ESTIMATION OF FRACTAL DIMENSION OF FRACTIONAL CALCULUS OF THE HoLDER CONTINUOUS FUNCTIONS

In the present paper, fractal dimension and properties of fractional calculus of certain continuous functions have been investigated. Upper Box dimension of the Riemann-Liouville fractional integral of continuous functions satisfying the Holder condition of certain positive orders has been proved to be decreasing linearly. If sum of order of the Riemann-Liouville fractional integral and the Holder condition equals to one, the Riemann-Liouville fractional integral of the function will be Lipschitz continuous. If the corresponding sum is strictly larger than one, the Riemann-Liouville fractional integral of the function is differentiable. Estimation of fractal dimension of the derivative function has also been discussed. Finally, the Riemann-Liouville fractional derivative of continuous functions satisfying the Holder condition exists when order of the Riemann-Liouville fractional derivative is smaller than order of the Holder condition. Upper Box dimension of the function has been proved to be increasing at most linearly.