RESEARCH AND MATHEMATICAL MODELING OF FRACTIONAL-DIFFERENTIAL RHEOLOGICAL MODELS

Deformation processes in media with fractal structure have been studied. At present, research on the construction of mathematical methods and models of interconnected deformation-relaxation and heat-mass transfer processes in environments with a fractal structure is at an early stage. There are a number of unsolved problems, in particular, the problem of correct and physically meaningful setting of initial and boundary conditions for nonlocal mathematical models of nonequilibrium processes in environments with fractal structure remains unsolved. To develop adequate mathematical models of heat and mass transfer and viscoelastic deformation in environments with fractal structure, which are characterized by the effects of memory, self-organization and spatial nonlocality, deterministic chaos and variability of rheological properties of the material, it is necessary to use non-traditional approaches. -differential operators. The presence of a fractional derivative in differential equations over time characterizes the effects of memory (eridity) or non-marking of modeling processes. The implementation of mathematical models can be carried out by both analytical and numerical methods. In particular, in this paper the integral form of fractional-differential rheological models is obtained on the basis of using the properties of the non-integer integral-differentiation operator and the Laplace transform method. The obtained analytical solutions of mathematical models of deformation in viscoelastic fractal media made it possible to obtain thermodynamic functions, creep nuclei and fractal-type relaxation. Developed software to study the effect of fractional differentiation parameters on the rheological properties of viscoelastic media.

NUMERICAL SIMULATIONS FOR THE VARIABLE ORDER TWO-DIMENSIONAL REACTION SUB-DIFFUSION EQUATION: LINEAR AND NONLINEAR

The applications and the fields that use the anomalous sub-diffusion equations cannot be easily listed due to their wide area. Sure, one of the main physical reasons for using and researching fractional order diffusion equations is to explain anomalous diffusion that occurs in transport processes through complex and/or disordered structures, such as fractal media. One of the important applications is their use in chemical reactions, where a single material continues to shift from a high concentration area to a low concentration area until the concentration across the space is equal. The mathematical model that describes these physical-chemical phenomena is called the reaction sub-diffusion equation. In our study, we try to solve the 2D variable order version of these equations (2DVORSE) (linear and nonlinear) by using an accurate numerical technique which is the variable weighted average finite difference method (WAFDM). We will analyze the stability of the resulting scheme by using a modified suitable version of the John Von Neumann procedure. Specific stability conditions that occur or some parameters in the resulting schemes are derived and checked. At the end of the study, numerical examples are simulated to check the stability and the accuracy of the proposed technique.

Analytical solution to local fractional Landau-Ginzburg-Higgs equation on fractal media

The main objective of the present article is to introduce a new analytical solution of the local fractional Landau-Ginzburg-Higgs equation on fractal media by means of the local fractional variational iteration transform method, which is coupling of the variational iteration method and Yang-Laplace transform method.

Dynamics of charged particles in fractal media

We consider the influence of the fractality of a medium on the dynamics of charged particles in the external magnetic field and show a significant increase in the sensitivity to weak effects with the fractality of a medium. The dynamical processes in the nonequilibrium nonconservative medium are analyzed in the framework of the Tsallis nonextensive thermodynamics with the use of Jackson’s derivatives.

Thermo-poromechanics of fractal media

This article advances continuum-type mechanics of porous media having a generally anisotropic, product-like fractal geometry. Relying on a fractal derivative, the approach leads to global balance laws in terms of fractal integrals based on product measures and, then, converting them to integer-order integrals in conventional (Euclidean) space. Proposed is a new line transformation coefficient that is frame invariant, has no bias with respect to the coordinate origin and captures the differences between two fractal media having the same fractal dimension but different density distributions. A continuum localization procedure then allows the development of local balance laws of fractal media: conservation of mass, microinertia, linear momentum, angular momentum and energy, as well as the second law of thermodynamics. The product measure formulation, together with the angular momentum balance, directly leads to a generally asymmetric Cauchy stress and, hence, to a micropolar (rather than classical) mechanics of fractal media. The resulting micropolar model allowing for conservative and/or dissipative effects is applied to diffusion in fractal thermoelastic media. First, a mechanical formulation of Fick’s Law in fractal media is given. Then, a complete system of equations governing displacement, microrotation, temperature and concentration fields is developed. As a special case, an isothermal model is worked out. This article is part of the theme issue ‘Advanced materials modelling via fractional calculus: challenges and perspectives’.