Fractional Calculus - Gauge-institute.org
Fractional Calculus - Gauge-institute.org
Fractional Calculus - Gauge-institute.org
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<strong>Gauge</strong> Institute Journal, Volume 5, No 1, February 2009H. Vic Dannon2.The Fundamental Theorem of the<strong>Fractional</strong> <strong>Calculus</strong>The meaning of the <strong>Fractional</strong> derivative follows from theFundamental Theorem of the <strong>Fractional</strong> <strong>Calculus</strong>.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 <strong>Fractional</strong> <strong>Calculus</strong> oforder 1 , and later of order 1 .2 3The Fundamental Theorem of the <strong>Fractional</strong> <strong>Calculus</strong> 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