Riemann-Liouville fractional derivatives of hidden variable recurrent fractal interpolation functions with function scaling factors and box dimension

Abstract

In [M.G. Ri and C.H. Yun, Smoothness and fractional integral of hidden variable recurrent fractal interpolation function with function vertical scaling factors, Fractals 29(6) (2021) 2150136], the authors proved that the partial and mixed Riemann-Liouville fractional integrals of bivariable hidden variable recurrent fractal interpolation function(HVRFIF) with function scaling factors are HVRFIFs. As its continuation, in this paper, we show that Riemann-Liouville fractional derivatives of one variable and bivariable HVRFIFs are HVRFIFs under certain conditions. We also derive the relationship between the order of fractional calculus and the upper box dimension of its graph. To do it, firstly, we prove that Riemann-Liouville fractional derivative of one variable HVRFIF is HVRFIF. Secondly, we obtain estimation of the upper box dimension of the graph of one variable HVRFIF and derive the relationship between the upper box dimension and the order of fractional calculus. Finally, in the similar way to one variable, we show that the partial and mixed Riemann-Liouville fractional derivatives of bivarible HVRFIF are HVRFIFs.