THE EFFECTS OF THE RIEMANN–LIOUVILLE FRACTIONAL INTEGRAL ON THE BOX DIMENSION OF FRACTAL GRAPHS OF HÖLDER CONTINUOUS FUNCTIONS
Junru Wu
Abstract
In this paper, the linearity of the dimensional-decrease effect of the Riemann–Liouville fractional integral is mainly explored. It is proved that if the Box dimension of the graph of an [Formula: see text]-Hölder continuous function is greater than one and the positive order [Formula: see text] of the Riemann–Liouville fractional integral satisfies [Formula: see text], the upper Box dimension of the Riemann–Liouville fractional integral of the graph of this function will not be greater than [Formula: see text]. Furthermore, it is proved that the Riemann–Liouville fractional integral of a Lipschitz continuous function defined on a closed interval is continuously differentiable on the corresponding open interval.
Topics & Concepts
MathematicsDifferentiable functionLipschitz continuityDimension (graph theory)Interval (graph theory)Fractional calculusGraphFractal dimensionMathematical analysisPure mathematicsFractalHölder conditionRiemann hypothesisRiemann integralIntegral equationCombinatoricsFourier integral operatorFractional Differential Equations SolutionsNonlinear Differential Equations AnalysisAdvanced Differential Equations and Dynamical Systems