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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
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Fractional Calculus - Gauge-institute.org

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