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FRACTAL DIMENSION ESTIMATION OF THE MARCHAUD FRACTIONAL DIFFERENTIAL OF CERTAIN CONTINUOUS FUNCTIONS

In this paper, we mainly investigate the fractional differential of a class of continuous functions. The upper Box dimension of the Marchaud fractional differential of continuous functions satisfying the Hölder condition increases at most linearly with the order of the fractional differential when they exist. Furthermore, if a continuous function satisfies the Lipschitz condition, the upper Box dimension of its Marchaud fractional differential is at most the sum of one and order of the fractional differential when it exists. From the point of view of the fractal dimension, it increases at most linearly with the fractional order.