Generalized Fractional Derivative, Fractional differential ring
GGeneralized Fractional Differential Ring
Zeinab Toghani, Luis GaggeroQueen Mary University of LondonUniversidad Autónoma del Estado de Morelos(CIICAp)
Abstract
There are many possible definitions of derivatives, here we present some and presentone that we have called generalized that allows us to put some of the others as aparticular case of this but, what interests us is to determine that there is an infinitenumber of possible definitions of fractional derivatives, all are correct as differentialoperators each of them must be properly defined its algebra.We introduce a generalized version of fractional derivative that extends the existingones in the literature. To those extensions it is associated a differentiable operator anda differential ring and applications that shows the advantages of the generalization.We also review the different definitions of fractional derivatives proposed by MicheleCaputo in [Cap67], Khalil, Al Horani, Yousef, Sababheh in [KAHYS14], Anderson andUlness in [AU15a], Guebbai and Ghiat in [GG16], Udita N. Katugampola in [Kat14],Camrud in [Cam16] and it is shown how the generalized version contains the previousones as a particular cases.
Keywords : Generalized Fractional Derivative, Fractional differential ring.
Fractional derivative was defined for responding to a question’ what does it mean d α fdt α if α = ’ in 1695. Following that, finding the right definition of fractional derivativehas attracted significant attention of researcher and in the last few years it has seen sig-nificantly progress in mathematical and non-mathematical journals(see [Las02], [Wei15],[Aga12], [DWH13], [CK15], [WZD17], [AHK14], [Nam80], [YZUHR13], [And16]). In fact,there are articles which in few months have gained hundreds of citations. In particu-lar in past three years several definitions of fractional derivative have been proposed(see[KAHYS14], [AU15b], [SdO17], [Cam16], [MR93], [OTM15], [Kat14], [IN16], [AAHK15],[AAHK15], [GG16], [And16], [SB17], [CF15], [DWH13], [Aga12], [Las00]). Since some ofprevious definitions do not satisfy the classical formulas of the usual derivative, it has beenproposed an ad hoc algebra associated to each definition. To unify that diversity, we pro-pose a version of fractional derivative that has the advantages that generalized the already Emails: [email protected], [email protected] . Corresponding Author: Luis Gaggero-Sager. a r X i v : . [ m a t h . F A ] F e b xisting in the literature and where the different algebras are unified under the notion offractional differential ring.The present paper is organized as follows: In the second section we give the previousdefinitions of fractional derivative and our generalized fractional derivative(GFD) definition,in the third section we introduce a fractional differential ring, in the forth section we givesome result of GFD, in the fifth section we give a definition of partial fractional differentialderivative, in the sixth section we give a definition of GFD when α ∈ ( n, n + 1] . Let α ∈ (0 , be a fractional number, we want to give a definition of generalized fractionalderivative of order α for a differentiable function f . We denote α − th derivative of f by D α ( f ) and we denote the first derivative of f by D ( f ) .We begin the present section listing previous definition of fractional derivative; later wepresent our proposal of generalized one showing how it contains the once already described.We finish the section providing some examples.1. The Caputo fractional derivative was defined by Michele Caputo in [Cap69]: D α ( f ) = 1Γ(1 − α ) (cid:90) ta f (cid:48) ( x )( t − x ) α dx. (1)2. The conformable fractional derivative was defined by Khalil, Al Horani, Yousef andSababheh in [KAHYS14] : D α f ( t ) = lim ε → f ( t + t − α ε ) − f ( t ) ε . (2)3. The conformable fractional derivative was defined by Anderson and Ulness in [AU15a]: D α f ( t ) = (1 − α ) | t | α f ( t ) + α | t | − α Df. (3)4. The fractional derivative was defined by Udita N.Katugampola in [Kat14]: D α f ( t ) = lim ε → f ( te εt − α ) − f ( t ) ε . (4)5. The fractional derivative was defined by Guebbai and Ghiat in [GG16] for an increasingand positive function f : D α f ( t ) = lim ε → (cid:32) f ( t + f ( t ) − αα ε ) − f ( t ) ε (cid:33) α . (5)6. The conformable ratio derivative was defined by Camrud in [Cam16] for a function f ( t ) ≥ with Df ( t ) ≥ : D α f ( t ) = lim ε → f ( t ) − α (cid:18) f ( t + ε ) − f ( t ) ε (cid:19) α . (6)2rom all these definitions, we propose a definition that unifies almost all of them. Definition 1.
Given a differentiable function f : [0 , ∞ ) → R , the generalized fractionalderivative (GFD) for α ∈ (0 , at point t is defined by : D α f ( t ) = lim ε → f ( t + w t,α t − α ε ) − f ( t ) ε , where w t,α is a function that may depend on α and t . Remark 1.
As a consequence of definition 1 we can see D α f ( t ) = w t,α t − α Df ( t ) . Definition 2.
A differentiable function f : [0 , ∞ ) → R is said to be α − generalizedfractional differentiable function over [0 , ∞ ) if it exists D α ( f )( t ) for all t ∈ [0 , ∞ ) for α ∈ (0 , .We denote C α [0 , ∞ ) the set of α − generalized differentiable functions with real values inthe interval [0 , ∞ ) in variable t . The set ( C α [0 , ∞ ) , + , . ) is a ring. In the following we wantto see the relation between GFD and the others definitions:1. The fractional derivative of Khalil, Al Horani, Yousef and Sababheh in [KAHYS14] isa particular case of GFD where w t,α = 1 .2. The fractional derivative of Anderson and Ulness in [AU15a] is a particular case ofGFD where w t,α = (1 − α ) t α f ( t ) + αt − α Dfαt − α . In this fractional derivative w t,α depends on α and t .3. The fractional derivatives of Guebbai and Ghiat in [GG16] and Camrud in [Cam16]are particular cases of GFD where w t,α = (cid:16) tDff (cid:17) α − .We are particularly interested in discussing GFD where w t,α = g ( t, α ) τ α − such that g :[0 , ∞ ) × (0 , → R is a function and τ is the characteristic of system with the properties w t,α = 1 if and only if α = 1 . If the system is periodic with period T , then we have τ = T . In the quantum systems τ isthe Bohr radius and in astronomy τ is the light year. The characteristic of system τ dependson the systems and the derivative. If t is time, τ is time too. If t is space, τ is space too. Infact the unit of t is τ , i.e., t = cτ where c is a constant. In the general τ = 1 . Example 3.
Let α, β ∈ (0 , . Let f, h be two functions in C α [0 , ∞ ) . We suppose w t,α = g ( t, α ) τ α − with τ = 1 .1. If g ( t, α ) is a function with g (0 ,
0) = 0 , then lim α → D α ( f ) = 0 .3. If g ( t, α ) = α , we have the chain rule D α ( f ◦ h ) = t α − α D α ( f ( h )) D α ( h ) . Example 4.
We want to present the corresponding figure to the generalized fractionalderivatives for α = for a trigonometric, using all the fractional derivative definitions thatwe have already mentioned in this article. It can be seen from all the figures that in principlethese definitions do not find a reason to discard them. That is, they have a fairly reasonablebehavior. We consider f ( t ) = sin(2 t ) , the figure of f can be seen in the following picture.Figure 1: red:Caputo, green:Khalil et al, blue:Anderson et al , orange:Guebbai et al,black:GFD when w t,α = α . In this section we want to stress out that instead of defining a new derivative, we focus onthe notion of differentiable operator and the ring that it carries with.
Definition 5.
Let R be a commutative ring with unity. A derivation on R is a map d : R → R that satisfies d ( a + b ) = d ( a ) + d ( b ) and d ( ab ) = d ( a ) b + ad ( b ) , ∀ a, b ∈ R . The pair ( R, d ) is called a differential ring(see [Rit50]). Theorem 6.
Let α ∈ (0 , . The ring C α [0 , ∞ ) with operator D α : C α [0 , ∞ ) → C α [0 , ∞ ) isa differential ring.Proof. Since C α [0 , ∞ ) is a commutative ring with unity f ( t ) = 1 and the derivation D α for α ∈ (0 , satisfies following properties from remark (1)1. D α ( af + bf ) = aD α ( f ) + bD α ( f ) , ∀ f , f ∈ C α [0 , ∞ ) , ∀ a, b ∈ R ,2. D α ( f f ) = f D α ( f ) + f D α ( f ) , ∀ f , f ∈ C α [0 , ∞ ) .4et α ∈ (0 , be a fractional number, let f , f ∈ C α [0 , ∞ ) be two functions, GFD hasthe following properties:1. D α ( f f ) = f D α f − f D α f f . D α ( f ◦ f ) = t α − w t,α D α ( f ( f )) D α ( f ) . D α + β ( f ) = w t,α w t,β tw t,α + β D α D β ( f ) . It is easy to see these properties from remark (1). If w t,α = t − α we have the equality D α ( f ◦ f ) = D α ( f ( f )) D α ( f ) . If w t,α + β w t,α w t,β = t we have the equality D α + β ( f ) = D α D β ( f ) , ∀ α, β ∈ (0 , . Parts 4 and 5 of the properties imply that we can create function spaces with differentalgebras using different expressions for w t,α .By considering previous properties of GFD we called C α [0 , ∞ ) a w t,α − generalizedfractional differential ring of functions and we denote it by ( C α [0 , ∞ ) , D α , w t,α ) . Let I ⊂ C α [0 , ∞ ) be an ideal. If D α ( I ) ⊂ I then the ideal I is called a w t,α − generalizedfractional differential ideal . By using the previous properties we can see the followingresult: Theorem 7.
Let α ∈ (0 , . Associated to any α and any w t,α there exists a fractionaldifferential ring. Let α ∈ (0 , be a fractional number and t ∈ [0 , ∞ ) then GFD has the following properties:1. D α ( t α αw t,α ) = 1 ,2. D α (sin( t α αw t,α )) = cos( t α αw t,α ) ,3. D α (cos( t α αw t,α )) = − sin( t α αw t,α ) ,4. D α ( e ( tααwt,α ) ) = e ( tααwt,α ) . Theorem 8. (Rolle’s theorem for α − Generalized Fractional Differentiable Functions)Let a > , let f : [ a, b ] → R be a function with the properties that1. f is continuous on [ a, b ] ,2. f is α − generalized fractional differentiable on ( a, b ) for some α ∈ (0 , ,3. f ( a ) = f ( b ) . hen, there exist c ∈ ( a, b ) such that D α f ( c ) = 0 .Proof. Since f is continuous on [ a, b ] and f ( a ) = f ( b ) , then the funtion f has a local extremein a point c ∈ ( a, b ) . Then D α f ( c ) = lim ε → + f ( c + w t,α c − α ε ) − f ( c ) ε = lim ε → − f ( c + w t,α c − α ε ) − f ( c ) ε . But two limits have different signs. Then D α f ( c ) = 0 . Theorem 9. (Mean Value Theorem for α − Generalized Fractional Differentiable Functions)Let a > and f : [ a, b ] → R be a function with the properties that1. f is continuous on [ a, b ] ,2. f is α − Generalized fractional differentiable on ( a, b ) for some α ∈ (0 , .Then, there exists c ∈ ( a, b ) such that D α f ( c ) = αw t,α ( f ( b ) − f ( a )) b − a .Proof. Consider function h ( t ) = f ( t ) − f ( a ) − αw t,α ( f ( b ) − f ( a )) b − a (cid:18) t α αw t,α − a α αw t,α (cid:19) , Then, the function h satisfies the conditions of the fractional Rolle’s theorem. Hence, thereexists c ∈ ( a, b ) such that D α h ( c ) = 0 . We have the result since D α h ( c ) = D α f ( c ) − αw t,α ( f ( b ) − f ( a )) b − a (1) = 0 In this section we introduce a partial fractional derivative of first and second order, also weintroduce a partial fractional differential ring.
Definition 10.
Let f ( t , · · · , t n ) : [0 , ∞ ) n → R be a function with n variables such that ∀ i ,there exists the partial derivative of f respect to t i . Let α ∈ (0 , be a fractional number.We define α − generalized partial fractional derivative (GPFD) of f with respect to t i at point t = ( t , . . . , t n ) ∂ α f ( t ) ∂t αi = lim ε → f ( t ,...,t i + w ti,α t − αi ε,...,t n ) − f ( t ) ε , where w t i ,α can be a function depend on α and t i . Remark 2.
As a consequence of definition (2) we can see for α ∈ (0 , and ≤ i ≤ n : ∂ α f∂t αi ( t ) = w t i ,α ( t i ) − α ∂f∂t i ( t ) . α ∈ (0 , and ≤ i ≤ n . A partial differentiable function f : [0 , ∞ ) n → R is said tobe a α − generalized partial fractional differentiable function respect to t i over [0 , ∞ ) if exists ∂ α f ( t ) ∂t αi for all t ∈ [0 , ∞ ) . We denote by C αi [0 , ∞ ) n the set of α − generalized partialfractional differentiable functions respect to t i with real values in the interval [0 , ∞ ) n invariable t = ( t , . . . , t n ) . The set ( C αi [0 , ∞ ) n , + , . ) is a ring. Theorem 11.
Let α ∈ (0 , and ≤ i ≤ n . The ring C αi [0 , ∞ ) n with operator ∂ α ∂t αi : C αi [0 , ∞ ) n → C αi [0 , ∞ ) n is a differential ring.Proof. Since the ring C αi [0 , ∞ ) n is a commutative ring with unity f ( t , . . . , t n ) = 1 and thederivation ∂ α ∂t αi for α ∈ (0 , satisfies the following properties from remark (2);1. ∂ α ( f + f ) ∂t αi = ∂ α ( f ) ∂t αi + ∂ α ( f ) ∂t αi f , f ∈ C α [0 , ∞ ) n , ∂ α ( f f ) ∂t αi = f ∂ α f ∂t αi + f ∂ α f ∂t αi f , f ∈ C α [0 , ∞ ) n .Let α ∈ (0 , and ≤ i ≤ n , let f , f ∈ C α [0 , ∞ ) n be two functions, then GPFD hasthe following properties from remark (2):1. ∂ α ( f f ) ∂t αi = f ∂αf ∂tαi − f ∂αf ∂tαi f , ∂ α f ◦ f ∂t αi = t α − i w ti,α ∂ α ( f ( f )) ∂t αi ∂ α ( f ) ∂t αi , ∂ α + β ( f ) ∂t αi = w ti,α w ti,β t i w ti,α + β ∂ α ∂t αi ∂ α ( f ) ∂t αi . By considering previous properties of GPFD we called the ring C αi [0 , ∞ ) n a w t i ,α − gener-alized partial fractional differential ring . We denote it by ( C αi [0 , ∞ ) n , ∂ α ∂t αi , w t i ,α ) .We can see the following result by using the previous properties. Theorem 12.
Let α ∈ (0 , and ≤ i ≤ n . Associated to any α and any w t i ,α there is apartial fractional differential ring. Example 13.
Let f ( t , t ) = t sin( t ) , let α ∈ [0 , . We have ∂ α f∂t α = w t,α ( t ) − α (3 t ) sin( t ) . Definition 14.
Let α ∈ (0 , be a fractional number. We define α − generalized partialfractional derivative of second order with respect to t i and t j at point t = ( t , · · · , t n ) is ∂ α f ( t ) ∂t αj ∂t αi = ∂ α ∂t αj ( ∂ α f ( t ) ∂t αi ) = lim ε → ∂αf∂tαi ( t ,...,t j + w tj,α t − αj ε,...,t n ) − ∂αf ( t ) ∂tαi ε . emark 3. As a consequence of definition (14) we can see for α ∈ (0 , and ≤ i, j ≤ n ; ∂ α f ( t ) ∂t αj ∂t αi = w t j ,α w t i ,α ( t j t i ) − α ∂∂t j ( ∂f∂t i ( t )) . A partial differentiable function of second order f : [0 , ∞ ) n → R is said to be a α − generalized fractional partial differentiable function of second order respect to t i and t j over [0 , ∞ ) if exists ∂ α f ( t ) ∂t αj ∂t αi for all t ∈ [0 , ∞ ) . We denote C α i,j [0 , ∞ ) n the set of α − generalizedpartial fractional differentiable functions of second order respect to t i and t j with real valuesin the interval [0 , ∞ ) n in variable t = ( t , . . . , t n ) . The set ( C α i,j [0 , ∞ ) n , + , . ) is a ring. α ∈ ( n, n + 1] In this section we define a fraction differential derivative for α ∈ ( n, n + 1] . Definition 15.
Let α ∈ ( n, n + 1] be a fractional number for n ∈ N , let f : [0 . ∞ ) → R be a n − differentiable. The generalized fractional derivative of order α is defined by D α f ( t ) = lim ε → f [ α ] − ( t + w t,α t [ α ] − α ε ) − f [ α ] − ( t ) ε . where [ α ] is the smallest integer greater than or equal to α. As a consequence of definition 15 we can see D α ( f ) = w t,α t [ α ] − α D [ α ] ( f ) , where α ∈ ( n, n + 1] .Let n < α ≤ n + 1 . A function f : [0 , ∞ ) → R is said to be α − generalized differen-tiable over [0 , ∞ ) if there exists D α ( f )( t ) for all t ∈ [0 , ∞ ) . We denote C α [0 , ∞ ) the setof α − generalized fractional differentiable functions with real values in the interval [0 , ∞ ) invariable t . The set ( C α [0 , ∞ ) , + , . ) is a ring. Theorem 16.
The ring ( C α [0 , ∞ ) , + , . ) with operator D α is not a differential ring for frac-tional number α ∈ ( n, n + 1] .Proof. Since D α ( f g ) (cid:54) = f D α g + gD α ( f ) for every f, g ∈ C α [0 , ∞ ) . α -Fractional Taylor Series There are some articles about fractional Taylor series see( [AU15b] , [Use08],[Mun04], [Yan11]).In this section we use GFD to define a fractional taylor series for a function f ∈ C r [0 , ∞ ) for every fractional number r .Let < α < , we define the α -fractional taylor series of f at real number x f ( x ) = f ( x ) + ∞ (cid:88) i =1 D i f ( x ) w x,α ( α + i − x − x ) α + i − , ( α + i − α ( α + 1) · · · ( α + i − .Let < α ≤ , we define the α -fractional taylor series of f at real number x f ( x ) = f ( x ) + Df ( x )( x − x ) + ∞ (cid:88) i =2 D i f ( x )( x − x ) α + i − w x,α ( α + i − , where ( α + i − α − α ( α + 1) · · · ( α + i − . Let n < α ≤ n + 1 such that α = n + A with < A < we define the α -fractional taylor series of f at real number x , f ( x ) = f ( x ) + n (cid:88) i =1 D i f ( x )( x − x ) i i ! + ∞ (cid:88) i = n +1 D i f ( x ) w x,α ( A + i − x − x ) A + i − , where α = n + A , ( A + i − A ( A + 1) · · · ( A + i − . There are some articles for applications of fractional differential derivative such as [YZUHR13],[Aga12] ,[AHK14], [WZD17]. In this section we solve some (partial)fractional differentialequations by using our definitions. At the first we solve the fractional differential equationswith the form aD α y + by = c, (7)where y = f ( t ) be a differentiable function and < α < .By substituting GFD in the equation (7) we have aw t,α t − α Dy + by = c = ⇒ Dy + bt α − aw t,α y = t α − caw t,α , the solutions of this equation have the form y ( t ) = cb + c e ( − btαawt,αα ) . Example 17.
We consider the partial fractional differential equation with boundary condi-tions u t + 2 √ xu x + u = x , t > u ( t,
0) = 0 ,u (0 , x ) = 0 , (8)where u ( x, t ) be a differentiable function respect to x and t , u ( x, t ) be a − partial fractionaldifferentiable function of first order respect to x , u t = ∂u∂t and √ u x = ∂ u∂x . For w t, = byusing remark (2) we can write u t + 2 √ xu x + u = x = ⇒ u t + 2 √ xw t,α x − α u x + u = x = ⇒ u t + 23 xu x + u = x . We solve this equation by taking Laplace transform of equation respect to t , we denote by U ( x, s ) the Laplace of u ( x, t ) respect to t , we have the following equation sU ( x, s ) − U ( x,
0) + xU x ( x, s ) + U ( x, s ) = x sU ( t,
0) = 0 ,U (0 , x ) = 0 . (9)9hen U x + 3 + 3 s x U = 3 x s = ⇒ U ( x, s ) = 3 x s (3 s + 7) + c ( s ) x − − s . By substituting U (0 , x ) = 0 we have c ( s ) = 0 , then U ( x, s ) = x ( s + s +7) ) . The solutionof equation is u ( x, t ) = x (1 − e − ) . Example 18.
We consider the partial fractional differential equation of second order; √ u xt + 2 ux = 0 , (10)where u ( x, t ) be a − fractional partial differentiable function of second order respect to t, x .For w x, = x , w t,, = √ t by using remark (3) we have x √ x √ t u xt + 2 √ tu = 0 . (11)We consider a solution of this differential equation with the form u ( x, t ) = f ( x ) g ( t ) suchthat f a function depends on x and g a function depends on t . By substituting u ( x, t ) in theequation (11) we have x √ x √ t Df.Dg + 3 √ tf g = 0 . (12)We can write the equation (12) a form that divide the functions of t and x : x √ x Dff = − √ tg √ t Dg , (13)two sides of the equality (13) is a constant k . We have (cid:40) Dff = k √ x → Lnf = k √ x + c → f = exp( k √ x + c ) Dgg = − k √ t → Lng = − √ t k + c → g = exp( − √ t k + c ) (14)The solution of the equation (10) has the form u ( x, t ) = exp( k √ x + c ) exp( − √ t k + c ) = c exp( k √ x + − √ t k ) . We define a generalized fractional derivative(GFD). We show that the previous derivativesare particular cases. We show how it is possible to have infinite fractional derivatives withtheir algebra. We present the fractional differential ring, the fractional partial derivativesand their applications.
Acknowledgement
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