On the local M-derivative
aa r X i v : . [ m a t h . C A ] A ug ON THE LOCAL M -DERIVATIVE J. VANTERLER DA C. SOUSA AND E. CAPELAS DE OLIVEIRA Abstract.
We introduce a new local derivative that generalizes the so-called alternative” f ractional ” derivative recently proposed. We denote this new differential operator by D α,βM , where the parameter α , associated with the order, is such that 0 < α < β > M is used to denote that the function to be derived involves a Mittag-Leffler function withone parameter.This new derivative satisfies some properties of integer-order calculus, e.g. linearity,product rule, quotient rule, function composition and the chain rule. Besides as in thecase of the Caputo derivative, the derivative of a constant is zero. Because Mittag-Lefflerfunction is a natural generalization of the exponential function, we can extend some of theclassical results, namely: Rolle’s theorem, the mean value theorem and its extension.We present the corresponding M -integral from which, as a natural consequence, newresults emerge which can be interpreted as applications. Specifically, we generalize theinversion property of the fundamental theorem of calculus and prove a theorem associatedwith the classical integration by parts. Finally, we present an application involving lineardifferential equations by means of local M -derivative with some graphs. Keywords : Local M -Derivative, Local M -Differential Equation, M -Integral, Mittag-LefflerFunction.MSC 2010 subject classifications. 26A06; 26A24; 26A33; 26A42. Introduction
The integral and differential calculus of integer-order developed by Leibniz and Newton wasa great discovery in mathematics, having numerous applications in several areas of physics,biology, engineering and others. But something intriguing and interesting to the mathemati-cians of the day was still to come. In 1695 [1, 2, 3], ℓ ’Hospital, in a letter to Leibniz, askshim about the possibility of extending the meaning of an integer-order derivative d n y/dx n to the case in which the order is a fraction. This question initiated the history of a newcalculus which was called non-integer order calculus and which nowadays is usually calledfractional calculus.Although fractional calculus emerged at the same time as the integer-order calculus proposedby Newton and Leibniz, it did not attract the attention of the scientific community andfor many years remained hidden. It was only after an international congress in 1974 thatfractional calculus began to be known and consolidated in numerous applications in severalfields such as mathematics, physics, biology and engineering. AND E. CAPELAS DE OLIVEIRA Several types of fractional derivatives have been introduced to date, among which theRiemann-Liouville, Caputo, Hadamard, Caputo-Hadamard, Riesz and other types [4]. Mostof these derivatives are defined on the basis of the corresponding fractional integral in theRiemann-Liouville sense.Recently, Khalil et al. [5] proposed the so-called conformable fractional derivative of order α , 0 < α <
1, in order to generalize classical properties of integer-order calculus. Someapplications of the conformable fractional derivative and the alternative fractional derivativeare gaining space in the field of the fractional calculus and numerous works using suchderivatives are being published of which we mention: the heat equation, the Taylor formulaand some inequalities of convex functions [6, 7]. More recently, in 2014, Katugampola [8]also proposed a new fractional derivative with classical properties, similar to the conformablefractional derivative.Given such a variety of definitions we are naturally led to ask which are the criteria thatmust be satisfied by an operator, differential or integral, in order to be called a fractionaloperator. In 2014, Ortigueira and Machado [9, 10] discussed the concepts underlying thosedefinitions and pointed out some properties that, according to them, should be satisfied bysuch operators (derivatives and integrals) in order to be called fractional. However, thereexist operators which one would like to call fractional [5, 8] even though they do not satisfythe criteria proposed by Ortigueira and Machado, and this led Katugampola [11] to criticizethose criteria.The main motivation for this work comes from the alternative fractional derivative recentlyintroduced [8] and some new results involving the conformable fractional derivative [5, 12, 13],all of which constitute particular cases of our results. In this sense, as an application of local M -derivative, we present the general solution of a linear differential equation with graphs.This paper is organized as follows: in section 2 we present the concepts of fractional deriva-tives in the Riemann-Liouville and Caputo sense, the definition of fractional derivative byKhalil et al. [5] and the alternative definition proposed by Katugampola [8], together withtheir properties. In section 3, our main result, we introduce the concept of an local M -derivative involving a Mittag-Leffler function and demonstrate several theorems. In section4 we introduce the corresponding M -integral, for which we also present several results; inparticular, a generalization of the fundamental theorem of calculus. In section 5, we presentthe relation between the local M -derivatives, introduced here, and the alternative proposedin [8]. In section 6, we present an application involving linear differential equations by meansof local M -derivative with some graphs. Concluding remarks close the paper.2. Preliminaries
The most explored and studied fractional derivatives of fractional calculus are the so-calledRiemann-Liouville and Caputo derivatives. Both types are fundamental in the study offractional differential equations; their definitions are presented below.
N THE LOCAL M -DERIVATIVE 3 Definition 1.
Let α ∈ C such that Re ( α ) > and m − < α ≤ m . The fractional derivativeof order α of a causal function f in the Riemann-Liouville sense, D αRL f ( t ) , is defined by [14, 15, 16](2.1) D αRL f ( t ) := D m J m − α f ( t ) , or (2.2) D αRL f ( t ) :=
1Γ ( m − α ) d m dt m (cid:20)Z t f ( τ ) ( t − τ ) m − α − dτ (cid:21) , m − < α < m,d m dt m f ( t ) , α = m where D m = d m /dt m is the usual derivative of integer-order m and J m − α is the fractionalintegral in the Riemann-Liouville sense. If α = 0 , we define D RL = I , where I is the identityoperator. Definition 2.
Let α ∈ C such that Re ( α ) > and m the smallest integer greater than orequal to Re ( α ) > , with m − < α ≤ m . The fractional derivative of order α of a causalfunction f in the Caputo sense, D αC f ( t ) , is defined by [14, 15, 16](2.3) D αC f ( t ) := J m − α D m f ( t ) , m ∈ N or(2.4) D αC f ( t ) :=
1Γ ( m − α ) Z t f ( m ) ( τ ) ( t − τ ) m − α − dτ, m − < α < m,d m dt m f ( t ) , α = m where D m is the usual derivative of integer-order m , J m − α is the fractional integral in theRiemann-Liouville sense and f ( m ) ( τ ) = d m f ( τ ) dτ m .We present now the definitions of two new types of ” f ractional ” derivatives. As we shallshow later, these definitions coincide, for a particular value of their parameters, with thederivative of order one of integer-order calculus. Definition 3.
Let f : [0 , ∞ ) → R and t > . Then the conformable fractional derivative oforder α of f is defined by [5](2.5) T α f ( t ) = lim ε → f ( t + εt − α ) − f ( t ) ε , ∀ t > α ∈ (0 , f is called α -differentiable if it has a fractional derivative. If f is α -differentiablein some interval (0 , a ), a > t → + f ( α ) ( t ) exists, then we define J. VANTERLER DA C. SOUSA AND E. CAPELAS DE OLIVEIRA T α (0) = lim t → + T α f ( t ) . Definition 4.
Let f : [0 , ∞ ) → R and t > . Then the alternative fractional derivative oforder α of f is defined by [8](2.6) D α f ( t ) = lim ε → f (cid:16) te εt − α (cid:17) − f ( t ) ε , ∀ t > α ∈ (0 , f is α -differentiable in some interval (0 , a ), a > t → + f ( α ) ( t ) exists, then wedefine D α (0) = lim t → + D α f ( t ) . In this work, if both the conformable and the alternative fractional derivatives of order α ofa function f exist, we will simply said that the function f is α -differentiable.Ortigueira and Machado [9, 10] proposed that an operator can be considered a fractionalderivative if it satisfies the following properties: (a) linearity; (b) identity; (c) compat-ibility with previous versions, that is, when the order is integer, the fractional deriva-tive produces the same result as the ordinary integer-order derivative; (d) the law of ex-ponents D α D β f ( t ) = D α + β f ( t ), for all α < β <
0; (e) generalized Leibniz rule D α ( f ( t ) g ( t )) = ∞ P i =0 (cid:0) αi (cid:1) D i f ( t ) D α − i g ( t ). Fractional derivatives in the Riemann-Liouville and in the Caputo sense satisfy such prop-erties, but neither the conformable nor the alternative fractional derivatives satisfy them.On the other hand, it is possible to find undesirable characteristics even in fractional deriva-tives that satisfy the criteria presented above, e.g.:(1) Most fractional derivatives do not satisfy D α (1) = 0 if α is not a natural number.An importante exception is the derivative in the Caputo sense.(2) Not all fractional derivatives obey the product rule for two functions: D α ( f · g ) ( t ) = f ( t ) D α g ( t ) + g ( t ) D α f ( t ) . (3) Not all fractional derivatives obey the quotient rule for two functions: D α (cid:18) fg (cid:19) ( t ) = f ( t ) D α g ( t ) − g ( t ) D α f ( t )[ g ( t )] (4) Not all fractional derivatives obey the chaim rule: D α ( f ◦ g ) ( t ) = g ( α ) ( t ) f ( α ) ( g ( t )) . Note that when α = n ∈ Z + we obtain the classical Leibniz rule. N THE LOCAL M -DERIVATIVE 5 (5) Fractional derivatives do not have a corresponding Rolle’s theorem.(6) Fractional derivatives do not have a corresponding mean value theorem.(7) Fractional derivatives do not have a corresponding extended mean value theorem.(8) The definition of the Caputo derivative assumes that function f is differentiable inthe classical sense of the term.In this sense, the new conformable and alternative fractional derivatives fit perfectly into theclassical properties of integer-order calculus, in particular in the case of order one.The objective of this work is to present a new type of derivative, the local M -derivative,that generalizes the alternative fractional derivative. The new definition seems to be anatural extension of the usual, integer-order derivative, and satisfies the eight propertiesmentioned above. Also, as in the case of conformable and alternative fractional derivatives,our definition coincides with the known fractional derivatives of polynomials. Finally, wewere able to define a corresponding integral for which we can prove the fundamental theoremof calculus, the inversion theorem and a theorem of integration by parts.3. Local M -derivative In this section we present the main definition of this article and obtain several results thatgeneralize equivalent results valid for the alternative fractional derivative and which bear agreat similarity to the results found in classical calculus.On the basis of this definition we could demonstrate that our local M -derivative is linear,obeys the product rule, the composition rule for two α -differentiable functions, the quotientrule and the chain rule. We show that the derivative of a constant is zero and present” f ractional ” versions of Rolle’s theorem, the mean value theorem and the extended meanvalue theorem. Further, the continuity of the ” f ractional ” derivative is demonstrated, as ininteger-order calculus.Thus, let us begin with the following definition, which is a generalization of the usual defi-nition of a derivative as a special limit. Definition 5.
Let f : [0 , ∞ ) → R and t > . For < α < we define the local M -derivativeof order α of function f , denoted D α,βM f ( t ) , by (3.1) D α,βM f ( t ) := lim ε → f ( t E β ( εt − α )) − f ( t ) ε , ∀ t > , where E β ( · ) , β > is the Mittag-Leffler function with one parameter [17, 18]. Notethat if f is α -differentiable in some interval (0 , a ), a >
0, and lim t → + D α,βM f ( t ) exists, then wehave D α,βM f (0) = lim t → + D α,βM f ( t ) . J. VANTERLER DA C. SOUSA AND E. CAPELAS DE OLIVEIRA Theorem 1.
If a function f : [0 , ∞ ) → R is α -differentiable at t > , < α ≤ , β > ,then f is continuous at t .Proof. Indeed, let us consider the identity(3.2) f (cid:0) t E β (cid:0) εt − α (cid:1)(cid:1) − f ( t ) = f (cid:0) t E β (cid:0) εt − α (cid:1)(cid:1) − f ( t ) ε ! ε. Applying the limit ε → ε → f (cid:0) t E β (cid:0) εt − α (cid:1)(cid:1) − f ( t ) = lim ε → f (cid:0) t E β (cid:0) εt − α (cid:1)(cid:1) − f ( t ) ε ! lim ε → ε = D α,βM f ( t ) lim ε → ε = 0 . Then, f is continuous at t .Using the definition of the one-parameter Mittag-Leffler function, we have(3.3) f (cid:0) t E β (cid:0) εt − α (cid:1)(cid:1) = f t ∞ X k =0 ( εt − α ) k Γ ( βk + 1) ! . Apply the limit ε → f is a continuous function, we havelim ε → f (cid:0) t E β (cid:0) εt − α (cid:1)(cid:1) = lim ε → f t ∞ X k =0 ( εt − α ) k Γ ( βk + 1) ! = f t lim ε → ∞ X k =0 ( εt − α ) k Γ ( βk + 1) ! = f ( t ) , (3.4)because when ε → k = 0, so that lim ε → ∞ X k =0 ( εt − α ) k Γ ( βk + 1) = 1 . We present here a theorem that encompasses the main classical properties of integer orderderivatives, in particular of order one. As for the chain rule, it will be verified by means ofan example, as we will see in the sequence.
Theorem 2.
Let < α ≤ , β > , a, b ∈ R and f, g α -differentiable at the point t > .Then:(1) (Linearity) D α,βM ( af + bg ) ( t ) = a D α,βM f ( t ) + b D α,βM g ( t ) . N THE LOCAL M -DERIVATIVE 7 Proof.
Using Definition 1, we have D α,βM ( af + bg ) ( t ) = lim ε → ( af + bg ) ( t E β ( εt − α )) − ( af + bg ) ( t ) ε = lim ε → af ( t E β ( εt − α )) + bg ( t E β ( εt − α )) − af ( t ) − bg ( t ) ε = lim ε → af ( t E β ( εt − α )) − af ( t ) ε + lim ε → bg ( t E β ( εt − α )) − bg ( t ) ε = a D α,βM f ( t ) + b D α,βM g ( t ) (2) (Product Rule) D α,βM ( f · g ) ( t ) = f ( t ) D α,βM g ( t ) + g ( t ) D α,βM f ( t ) .Proof. Using Definition 5, we have D α,βM ( f · g ) ( t ) = lim ε → f ( t E β ( εt − α )) g ( t E β ( εt − α )) − f ( t ) g ( t ) ε = lim ε → f ( t E β ( εt − α )) g ( t E β ( εt − α )) + f ( t ) g ( t E β ( εt − α )) −− f ( t ) g ( t E β ( εt − α )) − f ( t ) g ( t ) ε = lim ε → (cid:18) f ( t E β ( εt − α )) − f ( t ) ε (cid:19) lim ε → g (cid:0) t E β (cid:0) εt − α (cid:1)(cid:1) ++lim ε → (cid:18) g ( t E β ( εt − α )) − g ( t ) ε (cid:19) f ( t )= D α,βM ( f ) ( t ) lim ε → g (cid:0) t E β (cid:0) εt − α (cid:1)(cid:1) + D α,βM ( g ) ( t ) f ( t )= D α,βM ( f ) ( t ) g ( t ) + D α,βM ( g ) ( t ) f ( t ) , because lim ε → g ( t E β ( εt − α )) = g ( t ). (3) (Quotient rule) D α,βM (cid:18) fg (cid:19) ( t ) = g ( t ) D α,βM f ( t ) − f ( t ) D α,βM g ( t )[ g ( t )] .Proof. Using Definition 5, we have
J. VANTERLER DA C. SOUSA AND E. CAPELAS DE OLIVEIRA D α,βM (cid:18) fg (cid:19) ( t ) = lim ε → f ( t E β ( εt − α )) g ( t E β ( εt − α )) − f ( t ) g ( t ) ε = lim ε → g ( t ) f ( t E β ( εt − α )) − f ( t ) g ( t E β ( εt − α )) + f ( t ) g ( t ) − f ( t ) g ( t ) εg ( t E β ( εt − α )) g ( t )= lim ε → g ( t ) ( f ( t E β ( εt − α )) − f ( t )) ε − lim ε → f ( t ) ( g ( t E β ( εt − α )) − g ( t )) ε lim ε → g ( t E β ( εt − α )) g ( t )= g ( t ) D α,βM f ( t ) − f ( t ) D α,βM g ( t )[ g ( t )] , because lim ε → g ( t E β ( εt − α )) = g ( t ). (4) D α,βM ( c ) = 0 , where f ( t ) = c is a constant.Proof. The result follows directly from Definition 5. (5) If, furthermore, f is differentiable, then D α,βM ( f ) ( t ) = t − α Γ ( β + 1) df ( t ) dt . Proof.
We can write t E β (cid:0) εt − α (cid:1) = ∞ X k =0 ( εt − α ) k Γ ( βk + 1) = t + εt − α Γ ( β + 1) + t ( εt − α ) Γ (2 β + 1) + t ( εt − α ) Γ (3 β + 1) ++ · · · + t ( εt − α ) n Γ ( nβ + 1) + · · · = t + εt − α Γ ( β + 1) + O (cid:0) ε (cid:1) . (3.5) Using Definition 5 and introducing the change h = εt − α (cid:18)
1Γ ( β + 1) + O ( ε ) (cid:19) ⇒ ε = ht − α (cid:16) β +1) + O ( ε ) (cid:17) , N THE LOCAL M -DERIVATIVE 9 in Eq.(3.5), we conclude that D α,βM f ( t ) = lim ε → f (cid:18) t + εt − α Γ ( β + 1) + O (cid:0) ε (cid:1)(cid:19) − f ( t ) ε = lim ε → f ( t + h ) − f ( t ) ht α − β +1) (1 + Γ ( β + 1) O ( ε ))= t − α Γ ( β + 1) lim ε → f ( t + h ) − f ( t ) h β + 1) O ( ε )= t − α Γ ( β + 1) df ( t ) dt , with β > t > (6) (Chain rule) D α,βM ( f ◦ g ) ( t ) = f ′ ( g ( t )) D α,βM g ( t ) , for f differentiable at g ( t ) .Proof. If g ( t ) = a is a constant, then D α,βM ( f ◦ g ) ( t ) = D α,βM f ( g ( t )) = D α,βM ( f ) ( a ) = 0 . On the other hand, assume that g is not a constant in the neighborhood of a , thatis, suppose an ε > g ( x ) = g ( x ), ∀ x , x ∈ ( a − ε , a + ε ). Now,since g is continuous at a , for ε small enough we have D α,βM ( f ◦ g ) ( a ) = lim ε → f ( g ( a E β ( εt − α ))) − f ( g ( a )) ε (3.6) = lim ε → f ( g ( a E β ( εt − α ))) − f ( g ( a )) g ( a E β ( εt − α )) − g ( a ) g ( a E β ( εt − α )) − g ( a ) ε . Introducing the change ε = g (cid:0) a E β (cid:0) εt − α (cid:1)(cid:1) − g ( a ) ⇒ g (cid:0) a E β (cid:0) εt − α (cid:1)(cid:1) = g ( a ) + ε in Eq.(3.6), we conclude that D α,βM ( f ◦ g ) ( a ) = lim ε → f ( g ( a E β ( εt − α ))) − f ( g ( a )) ε lim ε → g ( a E β ( εt − α )) − g ( a ) ε = f ′ ( g ( a )) D α,βM g ( a ) , with a > Theorem 3.
Let a ∈ R , β > and < α ≤ . Then we have the following results:(1) D α,βM (1) = 0 . AND E. CAPELAS DE OLIVEIRA (2) D α,βM ( e at ) = t − α Γ ( β + 1) ae at .(3) D α,βM (sin ( at )) = t − α Γ ( β + 1) a cos ( at ) .(4) D α,βM (cos ( at )) = − t − α Γ ( β + 1) a sin ( at ) .(5) D α,βM (cid:0) t α α (cid:1) = 1Γ ( β + 1) .(6) D α,βM ( t a ) = a Γ ( β + 1) t a − α . Theorem 4.
Let < α ≤ , β > and t > . Then we have the following results:(1) D α,βM (cid:0) sin (cid:0) α t α (cid:1)(cid:1) = cos (cid:0) α t α (cid:1) Γ ( β + 1) . (2) D α,βM (cid:0) cos (cid:0) α t α (cid:1)(cid:1) = − sin (cid:0) α t α (cid:1) Γ ( β + 1) . (3) D α,βM (cid:16) e tαα (cid:17) = e tαα Γ ( β + 1) . The identities in Theorem 3 and Theorem 4 are direct consequences of item 5 of Theorem 2.We now prove the extensions, for α -differentiable functions in the sense of the local M -derivative defined in Eq.(3.1), of Rolle’s theorem and the mean value and extended meanvalue theorems. Theorem 5. (Rolle’s theorem for α -differentiable functions) Let a > and f : [ a, b ] → R bea function such that:(1) f is continuous on [ a, b ] ;(2) f is α -differentiable on ( a, b ) for some α ∈ (0 , ;(3) f ( a ) = f ( b ) .Then, there exists c ∈ ( a, b ) , such that D α,βM f ( c ) = 0 , β > .Proof. Since f is continuous on [ a, b ] and f ( a ) = f ( b ), there exists a point c ∈ ( a, b ) at whichfunction f has a local extreme. Then, D α,βM f ( c ) = lim ε → − f ( c E β ( εc − α )) − f ( c ) ε = lim ε → + f ( c E β ( εc − α )) − f ( c ) ε . As lim ε → ± E β ( εc − α ) = 1, the two limits in the right hand side of this equation have oppositesigns. Hence, D α,βM f ( c ) = 0. Theorem 6. (Mean value theorem for α -differentiable functions) Let a > and f : [ a, b ] → R be a function such that: N THE LOCAL M -DERIVATIVE 11 (1) f is continuous on [ a, b ] ;(2) f is α -differentiable on ( a, b ) for some α ∈ (0 , .Then, there exists c ∈ ( a, b ) such that D α,βM f ( c ) = f ( b ) − f ( a ) b α α − a α α , with β > . Theorem 7.
Consider the function (3.7) g ( x ) = f ( x ) − f ( a ) − Γ ( β + 1) f ( b ) − f ( a )1 α b α − α a α (cid:18) α x α − α a α (cid:19) . Function g satisfies the conditions of Rolle’s theorem. Then, there exists c ∈ ( a, b ) such that D α,βM f ( c ) = 0 . Applying the local M -derivative D α,βM to both sides of Eq.(3.7) and using thefact that D α,βM (cid:0) t α α (cid:1) = β +1) and D α,βM ( c ) = 0 , with c a constant, we conclude that D α,βM f ( c ) = f ( b ) − f ( a ) b α α − a α α . Theorem 8. (Extended mean value theorem for fractional α -differentiable functions) Let a > and f, g : [ a, b ] → R functions such that:(1) f, g are continuous on [ a, b ] ;(2) f, g are α -differentiable for some α ∈ (0 , .Then, there exists c ∈ ( a, b ) such that (3.8) D α,βM f ( c ) D α,βM g ( c ) = f ( b ) − f ( a ) g ( b ) − g ( a ) , with β > .Proof. Consider the function(3.9) F ( x ) = f ( x ) − f ( a ) − (cid:18) f ( b ) − f ( a ) g ( b ) − g ( a ) (cid:19) ( g ( x ) − g ( a )) . As F is continuous on [ a, b ], α -differentiable on ( a, b ) and F ( a ) = 0 = F ( b ), by Rolle’stheorem there exists c ∈ ( a, b ) such that D α,βM F ( c ) = 0 for some α ∈ (0 , AND E. CAPELAS DE OLIVEIRA the local M -derivative D α,βM to both sides of Eq.(3.9) and using the fact that D α,βM ( c ) = 0when c is a constant, we conclude that D α,βM f ( c ) D α,βM g ( c ) = f ( b ) − f ( a ) g ( b ) − g ( a ) . Definition 6.
Let β > , α ∈ ( n, n + 1] , for some n ∈ N and f n times differentiable (inthe classical sense) for t >
0. Then the local M -derivative of order n of f is defined by(3.10) D α,β ; nM f ( t ) := lim ε → f ( n ) ( t E β ( εt n − α )) − f ( n ) ( t ) ε , if and only if the limit exists.From Definition 6 and the chain rule, that is from item 5 of Theorem 2, by induction on n , wecan prove that D α,β,nM f ( t ) = t n +1 − α Γ ( β + 1) f ( n +1) ( t ), α ∈ ( n, n +1] and so f is ( n +1)-differentiablefor t >
0. 4. M -integral In this section we introduce the concept of M -integral of a function f . From this definitionwe can prove some results similar to classical results such as the inverse property, the funda-mental theorem of calculus and the theorem of integration by parts. Other results about the M -integral are also presented. In preparing this section we made extensive use of references[5, 8, 12, 13]. Definition 7. ( M -integral) Let a ≥ and t ≥ a . Let f be a function defined in ( a, t ] and < α < . Then, the M -integral of order α of a function f is defined by (4.1) M I α,βa f ( t ) = Γ ( β + 1) Z ta f ( x ) x − α dx, with β > . Theorem 9. (Inverse)
Let a ≥ and < α < . Also, let f be a continuous function suchthat there exists M I α,βa f . Then (4.2) D α,βM ( M I α,βa f ( t )) = f ( t ) , with t ≥ a and β > . N THE LOCAL M -DERIVATIVE 13 Proof.
Indeed, using the chain rule proved in Theorem 2 we have D α,βM (cid:0) M I α,βa f ( t ) (cid:1) = t − α Γ ( β + 1) ddt ( M I α,βa f ( t ))= t − α Γ ( β + 1) ddt (cid:18) Γ ( β + 1) Z ta f ( x ) x − α dx (cid:19) = t − α Γ ( β + 1) (cid:18) Γ ( β + 1) t − α f ( t ) (cid:19) = f ( t ) . (4.3)We now prove the fundamental theorem of calculus in the sense of the M -derivative men-tioned at the beginning of the paper. Theorem 10. (Fundamental theorem of calculus)
Let f : ( a, b ) → R be an α -differentiablefunction and < α ≤ . Then, for all t > a we have M I α,βa (cid:16) D α,βM f ( t ) (cid:17) = f ( t ) − f ( a ) , with β > .Proof. In fact, since function f is differentiable, using the chain rule of Theorem 2 and thefundamental theorem of calculus for the integer-order derivative, we have M I α,βa (cid:16) D α,βM f ( t ) (cid:17) = Γ ( β + 1) Z ta D α,βM f ( t ) x − α dx = Γ ( β + 1) Z ta x − α Γ ( β + 1) 1 x − α df ( t ) dt dx = Z ta df ( t ) dt dx = f ( t ) − f ( a )(4.4)If the condition f ( a ) = 0 holds, then by Theorem 9, Eq.(4.4), we have M I α,βa h D α βM f ( t ) i = f ( t ).Theorem 10 can be generalized to a larger order as follows. AND E. CAPELAS DE OLIVEIRA Theorem 11.
Let α ∈ ( n, n + 1] and f : [ a, ∞ ) → R be ( n + 1) -times differentiable for t > a .Then, ∀ t > a , we have M I α,βa (cid:16) D α,βM f ( t ) (cid:17) = f ( t ) − n X k =0 f ( k ) ( a ) ( t − a ) k k ! , with β > .Proof. Using the definition of M -integral and the chain rule proved in Theorem 2, we have M I α,βa (cid:16) D α,βM f ( t ) (cid:17) = M I n +1 a h ( t − a ) β − D α,βM f ( n ) ( t ) i = M I n +1 a h ( t − a ) β − ( t − a ) − β f ( n +1) ( t ) i = M I n +1 a (cid:0) f ( n +1) ( t ) (cid:1) . (4.5)Then, performing piecewise integration for the integer-order derivative in Eq.(4.5), we have M I α,βa (cid:16) D α,βM f ( t ) (cid:17) = f ( t ) − n X k =0 f ( k ) ( a ) ( t − a ) k k ! . As integer-order calculus has a result known as integration by parts, we shall now present,through a theorem, a similar result which we might call fractional integration by parts.We shall use the notation of Eq.(4.1) for the M -integral: M I α,βa f ( t ) = Γ ( β + 1) Z ta f ( x ) x − α dx = Z ta f ( x ) d α x, where d α x = Γ ( β + 1) x − α dx . Theorem 12.
Let f, g : [ a, b ] → R be two functions such that f, g are differentiable and < α < . Then (4.6) Z ba f ( x ) D α,βM g ( x ) d α x = f ( x ) g ( x ) | ba − Z ba g ( x ) D α,βM f ( x ) d α x, with β > . N THE LOCAL M -DERIVATIVE 15 Proof.
Indeed, using the definition of M -integral and applying the chain rule of Theorem 2and the fundamental theorem of calculus for integer-order derivatives, we have Z ba f ( x ) D α,βM g ( x ) d α x = Γ ( β + 1) Z ba f ( x ) x − α D α,βM g ( x ) dx = Γ ( β + 1) Z ba f ( x ) x − α x − α Γ ( β + 1) dg ( x ) dt dx = Z ba f ( x ) g ′ ( x ) dx = f ( x ) g ( x ) | ba − Γ ( β + 1) Z ba g ( x ) x − α x − α Γ ( β + 1) df ( x ) dt dx = f ( x ) g ( x ) | ba − Z ba g ( x ) D α,βM f ( x ) d α x, where d α x = Γ( β +1) x − α dx . Theorem 13.
Let < a < b and let f : [ a, b ] → R be a continuous function. Then, for < α < we have (4.7) (cid:12)(cid:12) M I α,βa f ( t ) (cid:12)(cid:12) ≤ ( M I α,βa | f ( t ) | ) , with β > .Proof. From the definition of M -integral of order α , we have (cid:12)(cid:12) M I α,βa f ( t ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) Γ ( β + 1) Z ta f ( x ) x − α dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ | Γ ( β + 1) | Z ta (cid:12)(cid:12)(cid:12)(cid:12) f ( x ) x − α (cid:12)(cid:12)(cid:12)(cid:12) dx = M I α,βa | f ( t ) | . Corollary 4.8.
Let f : [ a, b ] → R be a continuous function such that (4.9) N = sup t ∈ [ a,b ] | f ( t ) | . Then, ∀ t ∈ [ a, b ] and < α < , we have (4.10) (cid:12)(cid:12) M I α,βa f ( t ) (cid:12)(cid:12) ≤ Γ ( β + 1) N (cid:18) t α α − a α α (cid:19) , with β > . AND E. CAPELAS DE OLIVEIRA Proof.
By Theorem 13, we have (cid:12)(cid:12) M I α,βa f ( t ) (cid:12)(cid:12) ≤ M I α,βa | f ( t ) | = Γ ( β + 1) Z ta | f ( x ) | x α − dx ≤ Γ ( β + 1) N Z ta x α − dx = Γ ( β + 1) N (cid:18) t α α − a α α (cid:19) . (4.11) 5. Relation with alternative fractional derivative
In this section we discuss the relation between the alternative fractional derivative and thelocal M -derivative proposed here.Katugampola [8] proposed a new fractional derivative which he called alternative fractionalderivative, given by(5.1) D α f ( t ) = lim ε → f (cid:16) te εt − α (cid:17) − f ( t ) ε , with α ∈ (0 ,
1) and t > M -derivative Eq.(3.1)) is more general than thealternative fractional derivative Eq.(5.1).The definition in Eq.(3.1) contains the one parameter Mittag-Leffler function E β ( · ), whichcan be considered a generalization of the exponential function. Indeed, choosing β = 1 inthe definition of the one parameter Mittag-Leffler function [17, 18], we have(5.2) E β ( x ) = E ( x ) = ∞ X k =0 x k Γ ( k + 1) = e x . In particular, introducing x = εt − α in Eq.(3.1) and taking the limit ε → D α :(5.3) D α,βM f ( t ) = lim ε → f ( t E ( εt − α )) − f ( t ) ε = lim ε → f (cid:16) te εt − α (cid:17) − f ( t ) ε = D α f ( t ) . N THE LOCAL M -DERIVATIVE 17 Application
Fractional linear differential equations are important in the study of fractional calculus andapplications. In this section, we present the general solution of a linear differential equationby means of the local M -derivative. In this sense, as a particular case, we study an exampleand perform a graph analysis of the solution.The general first order differential equation based on the local M -derivative is representedby(6.1) D α,βM u ( t ) + P ( t ) u ( t ) = Q ( t ) . where P ( t ) , Q ( t ) are α − differentiable functions and u ( t ) is unknown.Using the item 5 of Theorem (2) in the Eq.(6.1), we have(6.2) ddt u ( t ) + Γ ( β + 1) t − α P ( t ) u ( t ) = Γ ( β + 1) t − α Q ( t ) . The Eq.(6.2) is a first order equation, whose general solution is given by u ( t ) = e − Γ( β +1) R P ( t ) t − α dt (cid:18) Γ ( β + 1) Z Q ( t ) t − α e Γ( β +1) R P ( t ) t − α dt dt + C (cid:19) , where C is an arbitrary constant.By definition of M -integral, we conclude that the solution is given by u ( t ) = e − M I α,βa ( P ( t )) (cid:16) M I α,βa (cid:16) Q ( t ) e M I α,βa ( P ( t )) (cid:17) + C (cid:17) . Now let us choose some values and functions and make an example using the linear differentialequation previously studied by means of the local M -derivative. Then, taking P ( t ) = − λ , Q ( t ) = 0, u (0) = u , a = 0, 0 < α ≤ β >
0, we have the following linear differentialequation(6.3) D α,βM u ( t ) = λu ( t ) , whose solution is given by u ( t ) = u e − λα Γ( β +1) t α = u E (cid:18) − λα Γ ( β + 1) t α (cid:19) , where E ( · ) is Mittag-Leffler function. AND E. CAPELAS DE OLIVEIRA Figure 1.
Analytical solution of the Eq.(6.3). We consider the values β = 0 . λ =1 and u =20. t u ( t ) α = 0.3 α = 0.6 α = 0.9 α = 1.0 Figure 2.
Analytical solution of the Eq.(6.3). We take the values β = 1 . λ =2 and u =20 t u ( t ) α = 0.3 α = 0.6 α = 0.9 α = 1.0 N THE LOCAL M -DERIVATIVE 19 Figure 3.
Analytical solution of the Eq.(6.3). We chose the values β = 1 . λ =2.5 and u =20. t u ( t ) α = 0.3 α = 0.6 α = 0.9 α = 1.0 Concluding remarks
We introduced a new derivative, the local M -derivative, and its corresponding M -integral.We could prove important results concerning integer order derivatives of this kind, in par-ticular, derivatives of order one. For α -differentiable functions in the context of local M -derivatives we could show that the derivative proposed here behaves well with respect to theproduct rule, the quotient rule, composition of functions and the chain rule. The local M -derivative of a constant is zero, differently from the case of the Riemann-Liouville fractionalderivative. Moreover, we present α -differentiable functions versions of Rolle’s theorem, themean value theorem and the extended mean value theorem.An M -integral was introduced and some results bearing relations to results in the calculus ofinteger order were obtained, among which the M -fractional versions of the inverse theorem,the fundamental theorem of calculus and a theorem involving integration by parts.We obtained a relation between our local M -derivative and the alternative fractional deriv-ative, presented in section 5 of the paper, as well as possible applications in several areas,particularly as we show, in the solution of a linear differential equation. We conclude fromthis result that the definition presented here can be considered a generalization of the so-called alternative fractional derivative [8].Possible applications of the local M -derivative and the corresponding M -integral are thesubject of a forthcoming paper [19]. AND E. CAPELAS DE OLIVEIRA Acknowledgment
We are grateful to Dr. J. Em´ılio Maiorino for several and fruitful discussions.
References [1] G. W. Leibniz,
Letter from Hanover, Germany to G.F.A L’Hospital, September 30, 1695, LeibnizMathematische Schriften . Olms-Verlag, Hildesheim, Germany, 301–302, (First published in 1849).[2] G. W. Leibniz,
Letter from Hanover, Germany to Johann Bernoulli, December 28, 1695, Leibniz Math-ematische Schriften . Olms-Verlag, Hildesheim, Germany, 1962, , (First published in 1849).[3] G. W. Leibniz,
Letter from Hanover, Germany to John Wallis, May 30, 1697, Leibniz MathematischeSchriften . Olms-Verlag, Hildesheim, Germany, 1962, , (First published in 1849).[4] E. Capelas de Oliveira and J. A. Tenreiro Machado, A Review of Definitions for Fractional Derivativesand Integral . Math. Probl. Eng., , (238459), (2014).[5] R. Khalil, M. Al Horani, A. Yousef and M. Sababheh,
A new definition of fractional derivative . J.Comput. Appl. Math, , 65–70, (2014).[6] D. R. Anderson, Taylors formula and integral inequalities for conformable fractional derivatives, Con-tributions in Mathematics and Engineering, in Honor of Constantin Caratheodory, Springer, to appear25-43. (2016).[7] Y. Cenesiz and A. Kurt,
The solutions of time and space conformable fractional heat equations withconformable Fourier transform .Acta Univ. Sapientiae, Mathematica, , 130–140, (2015).[8] U. N. Katugampola, A new fractional derivative with classical properties . arXiv:1410.6535v2, (2014).[9] J. A. Tenreiro Machado,
And I say to myself: What a fractional world !. Frac. Calc. Appl. Anal., ,635–654, (2011).[10] M. D. Ortigueira and J. A. Tenreiro Machado, What is a fractional derivative? . J. Comput. Phys., ,4–13, (2015).[11] U. N. Katugampola,
Correction to What is a fractional derivative? by Ortigueira and Machado [Journalof Computational Physics, Volume 293, 15 July 2015, Pages 413. Special issue on Fractional PDEs] . J.Comput. Appl. Math, , 1–2, (2015).[12] O. S. Iyiola and E. R. Nwaeze , Some new results on the new conformable fractional calculus withapplication sing DAlembert approach . Progr. Fract. Differ. Appl., , 115–122, (2016).[13] T. Abdeljawad, On conformable fractional calculus . J. Comput. Appl. Math, , 57–66, (2015).[14] I. Podlubny, Fractional Differential Equation, Mathematics in Science and Engineering, Academic Press,San Diego. , 1999.[15] R. Gorenflo and F. Mainardi,