ANALYSIS OF MIXED WEYL-MARCHAUD FRACTIONAL DERIVATIVE AND BOX DIMENSIONS

The calculus of the mixed Weyl-Marchaud fractional derivative has been investigated in this paper. We prove that the mixed Weyl-Marchaud fractional derivative of bivariate fractal interpolation functions (FIFs) is still bivariate FIFs. It is proved that the upper box dimension of the mixed Weyl-Marchaud fractional derivative having fractional order gamma = (p,q) of a continuous function which satisfies mu-Holder condition is no more than 3 - mu + (p + q) when 0 < p, q < mu < 1, p + q < mu, which reveals an important phenomenon about linearly increasing effect of dimension of the mixed Weyl-Marchaud fractional derivative. Furthermore, we deduce box dimension of the graph of the mixed Weyl-Marchaud fractional derivative of a continuous function which is defined on a rectangular region in Double-struck capital R-2 and also, we analyze that the mixed Weyl-Marchaud fractional derivative of a function preserves some basic properties such as continuity, bounded variation and boundedness. The results are new and compliment the existing ones.