The mathematical fractional modeling of TiO_2 nanopowder synthesis by sol–gel method at low temperature

Abstract

Titanium dioxide is a compound of oxygen and titanium with the formula TiO2 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 TiO2 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 anatase phase at a crystal size of 49.25 nm with success using titanium tetraisopropoxide (TTIP) as precursor by the sol–gel method. The powders obtained were analyzed by X-ray diffraction (XRD) with CuKα radiation (λ=0.15406 nm) and Fourier transform infrared spectroscopy (FTIR) in the wave number range 4000−400 cm−1, and on the other hand we present a mathematical model for the prediction of the TiO2 concentration as a function of time and the concentration of reactants by using the fractional order derivative more precise than the whole order derivative, we study the existence and the uniqueness of the solutions. In addition, we determine the points of equilibrium. Numerical simulations and their graphical representations are made to visualize the efficiency of this model.

The general Bernstein function: Application to χ‐fractional differential equations

“…□ Theorem 2. Let 𝑦(t) and 𝑦 m (t) represent the exact and approximate solutions of (5), respectively. The residual correction can be assigned to the estimation of the absolute error using the following procedure: [47] This error estimation can be used to check the accuracy of the results in the absence of knowledge of the exact 𝜒-FDE solution.…”

“…Similarly, the fractional integral of a function represents the integration of that function over a fractional order. Fractional calculus is a powerful tool for modeling [5] and understanding various physical, biological [6], and engineering processes such as viscoelasticity, diffusion, signal processing, and control systems [7][8][9]. Fractional calculus is used in many fields, including physics, engineering, finance, control theory, signal processing, and image processing.…”

DFT study on the electronic, structure, magnetic and optical properties of TiO2 anatase

“…TiO 2 is an N-type semiconductor, which can be formed in three crystalline phases, rutile (tetragonal structure), brookite (orthorhombic structure) and anatase (tetragonal structure). In general, TiO2 is preferred in the anatase form due to its strong photocatalytic activity, as it has a more negative conduction band edge potential, is photochemically stable, has a high speci c surface area, is non-toxic, and relatively inexpensive [7][8][9].…”

Stability characterization of a fractional-order viral system with the non-cytolytic immune assumption

“…Also, another advantage of this definition is that the Caputo derivative of a constant is zero. Memory effect is an essential characteristic of fractional-order derivatives which made fractional calculus and its applications widely used in many fields of science and engineering [11,12,13,14,15,16,17]. Obviously, this feature is very relevant for modeling the spread of infections [18,19,20,21,22,23,24].…”