Mathematical Modeling of the Fractional Differential Dynamics of the Relaxation Process of Convective Diffusion Under Conditions of Planned Filtration

2015 ◽  
Vol 51 (6) ◽  
pp. 886-895 ◽  
Author(s):  
V. M. Bulavatsky ◽  
V. A. Bogaenko
Keyword(s):  
Mathematical Modeling ◽  
Relaxation Process ◽  
Convective Diffusion ◽  
Fractional Differential

scholarly journals Mathematical Modeling of Solutes Migration under the Conditions of Groundwater Filtration by the Model with the k-Caputo Fractional Derivative

2018 ◽  
Vol 2 (4) ◽  
pp. 28 ◽  
Author(s):  
Vsevolod Bohaienko ◽  
Volodymyr Bulavatsky
Keyword(s):  
Caputo Derivative ◽  
Numerical Experiments ◽  
Convective Diffusion ◽  
Time Lag ◽  
Time Variable ◽  
Industrial Wastes ◽  
Diffusion Field ◽  
One Dimensional ◽  
Fractional Differential ◽  
Field Formation

Within the framework of a new mathematical model of convective diffusion with the k-Caputo derivative, we simulate the dynamics of anomalous soluble substances migration under the conditions of two-dimensional steady-state plane-vertical filtration with a free surface. As a corresponding filtration scheme, we consider the scheme for the spread of pollution from rivers, canals, or storages of industrial wastes. On the base of a locally one-dimensional finite-difference scheme, we develop a numerical method for obtaining solutions of boundary value problem for fractional differential equation with k-Caputo derivative with respect to the time variable that describes the convective diffusion of salt solution. The results of numerical experiments on modeling the dynamics of the considered process are presented. The results that show an existence of a time lag in the process of diffusion field formation are presented.

scholarly journals Mathematical modeling of the smoking dynamics using fractional differential equations with local and nonlocal kernel

2018 ◽  
Vol 11 (08) ◽  
pp. 994-1014 ◽  
Author(s):  
V. F. Morales-Delgado ◽  
J. F. Gómez-Aguilar ◽  
M. A. Taneco-Hernández ◽  
R. F. Escobar-Jiménez ◽  
V. H. Olivares-Peregrino
Keyword(s):  
Mathematical Modeling ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Differential

scholarly journals On the selection of fractional-differential model of convective diffusion with mass exchange

2020 ◽  
pp. 7-10
Author(s):  
Vsevolod Bohaienko ◽  
Anatolij Gladky
Keyword(s):  
Decision Making ◽  
Mass Exchange ◽  
Caputo Derivative ◽  
Heuristic Algorithms ◽  
Convective Diffusion ◽  
Concentration Field ◽  
Model Parameters ◽  
Differential Model ◽  
Fractional Differential

The paper considers two fractional-differential models of convective diffusion with mass exchange and proposes a decision-making algorithm for determining the optimal model at the time of concentration field observation. As for soils of fractal structure, direct experimental determination of model parameters’ values and type of mass exchange process is in many cases impossible, calibration and determination of the most adequate models is performed mainly solving inverse problems by, in particular, meta-heuristic algorithms that are computationally complex. In order to reduce the computational complexity, we study the qualitative differences between diffusion processes described by fractional-differential models with non-local mass exchange on the base of the Caputo derivative and local non-linear mass exchange based on the non-equilibrium sorption equation that corresponds to the description by the Caputo-Fabrizio derivative. We determine under which conditions both models within a given accuracy describe the same set of measurements at a certain moment of time. When the solutions are close at a certain initial stage of process development, the model with the Caputo derivative describes its faster approach to a steady state. Based on the obtained estimates of differences in solutions, a decision-making algorithm is proposed to determine the most accurate model and the values of its parameters concurrently with the acquisition of measurements. This algorithm’s usage reduces the time spent on solving inverse calibration problems.

scholarly journals Mathematical Modeling and Functional Order Pursuits with Separated Dynamics

2021 ◽  
Vol 2068 (1) ◽  
pp. 012002
Author(s):  
Mashrabjon Mamatov ◽  
Xakimjon Alimov
Keyword(s):  
Mathematical Modeling ◽  
Dynamic Systems ◽  
Sufficient Conditions ◽  
Order Equation ◽  
Matrix Functions ◽  
Pursuit Problem ◽  
Controlled Systems ◽  
Generalized Matrix ◽  
Fractional Differential ◽  
Qualitative Dynamics

Abstract This work is devoted to the study of the pursuit problem in controlled systems described by a fractional-order equation with divided dynamics. For fixed player controls, representations of solutions are established in the form of analogs of the Cauchy formula using generalized matrix functions. Sufficient conditions are obtained for the possibility of completing the pursuit. Specific types of fractional differential equations and models of fractional dynamical systems are considered. The qualitative dynamics, issues of stability and controllability of such systems are discussed. Considered, try which, the motion of the equation is described with irrational orders. Problems of the type under study are encountered in modeling the processes of economic growth and in problems of stabilizing dynamic systems.

scholarly journals Application Of More Complex Rheological Models In Continuum Mechanics

2015 ◽  
Vol 15 (2) ◽  
Author(s):  
Mária Minárová ◽  
Jozef Sumec
Keyword(s):  
Mathematical Modeling ◽  
Relaxation Process ◽  
Rheological Properties ◽  
Continuum Mechanics ◽  
Rheological Behavior ◽  
Constitutive Equations ◽  
Structural Materials ◽  
Rheological Models ◽  
Complex Models ◽  
Maxwell Models

Abstract The paper deals with mathematical modeling of the structural materials representing their rheological properties. The materials are modeled by more complex models (enhancement of Voigt and Maxwell models). The constitutive equations are derived; the relationships within the creep and relaxation process are developed. The rheological behavior of the materials is introduced.

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