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An Expansion Formula for Fractional Derivatives and its Application

Atanackovic, T., Stankovic, B. (2004)

Fractional Calculus and Applied Analysis

An expansion formula for fractional derivatives given as in form of a series involving function and moments of its k-th derivative is derived. The convergence of the series is proved and an estimate of the reminder is given. The form of the fractional derivative given here is especially suitable in deriving restrictions, in a form of internal variable theory, following from the second law of thermodynamics, when applied to linear viscoelasticity of fractional derivative type.

An extension of a theorem of Marcinkiewicz and Zygmund on differentiability

S. Mukhopadhyay, S. Mitra (1996)

Fundamenta Mathematicae

Let f be a measurable function such that Δ k ( x , h ; f ) = O ( | h | λ ) at each point x of a set E, where k is a positive integer, λ > 0 and Δ k ( x , h ; f ) is the symmetric difference of f at x of order k. Marcinkiewicz and Zygmund [5] proved that if λ = k and if E is measurable then the Peano derivative f ( k ) exists a.e. on E. Here we prove that if λ > k-1 then the Peano derivative f ( [ λ ] ) exists a.e. on E and that the result is false if λ = k-1; it is further proved that if λ is any positive integer and if the approximate Peano derivative...

Currently displaying 561 – 580 of 4583