Caputo derivatives

A new mathematical model of the nonstationary process of heat and moisture transfer in the two-dimensional region is constructed on the basis of the use of Caputo and Grunwald- Letnikov derivatives. An implicit finite-difference scheme for approximation of a mathematical model of noisothermal moisture transfer taking into account the fractional integro-differential apparatus is developed. The given algorithm of numerical realization of model allows to receive values of function of temperature and humidity for all points of area of partition.

Titanium dioxide is a compound of oxygen and titanium with the formula TiO$_2$ present in nature and manufactured on an industrial scale.  It is used in several fields and applications such as cosmetics, paint, food, photocatalyst, electrodes in lithium batteries, dye solar cells (DSSC), biosensors, etc., given its importance and its various fields of application, there are several methods of synthesis of TiO$_2$ such as the sol–gel method widely used to obtain nanoparticles.  In our study, on the one hand we synthesized titanium dioxide nanopowders crystallized in the