Fractional Calculus - Gauge-institute.org

Gauge Institute Journal, Volume 5, No 1, February 2009H. Vic DannonThe Fundamental Theorem ofthe Fractional Calculus, and theMeaning ofFractional DerivativesH. Vic Dannonvic0@comcast.netSeptember, 2008Abstract We obtain the Fundamental Theorem of theFractional Calculus in the Arithmetic Means Calculus, andinterpret Fractional Derivatives in the Arithmetic MeansCalculus.Our derivation and interpretation can be extended to Fractionalderivatives in any other Power Mean Calculus.In particular, we define the Fractional Product Integral, theFractional Product Derivative, and prove the FundamentalTheorem of the Fractional Product Calculus.Keywords Fractional Calculus, Fractional Derivative, Fractional Integral,Arithmetic Mean Calculus, Product Calculus, Convolution, Tautochrone,2000 Mathematics Subject Classification 26A33, 26A03, 26A06, 44A35,58E10,1

Gauge Institute Journal, Volume 5, No 1, February 2009H. Vic DannonContentsIntroduction………………………………………………………………..…………41.Fractional Integral and Derivative………………………………………….61.1 Liouville’s Fractional Integral of order −1….…………………….………....621.2 Liouville’s Fractional Derivative of order 1 ………........................... ..........622. The Fundamental Theorem of the Fractional Calculus………….....82.1 Operational Derivation of the Fundamental Theorem……………...............82.2 Direct Derivation of the Fundamental Theorem…………………………….103. The Meaning of the Fractional Derivative…………..…………………..123.1 Slope of the Convolution Transform…………….……………….…………...123.2 Rate of change of the Convolution Transform………………….……………123.3 Arithmetic mean of the Convolution Transform……….............................133.4 Fractional Calculus, and Arithmetic Mean Calculus…...………………….133.5 The Meaning of Higher Order derivatives of order 1 2 …………………..….133.6 The Applicability of the Fractional Calculus………………………………...144. The Tautochrone Problem and the Fundamental Theorem………..154.1 The Tautochrone Problem………………………………………………………154.2 The Tautochrone differential Equation……………… ………………………164.3 The Tautochrone Integral Equation…………...……………………………...174.4 The Tautochrone Fractional Equation………………………………………..174.5 The Tautochrone and the Fundamental Theorem…………………………..174.6 The Cycloid Differential Equation…………………………………………….194.7 The Cycloid parametric Equation……………………………………………..192

Gauge Institute Journal, Volume 5, No 1, February 2009H. Vic Dannonν( a ν)= ( −1) Γ ( a + ν) x− + ,and concluded [Ross] thatdνν( dx)xΓ ( a + ν)xΓ()a−aν− ( a+ν)= ( −1)Many years later, the meaning of Fractional Derivatives is stillunclear.The common perception is that the Fractional Derivative Methodrepresents a new kind of Calculus, such as the Product Calculus,that is based on the Geometric Mean [Dan1],But so far, the Fractional Derivative Method has been developedonly as a refinement of the Arithmetic Means Calculus, and is nota new kind of Calculus, such as the Product Calculus is.Here, We interpret the Fractional Derivative in the context of theArithmetic Means Calculus in which it was presented.We notethat it can be similarly developed, and interpreted in the ProductCalculus.We proceed with the simplest fraction q =1. The Fractionalintegral2and the derivative−12f−1D212d≡ f ,−( dx)12 121df( dx) 2≡ Df.5

Gauge Institute Journal, Volume 5, No 1, February 2009H. Vic Dannon12 1≡212df( dx ( − a))as the convolution of the function f () x , with the Kernel functionDfx −32Γ− ( )12, over the interval [ ax , ]u31= x3 x− 2−We use the notations( )2∫ x −u f( u) du ≡ ∗ f( x).Γ− ( ) Γ− ( )1 12 u=a211 df21−32≡ ≡21 1 ∫ −2Df ( x u) fudu ( )Γ− ( )( dx ( − a))2u=xu=a7

Gauge Institute Journal, Volume 5, No 1, February 2009H. Vic Dannon2.The Fundamental Theorem of theFractional CalculusThe meaning of the Fractional derivative follows from theFundamental Theorem of the Fractional Calculus.This theorem is nowhere stated in the standard literature, suchas the books by Miller and Ross [Mil], or Oldham and Spanier[Old], or Podlubny [Pod], and the meaning of the fractionalderivative remains there unknown.To keep it simple, we focus here on the Fractional Calculus oforder 1 , and later of order 1 .2 3The Fundamental Theorem of the Fractional Calculus of order 1 2is1−12 2( )( )Dfx ( ) = DF xWe first outline the proof of this theorem by an operationalderivation. Then, we establish it with a direct proof.2.1 Operational Derivation of the Fundamental TheoremProof Outline:8

Gauge Institute Journal, Volume 5, No 1, February 2009H. Vic DannonSince the Fractional Derivative, and Fractional Integral operatorsapply successively [Old] under the ruleα β α+βd d df ( x) =f( x),( dx ( −a) ) ( dx ( −a) ) ( dx ( −a))α β α+βwe have the Fundamental Theorem of the Fractional Calculusd1 12 2d−( dx ( −a) ) ( dx ( −a))−1 12 2f ( x) = f( x).Applying to both sides the Fractional Derivative operatorwe have,d12( dx ( − a))121 1 1 12 2 2 2d d d df ( x) =f( x)−( dx ( −a) ) ( dx ( −a) ) ( dx ( −a) ) ( dx ( −a))−1 1 1 12 2 2 2Composing the first two operators, we conclude thatThat is,−1 12 2d d df ( x) =f( x)( dx ( − a))−( dx ( −a) ) ( dx ( −a))( )1 12 21 12 2( − )DF x = D f ( x).This outline becomes the proof if we supplement it with the proofof the exponent rule [Old]. Instead, we give a direct derivation ofthe Fundamental Theorem.9

Gauge Institute Journal, Volume 5, No 1, February 2009H. Vic Dannon2.2 Direct Derivation of the Fundamental TheoremProof:( −1) d dDF2 ( x ) =f ( x )−12( dx ( − a)) ( dx ( − a))d 1−1= ( x −u) 2f( u)du( − ) Γ()∫( dx a)12u=xu=a−121 d−1= ( x −u) 2f( u)duΓ()dx∫12u=xu=aBy Leibnitz Rule for Differentiation of Integrals, [Spieg, p. 95],ddxu= ψ( x) u=ψ( x)∫ Gxudu ( , ) = ∫ ∂xGxudu ( , ) + ϕ'( xGx ) ( , ϕ( x)) −ψ'( xGx ) ( , ψ( x)).u= ϕ( x) u=ϕ( x)If we take−12Gxu (, ) = ( x− u) fu (), then Gxx (, ) is not defined. Toremove the singularity, we integrate by partsu= x u=xu=x−1 1 1⎡⎤( x − u) 2f( u) du = 2( x u) 2f( u) 2 ( x u) 2⎢ − ⎥ − − f '( u)du⎣⎦∫ ∫u=au= a u=aTherefore,u=x1 12 2=−2( x −a) f( a) −2 ( x −u) f '( u)du∫u=a10

Gauge Institute Journal, Volume 5, No 1, February 2009H. Vic Dannon⎧u x(1)1 d=⎫−1 1DF2( x) =− ⎪2( x a) 2f ( a) 2 ( x u) 2⎨ − + − f '( u)du ⎪⎬Γ()1 dx∫2⎪⎩u=a⎪⎭⎧1u=x⎫1 d1−=− ⎪( x a) 2f( a) 2( x u) 2⎨ − + − f '( u)du⎪⎬Γ()1dx∫2⎪⎩u=a⎪⎭If we take Gxu (, ) = 2( x− u) 2fu (), then by Leibnitz rule1⎧1u=x⎫1 1−−=− ⎪( x a) 2f( a) ( x u) 2⎨ − + − f '( u)du⎪⎬Γ()1∫ (H )2⎪⎩u=a⎪⎭Integrating by parts with respect to uu= x u=xu=x−1 1 3⎡ − ⎤( )2'( ) ( )2( ) 1−x − u f u du = x u f u ( x u) 2⎢ − ⎥ + − f( u)du⎣⎦∫ ∫2u=au= a u=aSubstituting into (H ),−u=x1 32 1−2=−( x − a) f( a) + ( x −u) f( u)duu=x( − ) 11()1 22 u=aSince Γ− (1+ 1) =−1Γ− (1),∫2u=a1−32 2DF ( x) =− ( x −u) f ( u)duΓ∫ .2 2 2( −1)1−3DF2( x) = ( x −u) 2f ( u)duΓ− ( )∫112u=xu=a= Dfx2().11

Gauge Institute Journal, Volume 5, No 1, February 2009H. Vic Dannon3.The Meaning of the FractionalDerivativeThe Fundamental Theorem of the Fractional Calculus of order 1 2is the key to the meaning of the Fractional Derivative of order 1 . 2The following are interpretations of the Fundamental Theorem ofthe Fractional Calculus of order 1 , stated as21−12 2( )Dfx ( ) = DF ( x).3.1 Slope of the Convolution TransformThe Fractional Derivative of order 1 of f ( x ) at x is the slope of the2tangent to the curve of the Convolution Transform12( −F) ( x )at x .3.2 Rate of Change of the Convolution TransformThe Fractional Derivative of order 1 2of f () x at x is the rate ofChange of12( −F) ( x ) at x .12

Gauge Institute Journal, Volume 5, No 1, February 2009H. Vic DannonIn [Dan1], we presented the Arithmetic Mean Calculus, andinterpreted the Fermat-Newton-Leibnitz Derivative in it.In terms of the Arithmetic Mean Calculus, we have3.3 Arithmetic Mean of the Convolution TransformThe Fractional Derivative of order 1 2of f () x at x is the ArithmeticMean of the Convolution Transform12( −F) ( x )over [ xx , + dx].More generally,3.4 Fractional Calculus, and Arithmetic Mean CalculusThe Fractional Calculus of order 1 of f ( x ) is the Arithmetic Mean2Calculus of the Convolution Transform12( −F) ( x ) .3.5 The Meaning of Higher Derivatives of order 1 2The Fundamental Theorem of the Fractional Calculus of order 1 2enables us to interpret Fractional Derivatives of order n +1as2higher derivatives of the Convolution Transform12( −F) ( x ) . Inparticular,3 1 12 2 2 −2( )D f( x) = DD f( x) = D F ( x)5 1 12 2 2 3 −2( )D f( x) = D D f( x) = D F ( x)13

Gauge Institute Journal, Volume 5, No 1, February 2009H. Vic DannonThe meaning of order n +1Fractional Derivative follows from21 1 12 2 + 1 2n + n n ( − )D f( x) = D D f( x) = D F ( x).3.6 The Applicability of the Fractional CalculusThe meaning of the Fractional Derivative enables us to evaluatethe applicability of the Fractional Calculus in simple terms:The Fractional Derivative of order 1 2may be useful in a modelthat involves the Convolution Transform12( − )F ( x). That is,If you can see the Convolution Transform underlying your model,you may be well served by the Fractional Derivative.The solution of the Tautochrone Problem in the next section withthe aid of the Fundamental Theorem, shows thatthe required Convolution Transform modeling and identification isfar from self-evident.14

Gauge Institute Journal, Volume 5, No 1, February 2009H. Vic Dannon4.The Tautochrone Problem and theFundamental TheoremThe Tautochrone Problem is associated with Abel [Abel]. Itssolution by Abel involves the use of the Fundamental Theorem.4.1 The Tautochrone ProblemA bead of mass m slips along a frictionless wire,from heightto heightη = y , at time t = 0η = 0, at time t = T .We want to find the smooth curve -if there is such curve- alongwhich the bead’s falling time is the same for any y . To that end,15

Gauge Institute Journal, Volume 5, No 1, February 2009H. Vic Dannonwe look for a differential equation that will describe such motion.4.2 The Tautochrone Differential Equationds( y − η )=−2gdtProof:The potential energymg( y − η)converts into the kinetic energy122⎛ds⎞m ⎜⎜⎝ dt ⎠ ⎟,wheres= s( η)is the arc-length of the curve. Therefore,ds=± 2( gy− η).dtSince the particle speed is in the direction of− y , we havedsdt=− 2( gy−η)s'( η)dη( y − η )=−2gdt.Integrating the right side over the time interval, and the left overthe height interval we obtain an integral equation for the curve ofthe bead’s motion:16

Gauge Institute Journal, Volume 5, No 1, February 2009H. Vic Dannony( α) = (1− cos2 α),a2Proof:By 5.6,Substitutingwe have,xη=ya= ∫ −1dη.ηη=0η = a sin 2 θ,2y asinα= ,aη− 1 cos1 = θ12sin θ− = sin θ,dη = 2asinθcosθdθ,θ=α2x = 2a ∫ cos θdθθ=0θ=α∫= a (1 + cos 2 θ)dθ2θ=0=a(2α + sin 2 α).2y = asinα= (1 − cos 2 α).a2The Tautochrone problem is treated by Nahin [Nahin] bytraditional methods.20

Gauge Institute Journal, Volume 5, No 1, February 2009H. Vic Dannon1 1 23 3 + 1 3n + n n ( − )D fx () = DDfx () = D F () xSimilarly,5 2 13 3 2 −3( )Dfx ( ) = DDfx ( ) = DF ( x),and2 2 13 3 + 1 3n + n n ( − )D f( x) = D D f( x) = D F ( x).23

Gauge Institute Journal, Volume 5, No 1, February 2009H. Vic Dannon6.The Fractional Product CalculusWe developed the Product Calculus, or Geometric Mean Calculusin [Dan1]. The Product Derivative and the Product Integral ofthat Calculus are related by the Fundamental Theorem of theProduct Calculus [Dan1]The Fractional Derivative, and the Fractional Integral can bedefined in the context of the Product Calculus to yield a FractionalProduct Calculus.We outline here the development of the Fractional ProductCalculus of order 1 , with its Fundamental Theorem. We keep the2notations of [Dan1]6.1 Definition of Fractional Product Integral of Order 1 211 122 D−f x −2(0) −ln ( )( )( D ) f( x) ≡e ≡ G ( x)6.2 Definition of Fractional Product Derivative of Order 1 21212(0) ln ( )D f x( D ) f( x) ≡ e24

Gauge Institute Journal, Volume 5, No 1, February 2009H. Vic Dannon6.3 The Fundamental Theorem of the Fractional ProductCalculus of Order 1 2Proof:−{ }1 12 2(0) (0) (0)( D ) f( x) = D ( D ) f( x)⎧−1⎫(0)Dlog ⎪( D ) 2 f( x)⎪1⎨⎬(0) (0) − 2⎪⎩⎪⎭D ( D ) f( x)= e= e= e= eD⎧⎪⎨⎪⎩12 log ( )log D −⎪ef x−1DD 2 log f ( x )1D 2 log f ( x )(0)12= ( D ) f( x)⎫⎪⎬⎪⎭This leads to the interpretation of the Fractional Productderivative of order 1 2 .6.4 The Geometric Mean of the Convolution TransformThe Fractional Product Derivative of order 1 of f ( x ) at x is the2Geometric Mean of12( −G) ( x )over [ xx , + dx].25

Gauge Institute Journal, Volume 5, No 1, February 2009H. Vic DannonReferences[Abel] Abel, Niels, “Solution de quelques problemes a l’aide d’integralesdefinies”, Oeuvres Completes, Christiania (Grondahl) 1 1881, pp. 16-18.[Dan1] Dannon, Vic, “Power Mean Calculus, Product Calculus, HarmonicMean Calculus, and Quadratic Mean Calculus”, Gauge Institute Journal ofMath and Physics, Vol. 4, No. 4, November 2008.[Lac] Lacroix, S., Traite du Calcul Differentiel et du Calcul Integral, 2 ndedition, 1819, pp. 409-410.[Liou] Liouville, Joseph, “Memoire sur le Theoreme des complementaires”, J.Reine Angew. Math., 11, 1834, pp. 1-19[Mil] Miller, Kenneth, & Bertram Ross, “An Introduction to the FractionalCalculus and Fractional Differential Equations”. Wiley, 1993.[Nahin] Nahin, Paul, “When Least is Best”, Princeton University Press, 2004.[Old] Oldham, Keith, & Spanier, Jerome, “The Fractional Calculus”,Academic Press, 1974.[Ross] Ross, Bertram, “Fractional Calculus”, Mathematics Magazine Vol. 50No.3, May 1977.[Pod] Podlubny, I., “Fractional Differential Equations” Academic Press, 1999.26