Non-local Integrals and Derivatives on Fractal Sets with Applications
aa r X i v : . [ m a t h . C A ] J a n Non-local Integrals and Derivatives on FractalSets with Applications
Alireza K. Golmankhaneh † , Dumitru Baleanu , ‡ August 13, 2018 Young Researchers and Elite Club, Urmia Branch,Islamic Azad university, Urmia, Iran. † E-mail address : [email protected] Department of MathematicsC¸ ankaya University, 06530 Ankara, Turkey Institute of Space Sciences,P.O.BOX, MG-23, R 76900, Magurele-Bucharest, Romania ‡ E-mail address : [email protected]
Abstract
In this paper, we discussed the non-local derivative on the fractalCantor set. The scaling properties are given for both local and non-local fractal derivatives. The local and non-local fractal dierentialequations are solved and compared and related physical models aresuggested.
Keywords:
Fractal calculus, Non-local fractal derivatives, Scale change,Cantor set, Fractal dimension
Fractional calculus became an important tool which was applied successfullyin many branches of science and engineering etc [1, 2, 3, 4, 5]. The modelsbased on fractional derivatives are crucial for describing the processes withmemory effect [6]. Local fractional has been defined on the real-line [7]. As itis well known the integer, fractional and complex order derivatives and inte-grals are defined on the real-line. Analysis on the fractal has been studied bymany researchers [8, 9, 10]. The fractals curves and the functions on fractalspace are not differentiable in the sense of standard calculus. As a result, bythis motivation recently in the seminal paper the F α -calculus is suggestedas a framework on the fractal sets and fractal curves [11, 12, 13, 14]. The F α -calculus is generalized and applied in physics as a new and useful tool formodelling processes on the fractals. Newtonian mechanics and Schr¨ o dinger1quation on the fractal sets and curves are given [15, 16, 17]. The gauge in-tegral is utilized to generalized the F α -calculus for the unbound and singularfunction [18]. The fractal grating is modeled by F α -calculus and correspond-ing diffraction is presented [18]. One of the important aspects of fractionalcalculus was transferred recently to the fractal derivatives. Namely, the con-cept of non-local fractal derivatives was introduced in [20]. In this manuscriptour main aim is to define the fractal non-local derivatives and study theirproperties.The plane of this work is as follows:In Section 2 we summarize the basic definitions and properties of the thelocal fractional derivatives. In Section 3 the scaling properties of local andnon-local derivatives are derived. More, in Section 4 we develop the the-ory of fractal local and non-local Laplace transformations. In Section 5 thecomparison of local and non-local linear fractal differential equations arepresented. In Section 6 we indicate some illustrative applications. Section 7contains our conclusion. In this section we recall some basic definitions and properties of the localfractal calculus (LFC) and non-local fractal calculus (NLFC) [11, 20].
In the seminal paper local F α -calculus is built on fractal Cantor set which isshown in Figure [1] [11]. Figure 1:
We present triadic Cantor set by iteration.
The integral staircase function S αF ( x ) of order α for the triadic Cantor set F is defined in [11] by S αF ( x ) = ( γ α ( F, a , x ) if x ≥ a − γ α ( F, a , x ) otherwise , (1)where a is an arbitrary real number. The graph of the integral staircasefunction is depicted in Figure [2]. 2 S . C ( x ) Figure 2:
We indicate the integral staircase function for a triadic Cantor set F . Then F α -derivative is defined for a function with this support as follows[11] D αF f ( x ) = ( F − lim y → x f ( y ) − f ( x ) S αF ( y ) − S αF ( x ) if x ∈ F, , otherwise , (2)if the limit exists. For more details we refer the reader to [11]. In this section, we review the non-local derivatives and basic definitions [20].
Definition 1.
A function f ( S αF ( x )) , x > C F,ρ , ρ ∈ ℜ ifthere exists a real number p > ρ , such f ( S αF ( x )) = S αF ( x ) p f ( S αF ( x )), where f ( S αF ( x )) ∈ C αF [ a, b ], and it is in the C nαF,ρ [ a, b ] if and only if( D αF ) n f ( S αF ( x )) ∈ C F,ρ , n ∈ N. (3)Here and subsequently, we define the fractal left-sided Riemann-Liouvilleintegral as follows a I βx f ( x ):= 1Γ αF ( β ) Z S αF ( x ) S αF ( a ) f ( t )( S αF ( x ) − S αF ( t )) α − β d αF t. (4)where S αF ( x ) > S αF ( a ). Definition 2.
The fractal left-sided Riemann-Liouville derivative is definedas a D βx f ( x ):= 1Γ αF ( n − β ) ( D αF ) n Z S αF ( x ) S αF ( a ) f ( t )( S αF ( x ) − S αF ( t )) − nα + β + α d αF t. (5)3 efinition 3. For A f ( x ) ∈ C αn [ a, b ] , nα − α ≤ β < αn the fractal left-sidedCaputo derivative is defined as Ca D βx f ( x ):= 1Γ αF ( n − β ) Z S αF ( x ) S αF ( a ) ( S αF ( x ) − S αF ( t )) nα − β − α ( D αF ) n f ( t ) d αF t. (6) Definition 4.
The fractal Gr¨unwald and Marchaud derivative of a function f ( x ) with support of fractal sets is defined as G D β f ( x ) = F − lim n →∞ αF ( − β ) (cid:18) S αF ( x ) n (cid:19) − β n − X k =0 Γ αF ( k − β )Γ αF ( k + 1) f (cid:18) S αF ( x ) − k S αF ( x ) n (cid:19) . Definition 5.
The generalized fractal standard Mittag-Leffler functions isdefined as [20] E αF,η ( x ) = ∞ X k =0 S αF ( x ) k Γ αF ( ηk + 1) , η > , ν ∈ ℜ . (7)The fractal two parameter η, ν Mittag-Liffler function is defined as E αF,η,ν ( x ) = ∞ X k =0 S αF ( x ) k Γ αF ( ηk + ν ) , η > , ν ∈ ℜ . (8) Definition 6.
For a given function f ( S αF ( x )) the fractal Laplace transformis denoted by F ( s ) and defined as [20] F αF ( S αF ( s )) = L αF [ f ( x )] = Z S αF ( ∞ ) S αF (0) f ( x ) e − S αF ( s ) S αF ( x ) d αF x, (9)where S αF ( s ) is limited by the values that the integral converges. The function f ( S αF ( x )) is F -continuous and has following conditionsup | f ( S αF ( x )) | e S αF ( c ) S αF ( x ) < ∞ , S αF ( c ) ∈ ℜ , S αF ( x ) > . (10)In view of the above conditions the fractal Laplace transform exists forall S αF ( s ) > S αF ( c ). We follow the notation as L αF [ f ( x )] = F αF ( S αF ( s )) and L αF [ g ( x )] = G αF ( S αF ( s )). Remark 1.
We denote that if we choose β = α then we have a D αx f ( x ) = D αF,x f ( x ) | x = S αF ( a ) . (11)4 Scale properties of fractal local and non-local fractal calculus
In this section we study the scale properties of the LFC and NLFC.
A function f ( S αF ( x )) is called fractal homogenous of degree- mα or invariantunder fractal rescalings if we have f ( S αF ( λx )) = λ mα f ( S αF ( x )) , (12)where for some m and for all λ . The fractals have self-similar properties,namely for the case of function with the fractal Cantor set support we choose m = 1 and λ = 1 / n , n = 1 , , ... then f ( S αF ( 13 n x )) = ( 13 n ) α f ( S αF ( x )) , (13)where α = 0 . f ( S αF ( x )) rescaling as follows D αF f ( S αF ( λx )) = λ mα − α f ( S αF ( x )) . (14) By a scale change of the fractal function f ( S αF ( x )), we mean converts x → λx ⇒ S αF ( λx ) = λ α S αF ( x ) , (15)and using Eq. (5) and choosing a = 0 we derive D βx ( f ( S αF ( λx ))) = λ βα D βλx ( f ( S αF ( λx ))) , (16)which is called scale change on the non-local fractal derivatives. Let us give some important lemmas which are useful for finding the fractalLaplace transforms of function f ( S αF ( x )). Lemma 1.
The fractal Laplace transform of the non-local fractal Caputoderivative of order mα − α < β ≤ mα , m ∈ N is L αF { C D βx f ( x ) } = ( S αF ( s )) mα F αF ( s ) − ( S αF ( s )) mα − α f ( S αF (0)) S αF ( s ) mα − β × − ( S αF ( s )) mα − α D αx f ( x ) | x = S αF (0) − . . . − D mα − αx f ( x ) | x = S αF (0) . (17)5 roof: We first compute the Laplace fractal transform of the fractal Caputofractional derivative of order β as follows L αF { C D βx f ( x ) } = L αF { I mα − βx ( D αx ) m f ( x ) } = L αF [( D αx ) m f ( x )] s mα − β (18)In view of Eq. (28) which completes the proof. Lemma 2.
For a given ζ , µ > , S αF ( a ) ∈ ℜ and S αF ( s ) ζ > | S αF ( a ) | the fractalLaplace transform is L α, − F (cid:20) S αF ( s ) ζ − µ S αF ( s ) ζ + S αF ( a ) (cid:21) = S αF ( x ) µ − E αF,ζ,µ ( − S αF ( a ) S αF ( x ) ζ ) . (19) Proof:
Using the series expansion we have S αF ( s ) ζ − µ S αF ( s ) ζ + S αF ( a ) = 1 S αF ( s ) µ
11 + S αF ( a ) S αF ( s ) ζ (20)= 1 S αF ( s ) µ ∞ X n =0 (cid:18) − S αF ( a ) S αF ( s ) ζ (cid:19) n = ∞ X n =0 ( − S αF ( a )) n S αF ( s ) nζ + µ (21)The inverse fractal Laplace transform of Eq. (20) leads to ∞ X n =0 ( − S αF ( a )) n S αF ( x ) nζ + µ − Γ αF ( nζ + µ )= S αF ( x ) µ − ∞ X n =0 ( − S αF ( a ) S αF ( x ) ζ ) n Γ αF ( nζ + µ )= S αF ( x ) µ − E αF,ζ,µ ( − S αF ( a ) S αF ( x ) ζ ) . (22) Lemma 3.
Suppose ζ ≥ µ > S αF ( a ) ∈ ℜ and S αF ( s ) ζ − µ > | S αF ( a ) | then wehave L α, − F (cid:20) S αF ( s ) ζ + S αF ( a ) S αF ( s ) µ ) n +1 (cid:21) = S αF ( x ) ζ ( n +1) − ∞ X k =0 − ( S αF ( a )) k Γ αF ( k ( ζ − µ ) + ( n + 1) ζ ) (cid:18) n + kk (cid:19) S αF ( x ) k ( ζ − µ ) . (23)6 roof: Let us use following expression1(1 + S αF ( x )) n +1 = ∞ X k =0 (cid:18) k + nk (cid:19) ( − S αF ( x )) k . (24)Therefore we can write 1( S αF ( s ) ζ + S αF ( a ) S αF ( s ) µ ) n +1 = 1( S αF ( s ) ζ ) n +1 S αF ( a ) S αF ( s ) ζ − µ ) n +1 = 1( S αF ( s )) n +1 ∞ X k =0 (cid:18) n + kk (cid:19) (cid:18) − S αF ( a ) S αF ( s ) ζ − µ (cid:19) k . The proof is complete.
Lemma 4.
For ζ ≥ µ, ζ > ξ, S αF ( a ) ∈ ℜ , S αF ( s ) ζ − µ > | S αF ( a ) | and | S αF ( s ) ζ + S αF ( a ) S αF ( s ) µ | we have L α, − F (cid:20) S αF ( s ) ξ S αF ( s ) ζ + S αF ( a ) S αF ( s ) µ + S αF ( b ) (cid:21) = S αF ( x ) ζ − ξ − ∞ X n =0 ∞ X k =0 ( − S αF ( b )) n ( − S αF ( a )) k Γ αF ( k ( ζ − µ ) + ( n + 1) ζ − ξ ) (cid:18) n + kk (cid:19) S αF ( x ) k ( ζ − µ )+ nζ . (25) Proof:
Since we can write S αF ( s ) ξ S αF ( s ) ζ + S αF ( a ) S αF ( s ) µ + S αF ( b )= S αF ( s ) ξ S αF ( s ) ζ + S αF ( a ) S αF ( s ) µ
11 + S αF ( b ) S αF ( s ) ζ + S αF ( a ) S αF ( s ) µ = ∞ X n =0 S αF ( s ) ξ ( − S αF ( b )) n S αF ( s ) ζ + S αF ( a ) S αF ( s ) µ , (26)according to the Lemma 3. the proof is complete. Some important formulas of the local fractal calculus are given elow : [11, 20]: L αF [ S αF ( x ) n ] = Γ αF ( n + 1) S αF ( s ) n +1 , L αF "Z S αF ( x ) S αF (0) f ( S αF ( t )) d αF t = L αF [ I αx f ( S αF ( t ))]= F αF ( s ) s , L αF [ S αF ( x ) n f ( S αF ( x ))] = ( − n ( D αF ) n F αF ( s ) , L αF "Z S αF ( x ) S αF (0) f ( S αF ( x ) − S αF ( t )) g ( S αF ( t )) d αF t = F αF ( S αF ( s )) G αF ( S αF ( s )) , (27)and L αF [( D αF ) n f ( S αF ( x ))]= ( S αF ( s )) nα F αF ( s ) − ( S αF ( s )) nα − f ( S αF (0)) − ( S αF ( s )) nα − D αF f ( x ) | x = S αF (0) − . . . − ( D αF ) n − f ( x ) | x = S αF (0) . (28) Remark 2.
If we choose α = 1 we obtain the standard result. The important formulas of the non-local fractal calculus are asfollows [20]: I βx ( S αF ( x )) η = Γ αF ( η + 1)Γ αF ( η + β + 1) ( S αF ( x )) η + β , D βx ( S αF ( x )) η = Γ αF ( η + 1)Γ αF ( η − β + 1) ( S αF ( x )) η − β . D βx ( c χ αF ) = c Γ αF (1 − β ) ( S αF ( x )) − β , L αF [ I βx f ( x )] = F αF ( S αF ( s )) S αF ( s ) β . (29)where c is constant. Remark 3.
If we choose β = α then we arrive at to the local fractal deriva-tive whose order is equal the dimension of the fractal. In this section, we compare the local and non-local fractal differential equa-tions. 8 xample 1.
Consider linear local fractal differential equation as D αF y ( x ) + y ( x ) = 0 , (30)with the initial-value y ( x ) | x = S αF (0) = 1 , (31) y ( x ) y ( x ) = e − x y ( x ) = e − S αF ( x ) Figure 3:
We plot the solution of Eq. (30).
Hence the solution to Eq. (30) is y ( x ) = e − S αF ( x ) , (32)where α = 0 . γ -dimension of the triadic Cantor set [11, 20].In Figure 3 we give the graph of Eq. (32). Example 2.
Consider linear non-local fractal differential equation as C D βx y ( x ) + y ( x ) = 0 , (33)with the initial condition y ( x ) | x = S αF (0) = 1 , D αF y ( x ) | x = S αF (0) = 0 . (34)In view of Eq. (17) we have L αF { C D βx f ( x ) } = ( S αF ( s )) α F αF ( s ) − S αF ( s ) α − β . (35)9 y ( x ) (a) If we choose β = 0 .
33 in Eq. (38) y ( x ) (b) If we choose β = 0 .
25 in Eq. (38)
Figure 4:
We draw the graph of Eq. (38).
Applying the fractal Laplace transformation on the both sides of Eq. (33)and using Eq. (17) we obtain( S αF ( s )) α F αF ( s ) − S αF ( s ) α − β + F αF ( s ) = 0 . (36)It follows that F αF ( s ) = S αF ( s ) β − α S αF ( s ) β , (37)using the fractal inverse Laplace transform Eq. (19) we arrive at the solutionof Eq. (33) as follows y ( x ) = S αF ( x ) α − E αF,β,α (cid:0) − S αF ( x ) β (cid:1) . (38)In Figure 4 we present the graph of Eq.( 38). In this section we give the applications and new models are given to thenon-local fractal derivatives [20].
Fractal Abel’s tautochrone:
As a first example we generalized Abel’sproblem which is the curve of quick descent on the fractal time-space. Usingthe conservation of energy in the fractal space the differential equation of themotion a particle is D αF,t s αF = d αF s αF d αF t = − q g αF ( S αF ( y ) − S αF ( y )) , (39)10here s αF is fractal arc length, and g αF fractal space gravitational constant,and y is the high particle from the reference of potential. As a result we have S αF ( T ) = − p g αF Z S αF ( B ) S αF ( A ) p ( S αF ( y ) − S αF ( η )) d αF s αF . (40)Let us consider s αF = h αF ( S αF ( η )) , (41)so that we have S αF ( T ) = − p g αF Z S αF (0) S αF ( y ) ( S αF ( y ) − S αF ( η )) − / D αF,η h αF ( η ) d αF η. (42)Utilizing D αF,η h αF ( S αF ( y )) = f ( S αF ( y )) we arrive at S αF ( T ) = − p g αF Z S αF (0) S αF ( y ) ( S αF ( y ) − S αF ( η )) − / f ( S αF ( y )) d αF η. (43)It follows p g αF Γ( ) S αF ( T ) = D / y f ( y ) . (44)The solution of Eq.(44) is called the fractal cycloid. Fractal models for the viscoelasticity:
We generalize the viscoelastic-ity models to the fractal mediums in the case of ideal solids and ideal liq-uids. Namely, the fractal ideal solids describe by σ αF ( t ) = E αF ǫ αF ( t ) , (45)which is called Hooke’s Law of fractal elasticity. Where σ αF is fractal stress, ǫ αF is fractal strain which occurs under the applied stress and E αF is the elasticmodulus of the fractal material.The fractal ideal fluid can model and describe by Newton’s Law of fractalviscosity as follows σ αF ( t ) = λ αF D αF ǫ αF ( t ) , (46)where λ αF is the viscosity of the fractal material. But in the nature we havereal martials which have properties between the ideal solids and ideal liquids.It is clear that in the Hooke’s Law of fractal elasticity Eq. (45) fractal stress isproportional to the 0-order derivative of the fractal strain and in the Newton’sLaw of fractal viscosity the stress is proportional to the α -order derivative ofthe fractal strain. Therefore, more general model is σ αF ( t ) = E αF ( χ αF ) β D βx ǫ αF ( t ) , χ αF = λ αF E αF , (47)11hich is called fractal Blair’s model. Here, we suggest the fractional non-local order fractal derivative β as an index of memory. Namely, if we choose β = 0 in the process is nothing forgotten and the case of β = α the processis memoryless. Hence if we choose 0 < β < α it shows the processes withmemory on the fractals.If we choose ǫ αF ( t ) = χ αF , (48)where χ αF is characteristic function of the triadic Cantor set. In Figure 5 weplot the ǫ αF ( t ). ε F α Figure 5:
We sketch ǫ αF ( t ) = χ αF which is characteristic function of the triadicCantor set. Utilizing Eq. (47) we obtain the fractal stress as follows σ αF ( t ) = E αF ( χ αF ) β αF (1 − β ) ( S αF ( t )) − β . (49) σ F α ( t ) σ αF ( t ) ≈ αF (1 − β ) ( S αF ( t )) − β σ αF ( t ) ≈ − β ) ( t ) − β Figure 6:
We sketch σ αF ( t ) for the fractal stress substituting β = 0 .
12n Figure 6 we show the graph of σ αF ( t ) fractal stress. Remark 4.
If we choose β = 0 and β = α in Eq. (47) we will have thefractal stress and the fractal strain relations for the cases of fractal idealsolids and the fractal ideal fluids, respectively. In this paper we generalized the fractal calculus involving the non-localderivatives. The scaling properties of the local and non-local derivativesare studied because they are important in physical applications. Using anillustrative example we compared the local and non-local linear fractal differ-ential equations. We also suggested some applications for the new non-localfractal differential equations.
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