fαminus 11 middot

zαf1

zxαi

+ middot middot middot +zαF1

zyαm

middot fαminus 1m middot

zαfm

zxαi

0

⋮

zαFm

zxαi

+zαFm

zyα1

middot fαminus 11 middot

zαf1

zxαi

+ middot middot middot +zαFm

zyαm

middot fαminus 1m middot

zαfm

zxαi

0

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

orzαF

zxαi

+ 1113944m

j1

zαF

zyαj

middot fαminus 1j middot

zαfj

zxαi

0

(48)

for all i 1 2 n In addition the MF of equation (48) is given as follows

zαF1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus

zαF1

zyα1

zαF1

zyαm

zαFm

zyα1

zαFm

zyαm

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

middot

fαminus 11 0

⋮ ⋮

0 fαminus 1m

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠middot

zαf1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

orzαFzxα

i

1113890 1113891

t

minus JαyF middot fαminus 1

middotzαfzxα

i

1113890 1113891

t

(49)

Since JαyF and fαminus 1 are regular matrices by hypothesis we

have

zαf1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus

fαminus 11 0

⋮ ⋮

0 fαminus 1m

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus 1

middot

zαF1

zyα1

zαF1

zyαm

zαFm

zyα1

zαFm

zyαm

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus 1

middot

zαF1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

orzαfzxα

i

1113890 1113891

t

minus fαminus 11113872 1113873

minus 1middot J

αyF1113872 1113873

minus 1middot

zαFzxα

i

1113890 1113891

t

(50)

which completes the proofWe will now show that eorem 16 can be used to

compute CoV of PDrs of systems with several equations andSRVs

Example 3 Consider a system of two equations and two realvariables

F1(x y z w) x2

+ y2

+ z2

+ w2

minus 6 0

F2(x y z w) x2

minus y2

+ z2

minus w2

0

⎧⎨

⎩ (51)

One solution of this equation is(x y z w) (1 1

2

radic

2

radic) Clearly F (F1 F2) is in Cα

which is OB B((1 12

radic

2

radic) ε0) with x y z wgt 0 for

some α isin (0 1] since

10 Journal of Mathematics

det JαzwF((

2

radic

2

radic 1 1))1113960 1113961 det

2z2minus α 2w2minus α

2z2minus α minus 2w2minus α⎛⎝ ⎞⎠⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

(112

radic

2

radic)

minus322propne 0 (52)

eorem 16 implies that there is aneighbourhood U sub R2 of (

2

radic

2

radic) niexist a UF f (f1 f2)

given by

z f1(x y)

w f2(x y)1113896 (53)

that satisfies

f1(1 1) 2

radic

f2(1 1) 2

radic

⎧⎨

⎩

F1 x y f1(x y) f2(x y)( 1113857 0

F2 x y f1(x y) f2(x y)( 1113857 0forall(x y) isin U1113896

(54)

Moreover f (f1 f2) is class Cα in U and we have

zαf1

zxα

zαf2

zxα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ minus

zαminus 1 0

0 wαminus 1⎛⎝ ⎞⎠

minus 1

middot2z2minus α 2w2minus α

2z2minus α minus 2w2minus α⎛⎝ ⎞⎠

minus 1

middot2x

2minus α

2x2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz1minus α 0

0 w1minus α

⎛⎜⎝ ⎞⎟⎠ middotzαminus 2

zαminus 2

wαminus 2

minus wαminus 2

⎛⎜⎝ ⎞⎟⎠

middot2x

2minus α

2x2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz

minus 1z

minus 1

wminus 1

minus wminus 1

⎛⎜⎝ ⎞⎟⎠ middot2x

2minus α

2x2minus α

⎛⎜⎝ ⎞⎟⎠ minus

x2minus α

z

0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

zαf1

zyα

zαf2

zyα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minuszαminus 1 0

0 wαminus 1⎛⎝ ⎞⎠

minus 1

middot2z2minus α 2w2minus α

2z2minus α minus 2w2minus α⎛⎝ ⎞⎠

minus 1

middot2y

2minus α

minus 2y2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz1minus α 0

0 w1minus α

⎛⎜⎝ ⎞⎟⎠ middotzαminus 2

zαminus 2

wαminus 2

minus wαminus 2

⎛⎜⎝ ⎞⎟⎠

middot2y

2minus α

minus 2y2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz

minus 1z

minus 1

wminus 1

minus wminus 1

⎛⎜⎝ ⎞⎟⎠ middot2y

2minus α

minus 2y2minus α

⎛⎜⎝ ⎞⎟⎠

0

minusy2minus α

w

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(55)

Finally we have

zαf1

zxα

zαf2

zxα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(112

radic

2

radic)

minus12

radic

0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

zαf1

zyα

zαf2

zyα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(112

radic

2

radic)

0

minus12

radic

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(56)

According to all of our numerical examples in this workwith the help of our investigation to all proposed theoremsand properties in this article the numerical examples showthat all our obtained results are valid and efficient As seen inall these examples the computations are simple and cost-

efficient which are important when scientific phenomenaare modelled in the sense of conformable calculus All in allthe simplicity of computations is always needed in modelingapplications in comparison to other complicated compu-tations that are needed using other approaches in otherfractional operators All our obtained results in this work areaccurate because our results coincide with the usual integer-order results

6 Conclusion

Anew investigation on the CoV ofmultivariable calculus hasbeen discussed in this work in detail e α-derivative of afunction of SVs and all related properties have been in-vestigatede CoV of CR for functions of SVs has also beenstudied e CoV of IFm has been presented in the lastpart of our results and numerical experiments have beenconducted to support our theoretical results e findings ofthis investigations show that all results formulated via CoD

Journal of Mathematics 11

coincide with the ones in the traditional integer case Allresults in this work can be potentially applied in modelingoceanographic phenomena such as Stommelrsquos box model ofthermohaline circulation and other related models wherethis studyrsquos analysis can be further extended or generalizedin 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

Authorsrsquo Contributions

MohammedK A Kaabar involved in actualization developedmethodology performed formal analysis validated and in-vestigated the study prepared the initial draft and supervisedand edited the original draft Francisco Martınez InmaculadaMartınez and Silvestre Paredes involved in actualizationdeveloped methodology performed formal analysis validatedand investigated the study and prepared the initial draftZailan Siri developed methodology performed formal anal-ysis validated and investigated the study and prepared theinitial draft All authors read and approved the final version

References

[1] M M A Khater M S Mohamed and R A M Attia ldquoOnsemi analytical and numerical simulations for a mathematicalbiological model the time-fractional nonlinear Kolmogor-ovndashPetrovskiindashPiskunov (KPP) equationrdquo Chaos Solitons ampFractals vol 144 Article ID 110676 2021

[2] M A Khater K S Nisar and M S Mohamed ldquoNumericalinvestigation for the fractional nonlinear space-time telegraphequation via the trigonometric Quintic B-spline schemerdquoMathematical Methods in the Applied Sciences vol 44 no 6pp 4598ndash4606 2021

[3] M M A Khater M S Mohamed and S K Elagan ldquoDiverseaccurate computational solutions of the nonlinearKleinndashFockndashGordon equationrdquo Results in Physics vol 23Article ID 104003 2021

[4] M F Li J R Ren and T Zhu ldquoFractional vector calculus andfractional special functionrdquo arXiv preprint10012889 2010

[5] D J Wollkind and B J Dichone Comprehensive AppliedMathematical Modeling in the Natural and Engineering Sci-ences Springer International Publishing Berlin Germany2017

[6] P J Olver Introduction to Partial Differential EquationsSpringer International Publishing Berlin Germany 2014

[7] J D Krehbiel and J B Freund ldquoStokes flow inside a sphere inan inviscid extensional flowrdquo Zeitschrift fur AngewandteMathematik und Physik vol 68 no 4 pp 68ndash81 2017

[8] A A Kilbas H M Srivastava and J J Trujillo ldquoeory andapplications of fractional differential equationsrdquo in North-Holland Mathematics StudiesVol 204 Elsevier AmsterdamNetherlands 2006

[9] K S Miller An Introduction to Fractional Calculus andFractional Differential Equations John Wiley and SonsHoboken NJ USA 1993

[10] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014

[11] R Khalil M Al Horani and M Abu Hammad ldquoGeometricmeaning of conformable derivative via fractional cordsrdquo eJournal of Mathematics and Computer Science vol 19 no 4pp 241ndash245 2019

[12] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journalof Computational and Applied Mathematics vol 279pp 57ndash66 2015

[13] O S Iyiola and E R Nwaeze ldquoSome new results on the newconformable fractional calculus with application usingDprimeAlambert approachrdquo Progress in Fractional Differentiationand Applications vol 2 no 2 pp 115ndash122 2016

[14] A Atangana D Baleanu and A Alsaedi ldquoNew properties ofconformable derivativerdquo Open Mathematics vol 13 pp 57ndash63 2015

[15] N Yazici and U Gozutok ldquoMultivariable conformablefractional calculusrdquo Filomat vol 32 no 2 pp 45ndash53 2018

[16] F Martınez I Martınez and S Paredes ldquoConformableEulerprimes theorem on homogeneous functionsrdquo ComputationalMathematical Methods vol 1 no 5 pp 1ndash11 2018

[17] A A Hyder and A H Soliman ldquoExact solutions of space-timelocal fractal nonlinear evolution equations a generalizedconformable derivative approachrdquo Results in Physics vol 17Article ID 103135 2020

[18] A A Hyder ldquoWhite noise theory and general improvedKudryashov method for stochastic nonlinear evolutionequations with conformable derivativesrdquo Advances in Dif-ference Equations vol 202019 pages 2020

[19] A H Soliman and A A Hyder ldquoClosed-form solutions ofstochastic KdV equation with generalized conformable deriv-ativesrdquo Physica Scripta vol 95 no 6 Article ID 065219 2020

[20] V E Tarasov ldquoNo violation of the Leibniz rule No fractionalderivativerdquo Communications in Nonlinear Science and Nu-merical Simulation vol 18 no 11 pp 2945ndash2948 2013

[21] R Khalil M Al Horani A Yousef and M SababhehldquoFractional analytic functionsrdquo Far East Journal of Mathe-matical Sciences vol 103 no 1 pp 113ndash123 2018

[22] S Uccedilar and N Y Ozgur ldquoComplex conformable derivativerdquoArabian Journal of Geosciences vol 12 no 201 2019

[23] F Martınez I Martınez I Martınez M K A Kaabar andS Paredes ldquoNew results on complex conformable integralrdquoAIMS Mathematics vol 5 no 6 pp 7695ndash7710 2020

[24] H Kiskinov M Petkova A Zahariev and M VeselinovaldquoSome results about conformable derivatives in Banach spacesand an application to the partial differential equationsrdquo inAIPConference Proceedingsvol 2333 no 1 AIP Publishing LLCArticle ID 120002 2021

[25] U Gozutok H A Ccediloban and Y Sagıroglu ldquoFrenet framewith respect to conformable derivativerdquo Filomat vol 33no 6 pp 1541ndash1550 2019

[26] J C Mayo-Maldonado G Fernandez-Anaya and O F Ruiz-Martinez ldquoStability of conformable linear differential systemsa behavioural framework with applications in fractional-ordercontrolrdquo IET Control eory amp Applications vol 14 no 18pp 2900ndash2913 2020

[27] M Al Horani and R Khalil ldquoTotal fractional differential withapplications to exact fractional differential equationsrdquo In-ternational Journal of Computer Mathematics vol 95 no 6-7pp 1444ndash1452 2018

[28] J E Marsden and M J Hoffman Vector Calculus WH Freeman and Company New York NY USA 4th edition1996

12 Journal of Mathematics

zαF1

zxαi

+zαF1

zyα1

middot fαminus 11 middot

zαf1

zxαi

+ middot middot middot +zαF1

zyαm

middot fαminus 1m middot

zαfm

zxαi

0

⋮

zαFm

zxαi

+zαFm

zyα1

middot fαminus 11 middot

zαf1

zxαi

+ middot middot middot +zαFm

zyαm

middot fαminus 1m middot

zαfm

zxαi

0

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

orzαF

zxαi

+ 1113944m

j1

zαF

zyαj

middot fαminus 1j middot

zαfj

zxαi

0

(48)

for all i 1 2 n In addition the MF of equation (48) is given as follows

zαF1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus

zαF1

zyα1

zαF1

zyαm

zαFm

zyα1

zαFm

zyαm

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

middot

fαminus 11 0

⋮ ⋮

0 fαminus 1m

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠middot

zαf1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

orzαFzxα

i

1113890 1113891

t

minus JαyF middot fαminus 1

middotzαfzxα

i

1113890 1113891

t

(49)

Since JαyF and fαminus 1 are regular matrices by hypothesis we

have

zαf1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus

fαminus 11 0

⋮ ⋮

0 fαminus 1m

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus 1

middot

zαF1

zyα1

zαF1

zyαm

zαFm

zyα1

zαFm

zyαm

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus 1

middot

zαF1

zxαi

⋮

zαFm

zxαi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

orzαfzxα

i

1113890 1113891

t

minus fαminus 11113872 1113873

minus 1middot J

αyF1113872 1113873

minus 1middot

zαFzxα

i

1113890 1113891

t

(50)

which completes the proofWe will now show that eorem 16 can be used to

compute CoV of PDrs of systems with several equations andSRVs

Example 3 Consider a system of two equations and two realvariables

F1(x y z w) x2

+ y2

+ z2

+ w2

minus 6 0

F2(x y z w) x2

minus y2

+ z2

minus w2

0

⎧⎨

⎩ (51)

One solution of this equation is(x y z w) (1 1

2

radic

2

radic) Clearly F (F1 F2) is in Cα

which is OB B((1 12

radic

2

radic) ε0) with x y z wgt 0 for

some α isin (0 1] since

10 Journal of Mathematics

det JαzwF((

2

radic

2

radic 1 1))1113960 1113961 det

2z2minus α 2w2minus α

2z2minus α minus 2w2minus α⎛⎝ ⎞⎠⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

(112

radic

2

radic)

minus322propne 0 (52)

eorem 16 implies that there is aneighbourhood U sub R2 of (

2

radic

2

radic) niexist a UF f (f1 f2)

given by

z f1(x y)

w f2(x y)1113896 (53)

that satisfies

f1(1 1) 2

radic

f2(1 1) 2

radic

⎧⎨

⎩

F1 x y f1(x y) f2(x y)( 1113857 0

F2 x y f1(x y) f2(x y)( 1113857 0forall(x y) isin U1113896

(54)

Moreover f (f1 f2) is class Cα in U and we have

zαf1

zxα

zαf2

zxα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ minus

zαminus 1 0

0 wαminus 1⎛⎝ ⎞⎠

minus 1

middot2z2minus α 2w2minus α

2z2minus α minus 2w2minus α⎛⎝ ⎞⎠

minus 1

middot2x

2minus α

2x2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz1minus α 0

0 w1minus α

⎛⎜⎝ ⎞⎟⎠ middotzαminus 2

zαminus 2

wαminus 2

minus wαminus 2

⎛⎜⎝ ⎞⎟⎠

middot2x

2minus α

2x2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz

minus 1z

minus 1

wminus 1

minus wminus 1

⎛⎜⎝ ⎞⎟⎠ middot2x

2minus α

2x2minus α

⎛⎜⎝ ⎞⎟⎠ minus

x2minus α

z

0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

zαf1

zyα

zαf2

zyα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minuszαminus 1 0

0 wαminus 1⎛⎝ ⎞⎠

minus 1

middot2z2minus α 2w2minus α

2z2minus α minus 2w2minus α⎛⎝ ⎞⎠

minus 1

middot2y

2minus α

minus 2y2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz1minus α 0

0 w1minus α

⎛⎜⎝ ⎞⎟⎠ middotzαminus 2

zαminus 2

wαminus 2

minus wαminus 2

⎛⎜⎝ ⎞⎟⎠

middot2y

2minus α

minus 2y2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz

minus 1z

minus 1

wminus 1

minus wminus 1

⎛⎜⎝ ⎞⎟⎠ middot2y

2minus α

minus 2y2minus α

⎛⎜⎝ ⎞⎟⎠

0

minusy2minus α

w

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(55)

Finally we have

zαf1

zxα

zαf2

zxα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(112

radic

2

radic)

minus12

radic

0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

zαf1

zyα

zαf2

zyα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(112

radic

2

radic)

0

minus12

radic

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(56)

According to all of our numerical examples in this workwith the help of our investigation to all proposed theoremsand properties in this article the numerical examples showthat all our obtained results are valid and efficient As seen inall these examples the computations are simple and cost-

efficient which are important when scientific phenomenaare modelled in the sense of conformable calculus All in allthe simplicity of computations is always needed in modelingapplications in comparison to other complicated compu-tations that are needed using other approaches in otherfractional operators All our obtained results in this work areaccurate because our results coincide with the usual integer-order results

6 Conclusion

Anew investigation on the CoV ofmultivariable calculus hasbeen discussed in this work in detail e α-derivative of afunction of SVs and all related properties have been in-vestigatede CoV of CR for functions of SVs has also beenstudied e CoV of IFm has been presented in the lastpart of our results and numerical experiments have beenconducted to support our theoretical results e findings ofthis investigations show that all results formulated via CoD

Journal of Mathematics 11

coincide with the ones in the traditional integer case Allresults in this work can be potentially applied in modelingoceanographic phenomena such as Stommelrsquos box model ofthermohaline circulation and other related models wherethis studyrsquos analysis can be further extended or generalizedin 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

Authorsrsquo Contributions

MohammedK A Kaabar involved in actualization developedmethodology performed formal analysis validated and in-vestigated the study prepared the initial draft and supervisedand edited the original draft Francisco Martınez InmaculadaMartınez and Silvestre Paredes involved in actualizationdeveloped methodology performed formal analysis validatedand investigated the study and prepared the initial draftZailan Siri developed methodology performed formal anal-ysis validated and investigated the study and prepared theinitial draft All authors read and approved the final version

References

[1] M M A Khater M S Mohamed and R A M Attia ldquoOnsemi analytical and numerical simulations for a mathematicalbiological model the time-fractional nonlinear Kolmogor-ovndashPetrovskiindashPiskunov (KPP) equationrdquo Chaos Solitons ampFractals vol 144 Article ID 110676 2021

[2] M A Khater K S Nisar and M S Mohamed ldquoNumericalinvestigation for the fractional nonlinear space-time telegraphequation via the trigonometric Quintic B-spline schemerdquoMathematical Methods in the Applied Sciences vol 44 no 6pp 4598ndash4606 2021

[3] M M A Khater M S Mohamed and S K Elagan ldquoDiverseaccurate computational solutions of the nonlinearKleinndashFockndashGordon equationrdquo Results in Physics vol 23Article ID 104003 2021

[4] M F Li J R Ren and T Zhu ldquoFractional vector calculus andfractional special functionrdquo arXiv preprint10012889 2010

[5] D J Wollkind and B J Dichone Comprehensive AppliedMathematical Modeling in the Natural and Engineering Sci-ences Springer International Publishing Berlin Germany2017

[6] P J Olver Introduction to Partial Differential EquationsSpringer International Publishing Berlin Germany 2014

[7] J D Krehbiel and J B Freund ldquoStokes flow inside a sphere inan inviscid extensional flowrdquo Zeitschrift fur AngewandteMathematik und Physik vol 68 no 4 pp 68ndash81 2017

[8] A A Kilbas H M Srivastava and J J Trujillo ldquoeory andapplications of fractional differential equationsrdquo in North-Holland Mathematics StudiesVol 204 Elsevier AmsterdamNetherlands 2006

[9] K S Miller An Introduction to Fractional Calculus andFractional Differential Equations John Wiley and SonsHoboken NJ USA 1993

[10] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014

[11] R Khalil M Al Horani and M Abu Hammad ldquoGeometricmeaning of conformable derivative via fractional cordsrdquo eJournal of Mathematics and Computer Science vol 19 no 4pp 241ndash245 2019

[12] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journalof Computational and Applied Mathematics vol 279pp 57ndash66 2015

[13] O S Iyiola and E R Nwaeze ldquoSome new results on the newconformable fractional calculus with application usingDprimeAlambert approachrdquo Progress in Fractional Differentiationand Applications vol 2 no 2 pp 115ndash122 2016

[14] A Atangana D Baleanu and A Alsaedi ldquoNew properties ofconformable derivativerdquo Open Mathematics vol 13 pp 57ndash63 2015

[15] N Yazici and U Gozutok ldquoMultivariable conformablefractional calculusrdquo Filomat vol 32 no 2 pp 45ndash53 2018

[16] F Martınez I Martınez and S Paredes ldquoConformableEulerprimes theorem on homogeneous functionsrdquo ComputationalMathematical Methods vol 1 no 5 pp 1ndash11 2018

[17] A A Hyder and A H Soliman ldquoExact solutions of space-timelocal fractal nonlinear evolution equations a generalizedconformable derivative approachrdquo Results in Physics vol 17Article ID 103135 2020

[18] A A Hyder ldquoWhite noise theory and general improvedKudryashov method for stochastic nonlinear evolutionequations with conformable derivativesrdquo Advances in Dif-ference Equations vol 202019 pages 2020

[19] A H Soliman and A A Hyder ldquoClosed-form solutions ofstochastic KdV equation with generalized conformable deriv-ativesrdquo Physica Scripta vol 95 no 6 Article ID 065219 2020

[20] V E Tarasov ldquoNo violation of the Leibniz rule No fractionalderivativerdquo Communications in Nonlinear Science and Nu-merical Simulation vol 18 no 11 pp 2945ndash2948 2013

[21] R Khalil M Al Horani A Yousef and M SababhehldquoFractional analytic functionsrdquo Far East Journal of Mathe-matical Sciences vol 103 no 1 pp 113ndash123 2018

[22] S Uccedilar and N Y Ozgur ldquoComplex conformable derivativerdquoArabian Journal of Geosciences vol 12 no 201 2019

[23] F Martınez I Martınez I Martınez M K A Kaabar andS Paredes ldquoNew results on complex conformable integralrdquoAIMS Mathematics vol 5 no 6 pp 7695ndash7710 2020

[24] H Kiskinov M Petkova A Zahariev and M VeselinovaldquoSome results about conformable derivatives in Banach spacesand an application to the partial differential equationsrdquo inAIPConference Proceedingsvol 2333 no 1 AIP Publishing LLCArticle ID 120002 2021

[25] U Gozutok H A Ccediloban and Y Sagıroglu ldquoFrenet framewith respect to conformable derivativerdquo Filomat vol 33no 6 pp 1541ndash1550 2019

[26] J C Mayo-Maldonado G Fernandez-Anaya and O F Ruiz-Martinez ldquoStability of conformable linear differential systemsa behavioural framework with applications in fractional-ordercontrolrdquo IET Control eory amp Applications vol 14 no 18pp 2900ndash2913 2020

[27] M Al Horani and R Khalil ldquoTotal fractional differential withapplications to exact fractional differential equationsrdquo In-ternational Journal of Computer Mathematics vol 95 no 6-7pp 1444ndash1452 2018

[28] J E Marsden and M J Hoffman Vector Calculus WH Freeman and Company New York NY USA 4th edition1996

12 Journal of Mathematics

det JαzwF((

2

radic

2

radic 1 1))1113960 1113961 det

2z2minus α 2w2minus α

2z2minus α minus 2w2minus α⎛⎝ ⎞⎠⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

(112

radic

2

radic)

minus322propne 0 (52)

eorem 16 implies that there is aneighbourhood U sub R2 of (

2

radic

2

radic) niexist a UF f (f1 f2)

given by

z f1(x y)

w f2(x y)1113896 (53)

that satisfies

f1(1 1) 2

radic

f2(1 1) 2

radic

⎧⎨

⎩

F1 x y f1(x y) f2(x y)( 1113857 0

F2 x y f1(x y) f2(x y)( 1113857 0forall(x y) isin U1113896

(54)

Moreover f (f1 f2) is class Cα in U and we have

zαf1

zxα

zαf2

zxα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ minus

zαminus 1 0

0 wαminus 1⎛⎝ ⎞⎠

minus 1

middot2z2minus α 2w2minus α

2z2minus α minus 2w2minus α⎛⎝ ⎞⎠

minus 1

middot2x

2minus α

2x2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz1minus α 0

0 w1minus α

⎛⎜⎝ ⎞⎟⎠ middotzαminus 2

zαminus 2

wαminus 2

minus wαminus 2

⎛⎜⎝ ⎞⎟⎠

middot2x

2minus α

2x2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz

minus 1z

minus 1

wminus 1

minus wminus 1

⎛⎜⎝ ⎞⎟⎠ middot2x

2minus α

2x2minus α

⎛⎜⎝ ⎞⎟⎠ minus

x2minus α

z

0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

zαf1

zyα

zαf2

zyα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minuszαminus 1 0

0 wαminus 1⎛⎝ ⎞⎠

minus 1

middot2z2minus α 2w2minus α

2z2minus α minus 2w2minus α⎛⎝ ⎞⎠

minus 1

middot2y

2minus α

minus 2y2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz1minus α 0

0 w1minus α

⎛⎜⎝ ⎞⎟⎠ middotzαminus 2

zαminus 2

wαminus 2

minus wαminus 2

⎛⎜⎝ ⎞⎟⎠

middot2y

2minus α

minus 2y2minus α

⎛⎜⎝ ⎞⎟⎠ minus14

middotz

minus 1z

minus 1

wminus 1

minus wminus 1

⎛⎜⎝ ⎞⎟⎠ middot2y

2minus α

minus 2y2minus α

⎛⎜⎝ ⎞⎟⎠

0

minusy2minus α

w

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(55)

Finally we have

zαf1

zxα

zαf2

zxα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(112

radic

2

radic)

minus12

radic

0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

zαf1

zyα

zαf2

zyα

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(112

radic

2

radic)

0

minus12

radic

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(56)

According to all of our numerical examples in this workwith the help of our investigation to all proposed theoremsand properties in this article the numerical examples showthat all our obtained results are valid and efficient As seen inall these examples the computations are simple and cost-

efficient which are important when scientific phenomenaare modelled in the sense of conformable calculus All in allthe simplicity of computations is always needed in modelingapplications in comparison to other complicated compu-tations that are needed using other approaches in otherfractional operators All our obtained results in this work areaccurate because our results coincide with the usual integer-order results

6 Conclusion

Anew investigation on the CoV ofmultivariable calculus hasbeen discussed in this work in detail e α-derivative of afunction of SVs and all related properties have been in-vestigatede CoV of CR for functions of SVs has also beenstudied e CoV of IFm has been presented in the lastpart of our results and numerical experiments have beenconducted to support our theoretical results e findings ofthis investigations show that all results formulated via CoD

Journal of Mathematics 11

coincide with the ones in the traditional integer case Allresults in this work can be potentially applied in modelingoceanographic phenomena such as Stommelrsquos box model ofthermohaline circulation and other related models wherethis studyrsquos analysis can be further extended or generalizedin 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

Authorsrsquo Contributions

MohammedK A Kaabar involved in actualization developedmethodology performed formal analysis validated and in-vestigated the study prepared the initial draft and supervisedand edited the original draft Francisco Martınez InmaculadaMartınez and Silvestre Paredes involved in actualizationdeveloped methodology performed formal analysis validatedand investigated the study and prepared the initial draftZailan Siri developed methodology performed formal anal-ysis validated and investigated the study and prepared theinitial draft All authors read and approved the final version

References

[1] M M A Khater M S Mohamed and R A M Attia ldquoOnsemi analytical and numerical simulations for a mathematicalbiological model the time-fractional nonlinear Kolmogor-ovndashPetrovskiindashPiskunov (KPP) equationrdquo Chaos Solitons ampFractals vol 144 Article ID 110676 2021

[2] M A Khater K S Nisar and M S Mohamed ldquoNumericalinvestigation for the fractional nonlinear space-time telegraphequation via the trigonometric Quintic B-spline schemerdquoMathematical Methods in the Applied Sciences vol 44 no 6pp 4598ndash4606 2021

[3] M M A Khater M S Mohamed and S K Elagan ldquoDiverseaccurate computational solutions of the nonlinearKleinndashFockndashGordon equationrdquo Results in Physics vol 23Article ID 104003 2021

[4] M F Li J R Ren and T Zhu ldquoFractional vector calculus andfractional special functionrdquo arXiv preprint10012889 2010

[5] D J Wollkind and B J Dichone Comprehensive AppliedMathematical Modeling in the Natural and Engineering Sci-ences Springer International Publishing Berlin Germany2017

[6] P J Olver Introduction to Partial Differential EquationsSpringer International Publishing Berlin Germany 2014

[7] J D Krehbiel and J B Freund ldquoStokes flow inside a sphere inan inviscid extensional flowrdquo Zeitschrift fur AngewandteMathematik und Physik vol 68 no 4 pp 68ndash81 2017

[8] A A Kilbas H M Srivastava and J J Trujillo ldquoeory andapplications of fractional differential equationsrdquo in North-Holland Mathematics StudiesVol 204 Elsevier AmsterdamNetherlands 2006

[9] K S Miller An Introduction to Fractional Calculus andFractional Differential Equations John Wiley and SonsHoboken NJ USA 1993

[10] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014

[11] R Khalil M Al Horani and M Abu Hammad ldquoGeometricmeaning of conformable derivative via fractional cordsrdquo eJournal of Mathematics and Computer Science vol 19 no 4pp 241ndash245 2019

[12] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journalof Computational and Applied Mathematics vol 279pp 57ndash66 2015

[13] O S Iyiola and E R Nwaeze ldquoSome new results on the newconformable fractional calculus with application usingDprimeAlambert approachrdquo Progress in Fractional Differentiationand Applications vol 2 no 2 pp 115ndash122 2016

[14] A Atangana D Baleanu and A Alsaedi ldquoNew properties ofconformable derivativerdquo Open Mathematics vol 13 pp 57ndash63 2015

[15] N Yazici and U Gozutok ldquoMultivariable conformablefractional calculusrdquo Filomat vol 32 no 2 pp 45ndash53 2018

[16] F Martınez I Martınez and S Paredes ldquoConformableEulerprimes theorem on homogeneous functionsrdquo ComputationalMathematical Methods vol 1 no 5 pp 1ndash11 2018

[17] A A Hyder and A H Soliman ldquoExact solutions of space-timelocal fractal nonlinear evolution equations a generalizedconformable derivative approachrdquo Results in Physics vol 17Article ID 103135 2020

[18] A A Hyder ldquoWhite noise theory and general improvedKudryashov method for stochastic nonlinear evolutionequations with conformable derivativesrdquo Advances in Dif-ference Equations vol 202019 pages 2020

[19] A H Soliman and A A Hyder ldquoClosed-form solutions ofstochastic KdV equation with generalized conformable deriv-ativesrdquo Physica Scripta vol 95 no 6 Article ID 065219 2020

[20] V E Tarasov ldquoNo violation of the Leibniz rule No fractionalderivativerdquo Communications in Nonlinear Science and Nu-merical Simulation vol 18 no 11 pp 2945ndash2948 2013

[21] R Khalil M Al Horani A Yousef and M SababhehldquoFractional analytic functionsrdquo Far East Journal of Mathe-matical Sciences vol 103 no 1 pp 113ndash123 2018

[22] S Uccedilar and N Y Ozgur ldquoComplex conformable derivativerdquoArabian Journal of Geosciences vol 12 no 201 2019

[23] F Martınez I Martınez I Martınez M K A Kaabar andS Paredes ldquoNew results on complex conformable integralrdquoAIMS Mathematics vol 5 no 6 pp 7695ndash7710 2020

[24] H Kiskinov M Petkova A Zahariev and M VeselinovaldquoSome results about conformable derivatives in Banach spacesand an application to the partial differential equationsrdquo inAIPConference Proceedingsvol 2333 no 1 AIP Publishing LLCArticle ID 120002 2021

[25] U Gozutok H A Ccediloban and Y Sagıroglu ldquoFrenet framewith respect to conformable derivativerdquo Filomat vol 33no 6 pp 1541ndash1550 2019

[26] J C Mayo-Maldonado G Fernandez-Anaya and O F Ruiz-Martinez ldquoStability of conformable linear differential systemsa behavioural framework with applications in fractional-ordercontrolrdquo IET Control eory amp Applications vol 14 no 18pp 2900ndash2913 2020

[27] M Al Horani and R Khalil ldquoTotal fractional differential withapplications to exact fractional differential equationsrdquo In-ternational Journal of Computer Mathematics vol 95 no 6-7pp 1444ndash1452 2018

[28] J E Marsden and M J Hoffman Vector Calculus WH Freeman and Company New York NY USA 4th edition1996

12 Journal of Mathematics

coincide with the ones in the traditional integer case Allresults in this work can be potentially applied in modelingoceanographic phenomena such as Stommelrsquos box model ofthermohaline circulation and other related models wherethis studyrsquos analysis can be further extended or generalizedin 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

Authorsrsquo Contributions

MohammedK A Kaabar involved in actualization developedmethodology performed formal analysis validated and in-vestigated the study prepared the initial draft and supervisedand edited the original draft Francisco Martınez InmaculadaMartınez and Silvestre Paredes involved in actualizationdeveloped methodology performed formal analysis validatedand investigated the study and prepared the initial draftZailan Siri developed methodology performed formal anal-ysis validated and investigated the study and prepared theinitial draft All authors read and approved the final version

References

[1] M M A Khater M S Mohamed and R A M Attia ldquoOnsemi analytical and numerical simulations for a mathematicalbiological model the time-fractional nonlinear Kolmogor-ovndashPetrovskiindashPiskunov (KPP) equationrdquo Chaos Solitons ampFractals vol 144 Article ID 110676 2021

[2] M A Khater K S Nisar and M S Mohamed ldquoNumericalinvestigation for the fractional nonlinear space-time telegraphequation via the trigonometric Quintic B-spline schemerdquoMathematical Methods in the Applied Sciences vol 44 no 6pp 4598ndash4606 2021

[3] M M A Khater M S Mohamed and S K Elagan ldquoDiverseaccurate computational solutions of the nonlinearKleinndashFockndashGordon equationrdquo Results in Physics vol 23Article ID 104003 2021

[4] M F Li J R Ren and T Zhu ldquoFractional vector calculus andfractional special functionrdquo arXiv preprint10012889 2010

[5] D J Wollkind and B J Dichone Comprehensive AppliedMathematical Modeling in the Natural and Engineering Sci-ences Springer International Publishing Berlin Germany2017

[6] P J Olver Introduction to Partial Differential EquationsSpringer International Publishing Berlin Germany 2014

[7] J D Krehbiel and J B Freund ldquoStokes flow inside a sphere inan inviscid extensional flowrdquo Zeitschrift fur AngewandteMathematik und Physik vol 68 no 4 pp 68ndash81 2017

[8] A A Kilbas H M Srivastava and J J Trujillo ldquoeory andapplications of fractional differential equationsrdquo in North-Holland Mathematics StudiesVol 204 Elsevier AmsterdamNetherlands 2006

[9] K S Miller An Introduction to Fractional Calculus andFractional Differential Equations John Wiley and SonsHoboken NJ USA 1993

[10] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014

[11] R Khalil M Al Horani and M Abu Hammad ldquoGeometricmeaning of conformable derivative via fractional cordsrdquo eJournal of Mathematics and Computer Science vol 19 no 4pp 241ndash245 2019

[12] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journalof Computational and Applied Mathematics vol 279pp 57ndash66 2015

[13] O S Iyiola and E R Nwaeze ldquoSome new results on the newconformable fractional calculus with application usingDprimeAlambert approachrdquo Progress in Fractional Differentiationand Applications vol 2 no 2 pp 115ndash122 2016

[14] A Atangana D Baleanu and A Alsaedi ldquoNew properties ofconformable derivativerdquo Open Mathematics vol 13 pp 57ndash63 2015

[15] N Yazici and U Gozutok ldquoMultivariable conformablefractional calculusrdquo Filomat vol 32 no 2 pp 45ndash53 2018

[16] F Martınez I Martınez and S Paredes ldquoConformableEulerprimes theorem on homogeneous functionsrdquo ComputationalMathematical Methods vol 1 no 5 pp 1ndash11 2018

[17] A A Hyder and A H Soliman ldquoExact solutions of space-timelocal fractal nonlinear evolution equations a generalizedconformable derivative approachrdquo Results in Physics vol 17Article ID 103135 2020

[18] A A Hyder ldquoWhite noise theory and general improvedKudryashov method for stochastic nonlinear evolutionequations with conformable derivativesrdquo Advances in Dif-ference Equations vol 202019 pages 2020

[19] A H Soliman and A A Hyder ldquoClosed-form solutions ofstochastic KdV equation with generalized conformable deriv-ativesrdquo Physica Scripta vol 95 no 6 Article ID 065219 2020

[20] V E Tarasov ldquoNo violation of the Leibniz rule No fractionalderivativerdquo Communications in Nonlinear Science and Nu-merical Simulation vol 18 no 11 pp 2945ndash2948 2013

[21] R Khalil M Al Horani A Yousef and M SababhehldquoFractional analytic functionsrdquo Far East Journal of Mathe-matical Sciences vol 103 no 1 pp 113ndash123 2018

[22] S Uccedilar and N Y Ozgur ldquoComplex conformable derivativerdquoArabian Journal of Geosciences vol 12 no 201 2019

[23] F Martınez I Martınez I Martınez M K A Kaabar andS Paredes ldquoNew results on complex conformable integralrdquoAIMS Mathematics vol 5 no 6 pp 7695ndash7710 2020

[24] H Kiskinov M Petkova A Zahariev and M VeselinovaldquoSome results about conformable derivatives in Banach spacesand an application to the partial differential equationsrdquo inAIPConference Proceedingsvol 2333 no 1 AIP Publishing LLCArticle ID 120002 2021

[25] U Gozutok H A Ccediloban and Y Sagıroglu ldquoFrenet framewith respect to conformable derivativerdquo Filomat vol 33no 6 pp 1541ndash1550 2019

[26] J C Mayo-Maldonado G Fernandez-Anaya and O F Ruiz-Martinez ldquoStability of conformable linear differential systemsa behavioural framework with applications in fractional-ordercontrolrdquo IET Control eory amp Applications vol 14 no 18pp 2900ndash2913 2020

[27] M Al Horani and R Khalil ldquoTotal fractional differential withapplications to exact fractional differential equationsrdquo In-ternational Journal of Computer Mathematics vol 95 no 6-7pp 1444ndash1452 2018

[28] J E Marsden and M J Hoffman Vector Calculus WH Freeman and Company New York NY USA 4th edition1996

12 Journal of Mathematics