Fractional Calculus - Gauge-institute.org

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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
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Fractional Calculus - Gauge-institute.org

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