fractional calculus

Generalized Equations for Ion Transport in Aqueous Solutions Through Membrane Structures: І. Electromagnetic and Subdiffusion Processes

A system of generalized transport equations for ions and water molecules in the system "initial aqueous solution of electrolyte – membrane – filtrate" is obtained, which describes diffusion electromagnetic processes through Maxwell's equations.  The method of non-equilibrium statistical operator Zubarev is used.  Generalized transport equations for ions of the $\bar{a}$ variety, which must be retained by the membrane for the purpose of selectivity, are identified.  Using the method of fractional calculus and modeling the corresponding memory function, a generalized subd

Numerical solution for fractional differential equations by using Jacobi–Gauss–Radau collocation method

This study proposes a novel numerical approach for addressing both linear and nonlinear initial fractional order differential equations (FDEs) through the implementation of the Jacobi–Gauss–Radau (JGR) integrated with Caputo fractional derivatives.  The problem is effectively transformed into a simplified system of FDEs, encompassing the unknown coefficients, by employing shifted JGR points for the FDEs and their initial conditions.  For the purpose of investigating the effectiveness and accuracy of the introduced method, some numerical illustrations are provided for various linear and nonl

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

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

A nonlinear fractional partial differential equation for image inpainting

Image inpainting is an important research area in image processing.  Its main purpose is to supplement missing or damaged domains of images using information from surrounding areas.  This step can be performed by using nonlinear diffusive filters requiring a resolution of partial differential evolution equations.  In this paper, we propose a filter defined by a partial differential nonlinear evolution equation with spatial fractional derivatives.  Due to this, we were able to improve the performance obtained by known inpainting models based on partial differential equations and extend certa