ON THE BOX DIMENSION OF WEYL–MARCHAUD FRACTIONAL DERIVATIVE AND LINEARITY EFFECT.

This paper intends to estimate the box dimension of the Weyl–Marchaud fractional derivative (Weyl–M derivative) for various choices of continuous functions on a compact subset of ℝ. We show that the Weyl–M derivative of order γ of a continuous function satisfying Hölder condition of order μ also satisfies Hölder condition of order μ − γ and the upper box dimension of the Weyl–M derivative increases at most linearly with the order γ. Moreover, the upper box dimension of the Weyl–M derivative of a continuous function satisfying the Lipschitz condition is not more than the sum of the box dimension of the function itself and order γ. Furthermore, we prove that the box dimension of the Weyl–M derivative of a certain continuous function which is of bounded variation is one. [ABSTRACT FROM AUTHOR]