TOTAL and DIRECTIONAL FRACTIONAL DERIVATIVES A. Tallafha1 §, S. Al Hihi2 1Department of Mathematics the University of Jordan Am

International Journal of Pure and Applied Mathematics Volume 107 No. 4 2016, 1037-1051 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: 10.12732/ijpam.v107i4.21 ijpam.eu

TOTAL AND DIRECTIONAL FRACTIONAL DERIVATIVES

A. Tallafha1 §, S. Al Hihi2 1Department of Mathematics The University of Jordan Amman, JORDAN Department of Mathematics Al-Balqa Applied University Al-Salt, JORDAN

Abstract: Vector calculus is an important subject in mathematics with applications in all areas of applied sciences. Till now researchers deal with the partial fractional derivative as the fractional derivative with respect to x, y,... . In this paper we shall define total and directional fractional derivative of functions of several variables, we set some basics about fractional vector calculus then we use our definition to modify the definition of conformal fractional derivative obtained by R. Khalil et al [6].

AMS Subject Classification: 26A33 Key Words: fractional derivative, conformal fractional derivative, total fractional deriva- tive, directional fractional derivative

1. Introduction

The subject of fractional derivative is as old as calculus. The most popular definitions of fractional derivative are:

(i) Riemann-Liouville definition [6]: If n is a positive integer and α ∈ Received: October 18, 2015 c 2016 Academic Publications, Ltd. Published: May 9, 2016 url: www.acadpubl.eu §Correspondence author 1038 A. Tallafha, S. Al Hihi

[n 1, n), the αth derivative of f given is by − t n α 1 d f (x) Da (f)(t) = dx. Γ(n α) dtn Z (t x)α−n+1 − a −

(ii) Caputo definition [5]. If n is a positive integer and α [n 1, n) , the ∈ − αth derivative of f is

t (n) α 1 f (x) Da (f)(t) = dx. Γ(n α) Z (t x)α−n+1 − a −

All definitions of fractional derivatives do not satisfy the known product rule, quotient rule, chain rule. In [6] Khalil etall gave a new definition of fractional derivative ”conformable fractional derivative” of a function f . This definition seems to be a natural extension of the usual definition of derivative. Many theorems which are proved using the classical definitions are still valid using the new definition as product rule, quotient rule, chain rule. Many authors used the new definition to solve a fractional differential equations as in [3] and [4] , since the computation using the new definition is more easier than using Riemann-Liouville or Caputo definition of fractional derivative. Thabet Abdeljawd in [1] define the left and right conformable fractional derivative, so the connection can be mad between the conformable fractional derivative and the classical definition.

In this paper we shall define the concepts of directional fractional derivative and total fractional derivative of functions of several variables, these definitions announce the born of fractional vector calculus. Also we set some basics about fractional vector calculus, beside we shall give a simple modification of the definition given in [6] as an application of the new definition.

2. Basics

The concept of conformable fractional derivative is recently introduced by R. Khalil etall in 2014 by imitating the usual definition of derivative. TOTAL AND DIRECTIONAL FRACTIONAL DERIVATIVES 1039

Definition 1. [6]. Let f : [0, ) R, t > 0, and α (0, 1]. Then the ∞ −→ (α) ∈ fractional derivative of f of order α, Tα or f is defined as:

1−α (α) f t+ t f (t) f (t) = Tα (f)(t) = lim ∈ − . ∈→0
∈ If f is α differentiable in some (0, a) , a > 0, and lim f (α) (t) exists, then − t→0+ f (α) (0) is defined by f (α) (0) = lim f (α) (t) . t→0+ p p−α It is clear that Tα (t ) = pt . Further this definition coincides with the classical definition of Riemann-Liouville definition and Caputo definition on polynomials ( up to constant multiple), also if α = 1 we have the classical definition of derivative. Definition 2. [6]. Let α (n, n + 1) , and f be an n differentiable ∈ − function at t > 0.Then Tα (f)(t) is defined by

f (⌈α⌉−1) t+ t(⌈α⌉−α) f (⌈α⌉−1) (t) Tα (f)(t) = lim ∈ − , ∈→0
∈ where α is the smallest integer greater than or equal to α. ⌈ ⌉ Remark 1. The form of Definition 2 given in this form in [5] . Now since α (n, n + 1) , then α = n + 1. So ∈ ⌈(n)⌉ (⌈α⌉−α) (n) f (t+∈t )−f (t) (n) Tα (f)(t) = lim = Tn+1−α f . ∈→0 ∈
(⌈α⌉−α)f ⌈α⌉(t) As a consequence of Definition 2 one can show that Tα (f)(t) = t . Also Tα satisfies the following theorem. Theorem 1. [6]. Let α (0, 1] and f.g be α- differentiable at t. Then: ∈ 1. Tα (af + bg) = aTα (f) + bTα (g) for all a, b R. ∈ p p−1 2. Tα (t ) = pt for any p R. ∈ 3. Tα (fg) = fTα (g) + gTα (f) .

f gTα(f)−fTα(g) 4. Tα g = g2 .   5. Tα (λ) = 0, for all constant function λ. 1−α df 6. If, in addition f is differentiable, then Tα (f)(t) = t dt . In the following example the fractional derivatives of well known functions are given. 1040 A. Tallafha, S. Al Hihi

Example 1. Let α (0, 1] and c, b R. Then: ∈ ∈ ct 1−α ct 1. Tα e = ct e .
1−α 2. Tα (sin bt) = bt cos bt. 1−α 3. Tα (cos bxt) = bt sin bt. − 1 α 4. Tα α t = 1. Using the
new definition of conformable fractional derivative, most of the important Theorems in calculus as Roll’s theorem, Mean Value theorem and differentiability implies continuity still valid.

3. Directional Fractional Derivative

So far researchers deal with the partial fractional derivative as the fractional derivative with respect to x or y, etc. In this section we shall generalize the definition given in [6] ,to cover the concepts of directional fractional derivative. n n Let f be any function on R , with domain Df and u R be any vector. ∈ Put u⊥ = v Rn : v is orthogonal to u and u be the Euclidean norm of u. { ∈ } k k Finally, let c u be the dot product of c and u. By A B we mean the points in · \ the set A not in the set B. n m n Definition 3. Let f : Df R R be any function, α (0, 1] , u R ⊆ → ∈ ∈ ◦ be a non-zero vector and c in the interior of Df c Df .  ∈  1. If c D◦ u⊥ and there exists a vector Lα Rm that satisfies for each > ∈ f \ u ∈ ∈ 0,there is a δ ( ) > 0 such that for all t R satisfying 0 < t < δ ( ) ,we have 1 ∈ 1−α α ∈ α | | ∈ t f c + t c u u f (c) Lu < , then Lu is called the fractional   | · |  −  − ∈ directional derivative of f of order α in the direction of u at c and denoted by

Dαf (c). If there is no such a vector then we say that f is not α differentiable u − at c in the direction of u. ◦ ⊥ α α 2. If c Df u , then Du f (c) is defined by limDu f (d) , d Df if the ∈ ∩ d→c ∈ limit exist. If the limit does not exist we say that f not α differentiable at c − in the direction of u. ⊥ Using the above definition it is clear that if c Df u and ∈ \ f c + t c u 1−α u f (c) lim  | · |  − t→0 t TOTAL AND DIRECTIONAL FRACTIONAL DERIVATIVES 1041 exists then

f c + t c u 1−α u f (c) α α | · | − Lu = Du f (c) = lim   . t→0 t ⊥ If c Df u , then ∈ ∩ α α Du f (c) = limDu f (d) , d Df . d→c ∈ In the above definition if c D◦ u⊥ then we can’t use (1) to define Dαf (c) , ∈ f ∩ u so we use (2) . The case c u⊥ corresponds to the case t = 0 in Definition 1. ∈ Also if f is continuous and c u⊥, Then Dαf (c) = 0. ∈ u If one is interested in the fractional directional derivative at a boundary ◦ point of Df , that is c Df D , we suggest the following definition. ∈ \ f n m Definition 4. Let f : Df R R be a continuous function, α n ⊆ → ∈ (0, 1) , and u R be a non-zero vector. Let c be a boundary point of Df ◦ ∈ α α and cn D be such that cn c. Then Du f (c) = lim Du f (cn), if the limit ∈ f → n→∞ exists. And f not α differentiable at c in the direction of u if the limit does − not exist. Note that, since f is continuous the limit is independent of the choice of cn. As consequence of the above definitions we have. Lemma 1. 1. If α = 1, then the above definition coincides with classical definition.

2. If the directional derivative of f in the direction of u at a point c , Duf (c) α 1−α exists, then D f (c) = c u Duf (c) . u | · | Now one can easily prove the following theorem. Theorem 2. Let α (0, 1) and f, g be two α differentiable at c n ∈ − ∈ Df Dg R in the direction of a non-zero vectoru. Then: ∩ ⊆ 1. Dα (f g) = Dα (f) Dα (g) . u ± u ± u α α 2. Du (γf) = γDu (f) , for all constant γ. 3. Dα (k) = 0, for all constant functions k : Rn Rm. u → Example 2. Let f (x, y) = xy and u = (2, 1). Then

1 f ((1, 2) + 2 (2, 1)) f ((1, 2)) D 2 f ((1, 2)) = lim ∈ − (2,1) ∈→0 ∈ 1042 A. Tallafha, S. Al Hihi

f ((1 + 4 , 2 + 2 )) f ((1, 2)) = lim ∈ ∈ − ∈→0 ∈ (1 + 4 ) (2 + 2 ) 2 10 +8 2 = lim ∈ ∈ − = lim ∈ ∈ = 10. ∈→0 ∈→0 ∈ ∈ α 1− 1 Clearly Duf (c) = (2, 1) (2, 1) = 5, so D f (c) = 10 = c u 2 Duf (c) . · u | · | 1 2 Example 3. Let f (x, y) = √x + y. To find D(1,1)f ((0, 1)) we shall find 1 2 1 2 D f (cn) , where cn (x, y): x > 0 , and cn = c , c (0, 1) . (1,1) ∈ { } n n → Now
1 1 2 1 2 2 1 2 1 f cn, cn + cn + cn (1, 1) f cn, cn 2 ∈ − D f (cn) = lim 



(1,1) ∈→0 ∈ 1 1 c1 + (c1 + c2 ) 2 + c1 + c2 2 c1 q n ∈ | n n | ∈ n n − n = lim
p ∈→0 1 ∈ c1 + c2 2 1 = n n + c1 + c2 2 . 1
n n 2 cn
p 1 Taking the limit as n , we have D 2 f ((0, 1)) does not exist. → ∞ (1,1) ⊥ α From Lemma 1. It is clear that if Duf (c) exist and c u , then D f (c) = 0. ∈ u In the following example we shall find Dαf (c) , c u⊥. u ∈ 1 2 Example 4. Let f (x, y) = √x + y.To find D(1,0)f ((0, 1)) we shall find 1 D 2 f ((a, b)) , (a, b) (x, y): x > 0 , (1,0) ∈ { } 1 f ((a, b) + √a (1, 0)) f ((a, b)) D 2 f ((a, b)) = lim ∈ − (1,0) ∈→0 ∈ f ((a+ √a, b)) f ((a, b)) a+ √a √a = lim ∈ − = lim ∈ − ∈→0 ∈→0 p ∈ ∈ √a 1 = = . 2√a 2

4. Total Fractional Derivative

In this section we shall define total fractional derivative of functions on Rn. Using the new definition the most important theorems of functions of several variables are still valid. TOTAL AND DIRECTIONAL FRACTIONAL DERIVATIVES 1043

n m Definition 5. .[2]. Let f : Df R R be any function, and c an ⊆ → interior point ofDf . We say that f is differentiable at c if there exist a linear n m mapTc : R R such that f (c + v) = f (c) + Tc (v) + v Ec (v) , where → k k limEc (v) = 0 . v→0 To define total fractional derivative of f, we replace v by c v 1−α v. | · | n m Definition 6. f : Df R R be any function, α (0, 1] and c an ⊆ → ∈ interior point of Df . We say that the α fractional derivative of f at c exist, if there exist a linear map T α : Rn Rm such that c → 1−α α 1−α 1−α f c + c v v = f (c) + Tc c v v + c v v Ec (v) , ( )  | · |  | · |  | · | ∗

where limEc (v) = 0. v→0 Obviously ( ) can be expressed more compactly by writing ∗ 1−α α 1−α f c + c v v f (c) Tc c v v lim  | · |  − − | · |  = 0. 1−α kvk→0 c v v | · |

Alternatively ( ) can be rephrased as. For any > 0, there exist δ ( ) > 0 ∗ ∈ ∈ such that if v Rn and v δ ( ) , then ∈ k k ≤ ∈ 1−α α 1−α 1−α f c + c v v f (c) Tc c v v c v v . ( )  | · |  − − | · |  ≤∈ | · | ∗∗

α Such a linear map Tc is called the fractional total derivative of f of order α at c and we shall denote it by Dαf (c) and Dαf (c)(v) for the value of the α α α linear map Tc at v. That is D f (c)(v) = Tc (v) . From the above definition easily one can show. Lemma 2. 1. If α = 1, then Dαf (c) is Df (c) . α 2. If v = 0, then Tc (v) = 0. 3. If v c⊥, then ( ) is true so we assume v / c⊥. ∈ ∗ ∈ 4. If f is constant, then for z Rn 0 , we have T α (v) v , where ∈ \{ } k c k ≤∈ k k v = δ(∈) z, so T α (z) z for all > 0. Hence T α (z) = 0. kzk k c k ≤∈ k k ∈ c Now one can easily prove the following lemma. n m Lemma 3. Let f : Df R R be any function, α (0, 1) and c an ⊆ → ∈ interior point of Df . 1044 A. Tallafha, S. Al Hihi

1. If f is linear then the fractional total derivative of f of order α at c is α the function f. That is Tc (v) = f (v) . 2. If the total derivative of f at c, Df (c) exists then

Dαf (c)(v) = Df (c) c v 1−α v = c v 1−α Df (c)(v) . | · |  | · | Moreover we have the following. n m α Lemma 4. Let f : Df R R , if D f (c) exists then it is unique. ⊆ → α α Proof. Let f has T1c,T2c as fractional total derivative at c, then for any > 0, there exist δ ( ) > 0 such that if v Rn and v δ ( ) , then ∈ ∈ ∈ k k ≤ ∈ 1−α 1−α 1−α f c + c v v f (c) Tic c v v c v v ,  | · |  − − | · |  ≤∈ | · | for v δ ( ) , i = 1, 2. Hence k k ≤ ∈ α 1−α α 1−α 0 T1c c v v T2c c v v ≤ | · |  − | · |  α 1−α 1−α = T1c c v v f c + c v v f (c) | · |  −   | · |  −  1−α α 1−α + f c + c v v f (c) T2c c v v   | · |  −  − | · | 

2 c v 1−α v . ≤ ∈ | · |

Therefore 0 T α (v) T α (v) 2 v for all v Rn. ≤ k 1c − 2c k ≤ ∈ k k ∈ To compleat the proof, we have two cases: (i) If z Rn 0 , let z = δ(∈) z Rn, which implies z δ ( ) and ∈ \{ } 0 kzk ∈ k 0k ≤ ∈ hence T α (z ) T α (z ) 2 z . k 1c 0 − 2c 0 k ≤ ∈ k 0k So T α (z) T α (z) 2 z , k 1c − 2c k ≤ ∈ k k for all > 0, and consequently T α (z) = T α (z) . ∈ 1c 2c α α (ii) If z = 0, then by linearity of T1c,T2c the result follows.

It is known that a differentiable function is continuos. The following lemma show that this is still true if Dαf (c) exist for some α (0, 1) , c D◦. ∈ ∈ f TOTAL AND DIRECTIONAL FRACTIONAL DERIVATIVES 1045

n m α Lemma 5. Let f : Df R R be a given function. If D f (c) exist ⊆ → for some α (0, 1) , c D◦, then there exist strictly positive numbers δ, K such ∈ ∈ f that if v δ, then k k ≤ f c + c v 1−α v f (c) K c v 1−α v .  | · |  − ≤ | · |

In particular f is continuous at c.

Proof. Since every linear map on a finite dimensional normed space is bounded, there exists a positive constant M such that T α (u) M u . k c k ≤ k k Now by Definition 3, it follows that for = 1, there exist 0 < δ (1) such that ∈ for v Rn with v < δ (1) . Then ∈ k k

f c + c v 1−α v f (c)  | · |  − 1−α α 1−α α 1−α f c + c v v f (c) Tc c v v + Tc c v v ≤  | · |  − − | · |  | · |  1−α α 1− α c v v + Tc c v v ≤ | · | | · | 

c v 1−α v + M c v 1−α v . ≤  | · | | · | 

Hence

f c + c v 1−α v f (c) (M + 1) c v 1−α v .  | · |  − ≤ | · |

Now let u be a non-zero victor v = tu, if t δ(1) , then, | | ≤ kuk

f c + c tu 1−α tu f (c) (M + 1) c tu 1−α tu .  | · |  − ≤ | · |

Hence

lim f (c + tw) f (c) = 0. t→0 k − k We know that if f is differentiable at c, then the directional derivative of f in the direction of u at c, Duf (c) exist and Duf (c) = Df (c)(u) . So we shall end this section with the following Theorem. n m α Theorem 3. Let f : Df R R , if D f (c) exist for some α (0, 1) , ◦ Rn ⊆ → ∈ c Df , and u be a non zero vector. Then the fractional directional ∈ ∈ α α derivative of f of order α in the direction of u at c, Du f (c) exist and Du f (c) = c u 1−α Dαf (c)(u) . | · | 1046 A. Tallafha, S. Al Hihi

Proof. For c u⊥, the result follows. Suppose c / u⊥. Since the fractional ∈ ∈ derivative of order α of f at c exist, then given > 0, there exist 0 < δ ( ) , ∈ ∈ such that

f c + c v α−1 v f (c) c v α−1 Dαf (c)(v) c v α−1 v  | · |  − − | · | ≤∈ | · | k k provided v δ ( ) . k k ≤ ∈ Let u be a non zero vector, thus if 0 < t δ(∈) , we have | | ≤ kuk

f c + t c tu α−1 u f (c) c tu α−1 Dαf (c)(tu) c tu α−1 tu ,  | · |  − − | · | ≤∈ | · | k k thus f c + t t α−1 c u α−1 u f (c)  | | | · |  − Dαf (c)(u) u . α−1 α−1 − ≤∈ k k t t c u | | | · |

Therefore f (c + ru) f (c) − Dαf (c)(u) u . r − ≤∈ k k

α α This shows that Duf (c) exist and Duf (c) = D f (c)(u). Hence Du f (c) = 1−α 1−α α c u Duf (c) = c u D f (c)(u) . | · | | · | n m ◦ Corollary 1. Let f : Df R R , c D .. ⊆ → ∈ f If Dαf (c) exist for some α (0, 1) , then Dαf (c) the conformable fractional ∈ i partial derivative of f at the variable xi.exist. n If u = (u , u , ..., un) R , we have 1 2 ∈ n α α 1−α α D f (c) = D f (c)(u) = c u ujD f (c) . u | · | j Xj=1

α Proof. Let u = ei, by Theorem 3 Di f (c) exist for all i = 1, ...n, and if n α u = (u , u , ..., un) R then linearity of D f (c) implies 1 2 ∈ n α 1−α α D f (c) = c u D f (c)  ujej u | · | Xj=1 n   n 1−α α 1−α α = c u ujD f (c)(ej) = c u ujD f (c) . | · | | · | j Xj=1 Xj=1 TOTAL AND DIRECTIONAL FRACTIONAL DERIVATIVES 1047

n m α Definition 7. Let f : Df R R , if D f (c) exist for some α ⊆ → ∈ (0, 1) , c D◦. Then αf the fractional gradient of f at c is defined by ∈ f ▽c αf = (Dαf (c) ,Dαf (c) , ...Dαf (c)) . ▽c 1 2 n n m From the theory of functions of several variables for f : Df R R ⊆ → where both n > 1, m > 1. In this case we can represent y = f (x) by a system of m functions of n variables

y1 =f1 (x1, x2, ..., xn) ,

y2 =f2 (x1, x2, ..., xn) , . .

ym =fm (x1, x2, ..., xn) .

If f is differentiable at a point c,then Df (c) is the linear mapping of Rn into Rm determined by the n m matrix ×

D1f1 (c) D2f1 (c) ... Dnf1 (c)  D1f2 (c) D2f2 (c) ... Dnf2 (c)  . . . . .  . . .. .     D f (c) D f (c) ... D f (c)   1 m 2 m n m  In the following theorem we shall show that a similar result is obtained if we use total fractional derivative. n m Theorem 4. Let f : Df R R where n > 1, m > 1. ⊆ → If Dαf (c) exist for some α (0, 1) , c D◦.. Then Dαf (c) is the linear ∈ ∈ f mapping of Rn into Rm determined by the n m matrix × α α α D1 f1 (c) D2 f1 (c) ... Dn f1 (c) α α α  D1 f2 (c) D2 f2 (c) ... Dn f2 (c)  . . . . .  . . .. .     Dαf (c) Dαf (c) ... Dαf (c)   1 m 2 m n m  Proof. Let us present y = f (x) by the system,

y1 =f1 (x1, x2, ..., xn) ,

y2 =f2 (x1, x2, ..., xn) , . .

ym =fm (x1, x2, ..., xn) . 1048 A. Tallafha, S. Al Hihi

α ◦ α If D f (c) exist for some α (0, 1) , c D , then D fi (c) , exist for i = ∈ ∈ f 1, ...m. α n Now D f (c) maps the point (u1, u2, ...un) of R into the point w = (w1, m w2, ...wm) of R given by

α α α w1 =D1 f1 (c) u1 + D2 f1 (c) u2 + + Dn f1 (c) un, α α ··· α w =D f (c) u + D f (c) u + + D f (c) un, 2 1 2 1 2 2 2 ··· n 2 . . α α α wm =D fm (c) u + D fm (c) u + + D fm (c) un. 1 1 2 2 ··· n So the fractional derivative Dαf (c) determined by the n m matrix whose × elements are α α α D1 f1 (c) D2 f1 (c) ... Dn f1 (c) α α α  D1 f2 (c) D2 f2 (c) ... Dn f2 (c)  . . . . .  . . .. .     Dαf (c) Dαf (c) ... Dαf (c)   1 m 2 m n m  This matrix is as usual called the fractional Jacobi matrix and denoted by α Jf (c).

5. Application

In his paper [5] U. Katugampola, wrote ” one of the limitation of this version of the fractional derivative is that it assumes that the variable t > 0. So the question is, wether we can relax this condition on a special class of functions?, if so, what it is?.” Using our definition of total fractional derivatives we shall give a simple modification of the definition of conformable fractional derivative of f, in this modification we don’t assume t > 0. Also we shall show that all the results ob- tained in [6] , are still valid using the modified conformable fractional derivative of f. It is known that if f is a function of several variable, the directional deriva- tive of f in the direction of a vector u, at a point c is defined by Duf (c) = f(c+hu)−f(c) ′ lim , so if we let u = 1 and c = x0, then we obtain f (x0) . Using h→0 h this idea we can modify the definition given in [6] . Let f : Df R R, α (0, 1) and c a non zero interior point of Df . ⊆ −→ ∈ Let u = 1 in 3. Then we have, TOTAL AND DIRECTIONAL FRACTIONAL DERIVATIVES 1049

Definition 8. Let f : R R be any function,and t R 0 , α −→ ∈ \{ } ∈ (0, 1) .Then define the modified conformable fractional derivative of f of order 1−α α (α) f(c+∈|c| )−f(c) α, Tα or f by. f (c) = Tα (f)(c) = lim . If f is α differ- ∈→0 ∈ − entiable in some ( a, a) 0 , a > 0, and lim f (α) (c) exist, then f (α) (0) is − \{ } c→0+ defined by f (α) (0) = lim f (α) (c) . c→0+ Clearly, if α = 1 the modified definition coincide with the classical definition of derivative. Example 5. Let n = 3, then for u = i = (1, 0, 0), we have:

1−α 1−α α f(c+t|c·u| u)−f(c) f(c1+t|c1| ,c2,...,cn)−f(c) (α) 1. Di f (c) = lim = lim = fx (c) , t→0 t t→0 t the modified partial conformable fractional derivatives f with respect to x. α α 2. Similarly Dj f (c) ,Dk f (c) the modified partial conformable fractional derivative of f with respect to y, z respectively. Rn α Remark 2. Let e1, e2, ..., en be the standard basis of . Then Dei f (c) = α { } Di f (c) is the modified conformable fractional partial derivative of f with re- spect to the variable xi. Definition 9. Let α (1, ) , f : R R be ( α 1)differentiable ∈ ∞ −→ ⌈ ⌉ − at t R 0 ,where α is the smallest integer greater than or equal to α. ∈ \{ } ⌈ ⌉ α Then the modified conformable fractional derivative of f of order α, Tα or f is defined by

f (⌈α⌉−1) t+ t (⌈α⌉−α) f (⌈α⌉−1) (t) (α) ∈ | | − f (t) = Tα (f)(t) = lim   . ∈→0 ∈ Remark 3. As a consequence of Definition 9 it is easy to show that for α (1, ). ∈ ∞ (⌈α⌉−1) 1. Tα (f)(t) = T1+α−⌈α⌉ f (t) .
(⌈α⌉−α) ⌈α⌉ 2. If f is α -differentiable at t R 0 , then Tα (f)(t) = t f (t) . ⌈ ⌉ ∈ \{ } | | Theorem 5. If f : R R, if the modified conformable the fractional −→ derivative of f of order α exists at t R 0 , α (0, 1] , then f is continuous 0 ∈ \{ } ∈ at t0

h Proof. We want to show that limf (t0 + h) = f (t0) . Let = 1−α , then h→0 ∈ |t0| 1050 A. Tallafha, S. Al Hihi

1−α limf (t0 + h) f (t0) = lim f t0+ t0 f (t0) h→0 − ∈→0  ∈ | |  − 1−α f t0+ t0 f (t0) ∈ | | − (α) = lim   = f (t0) .0 = 0. ∈→0 ∈ ∈ It is clear that if α (1, ) and Tα (f)(t) exists then f is continuous. The ∈ ∞ following theorem is an easy consequence of the modified definition. Theorem 6. Let α (0, 1] and f.g be α- differentiable at t R 0 . ∈ ∈ \{ } Then:

1.- Tα (af + bg) = aTα (f) + bTα (g) for all a, b R. ∈ p p−α 1 t > 0 2. Tα ( t ) = p t signt. for all p R, where signt = . | | | | ∈  1 t < 0  − 3. Tα (fg) = fTα (g) + gTα (f) .

f gTα(f)−fTα(g) 4. Tα g = g2 .   5. Tα (λ) = 0, for all constant function λ. 1−α df 6. If, in addition f is differentiable, then Tα (f)(t) = t . | | dt In the following example the fractional derivative of a well known functions are given using the modified definition. Example 6. Let α (0, 1] and c, b R. Then: ∈ ∈ ct 1−α ct 1. Tα e = c t e . | |
1−α 2. Tα (sin bx) = b t cos bt. | | 1−α 3. Tα (cos bx) = b t sin bt. − | | 1 α 4. Let t = 0, then Tα t = signt. 6 α | | Using the modified definition,
the proof of Theorem 2.3 in [6] (Rolle’s The- orem for conformable fractional derivative) is the same, but Theorem 2.4 in [6] (Mean Value Theorem for Conformable Fractional Differentiable Functions) needs a simple modification. Theorem 7. (Rolle’s Theorem for Conformable Fractional Differentiable Functions). Let 0 / [a, b] , and f :[a, b] R be a given function that satisfies: ∈ → (i) f is continuous on [a, b] . (ii) f is α differentiable for some α (0, 1] . − ∈ (iii) f (a) = f (b) . TOTAL AND DIRECTIONAL FRACTIONAL DERIVATIVES 1051

Then, there exists c [a, b] , such that f (α) (c) = 0. ∈ Theorem 8. (Mean Value Theorem for Conformable Fractional Differ- entiable Functions). Let 0 / [a, b] ,and f :[a, b] R be a given function that ∈ → satisfies, (i) f is continuous on [a, b] . (ii) f is α differentiable for some α (0, 1] . − ∈ (α) f(b)−f(a) Then, there exists c [a, b] , such that f (c) = sgnc 1 α 1 α . ∈ α |b| − α |a| Proof. Consider the function f (b) f (a) 1 1 g (x) = f (x) f (a) − x α a α . − − 1 b α 1 a α α | | − α | |  α | | − α | | Then the function g satisfies the condition of 7. Hence there exists c [a, b] , (α) (α) f(b)−f(a) ∈ such that 0 = g (c) = f (c) 1 α 1 α signc. − α |b| − α |a|

References [1] T. Abdeljawad, On conformable fractional calculus, Journal of Computational and Ap- plied Mathematics, 279 (2015), 57-66. [2] R.G. Bartle, The Elements of Real Analysis, John Wiley and Sons, New York (1976). [3] M. Abu Hammad, R. Khalil, Conformable fractional heat differential equation, Interna- tional Journal of Pure and Applied Mathematics, 94, No. 2 (2014), 215-221. [4] M. Abu Hammad, R. Khalil, Legender fractional differential equation and Legender frac- tional polynomials, International Journal of Applied Mathematical Research, 3, No. 3 (2014), 214-219. [5] U. Katugampola, A new fractional derivative with classical properties, arXiv:1410.6535v1, 2014. [6] R. Khalil, M. Al Horani, A Yousef, M. Sababheh, A new definition of fractional derivative, Journal of Computational Applied Mathematics, 264 (2014), 65-70. [7] K.S. Miller, An Introduction to Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York (1999). 1052