scholarly journals ON BOUNDARY VALUE PROBLEM FOR GENERALIZED ALLER EQUATION

2021 ◽  
Vol 26 (2) ◽  
pp. 7-14
Author(s):  
S. Kh. Gekkieva ◽  
M. M. Karmokov ◽  
M. A. Kerefov
Keyword(s):  
Mathematical Modeling ◽  
Mathematical Model ◽  
Porous Media ◽  
Fractional Order ◽  
Moisture Transport ◽  
Fractional Derivatives ◽  
Fractal Structure ◽  
Goursat Problem ◽  
Order Equation ◽  
Fluid Filtration

The mathematical models of fluid filtration processes in porous media with a fractal structure and memory are based on differential equations of fractional order in both time and space variables. The dependence of the soil water content can significantly affect the moisture transport in capillary-porous media. The paper investigates the generalized Aller equation widely used in mathematical modeling of the processes related to water table dynamics in view of fractal structure. As a mathematical model of the Aller equation withRiemann Liouville fractional derivatives, a loaded fractional order equation is proposed, and a solution to the Goursat problem has been written out for this model in explicit form.

Mathematical Modeling and Optimization of Fluidized Layer Carbonitriding Process for 1C 25 Steel

2017 ◽  
Vol 1143 ◽  
pp. 180-187
Author(s):  
Marian Iulian Neacsu ◽  
Sorin Dobrovici
Keyword(s):  
Mathematical Modeling ◽  
Mathematical Model ◽  
Software Package ◽  
Order Equation ◽  
Graphical Interface ◽  
Technological Parameters ◽  
Fluidized Layer ◽  
The Mathematical Model ◽  
Hardness Values ◽  
Temperature Depth

This paper presents the experiment-based mathematical modelling of fluidized bed carbonitriding process for 1C 25 steel meant to optimize this type of thermochemical processing.Based on experimental results, the mathematical model was developed, which is a second order equation with three unknown terms (parameters): temperature, depth of carbonitrided layer, the percentage of ammonia.The mathematical model allowed the simulation of the fluidized layer carbonitriding process according to its parameters and the thermal energy optimization for obtaining HV hardness values in the range 300-400 MPa.Using the software package Matlab a graphical interface was done, through which all the combinations of technological parameters of the carbonitriding process are determined, leading to obtaining values of microhardness between 300 and 400 MPa, as well as the amount of energy consumed for each variant. The variant consuming the lowest energy is considered optimal.

scholarly journals Mathematical modeling of bioprocesses with the use of fractional order derivatives

2021 ◽  
Vol 8 (1) ◽  
pp. 1
Author(s):  
Vladimira Rumenova Suvandzhieva
Keyword(s):  
Mathematical Modeling ◽  
Mathematical Model ◽  
Fractional Order ◽  
Classical Case ◽  
Integer Order ◽  
Wolfram Mathematica ◽  
Specific Behaviour ◽  
Mathematica Software ◽  
The Mathematical Model ◽  
Derivatives Of

This work brings together two recently discussed topics: mathematical modeling of a bioreactor and working with derivatives of non-integer order. Generally, it turns out that it is reasonable to replace the integer order derivatives in some of the already well known mathematical models describing bioprocesses with fractional order ones. However, the specific structure of such type of derivatives makes the study of the properties of the models a real challenge. This work contains primary results for modeling of a bioreactor with appropriately selected numerical approximations. Different scenarios are taken into consideration: starting from the simplest one - without mortality and then complicating by adding nonzero mortality term. In the classical case the solution of the system of differential equations describing the process has a specific behaviour in terms of monotonicity. Therefore, the focus of the further examinations is to find out whether it is possible to generalize the model into a fractional order one such that the key properties considering monotonicity still hold. The results show that the latter requires certain dependencies between the orders of the derivatives in the mathematical model. The hypothesis is based on two types of experiments which are described in detail. Lotka-Volterra and Monod specific growth rate are used in the mathematical model. The paper contains figures which illustrate the results from different numerical computations performed via Wolfram Mathematica software.

scholarly journals Mathematical Modelling of Non-Isothermal Moisture Transfer and Rheological Behavior in Cappilary-Porous Materials with Fractal Structure During Drying

2014 ◽  
Vol 7 (4) ◽  
pp. 111 ◽  
Author(s):  
Yaroslav Sokolowskyi ◽  
Volodymyr Shymanskyi
Keyword(s):  
Mathematical Model ◽  
Finite Difference Method ◽  
Fractional Order ◽  
Rheological Behavior ◽  
Strain State ◽  
Fractal Structure ◽  
Moisture Transfer ◽  
The Finite Difference Method ◽  
Predictor Corrector ◽  
The Mathematical Model

The mathematical model of non-isothermal moisture transfer and rheological behavior of wood during drying with taking into account the fractal structure of this material is regarded in the article. The mathematical tools of integration and differentiation of fractional order for description the mathematical model of this process was used. For finding the numerical solution of this problem the finite-difference method predictor-corrector was used. Results show the practicability of using the mathematical tools of integration and differentiation of fractional order to calculate the temperature and humidity fields and the stress-strain state during drying timber.

A MATHEMATICAL MODEL OF MICROBIAL ENHANCED OIL RECOVERY IN POROUS MEDIA

2017 ◽  
Vol 8 (4) ◽  
pp. 263-272
Author(s):  
Jianlong Xiu ◽  
Tianyuan Wang ◽  
Ying Guo ◽  
Qingfeng Cui ◽  
Lixin Huang ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Porous Media ◽  
Enhanced Oil Recovery ◽  
Oil Recovery ◽  
Microbial Enhanced Oil Recovery

The properties of Bessel-type fractional derivatives and integrals

2016 ◽  
pp. 3973-3982
Author(s):  
V. R. Lakshmi Gorty
Keyword(s):  
Fractional Order ◽  
Real Line ◽  
Computer Algebra System ◽  
Fractional Derivatives ◽  
Graphical Representation ◽  
Fractional Integrals ◽  
The Real ◽  
Fractional Derivatives And Integrals ◽  
Half Axis ◽  
Derivatives Of

The fractional integrals of Bessel-type Fractional Integrals from left-sided and right-sided integrals of fractional order is established on finite and infinite interval of the real-line, half axis and real axis. The Bessel-type fractional derivatives are also established. The properties of Fractional derivatives and integrals are studied. The fractional derivatives of Bessel-type of fractional order on finite of the real-line are studied by graphical representation. Results are direct output of the computer algebra system coded from MATLAB R2011b.

Application of the method of mathematical modeling for forecasting channel deformations at the underwater crossing of the main pipeline

2020 ◽  
Vol 10 (6) ◽  
pp. 599-609
Author(s):  
Valery А. Gruzdev ◽  
◽  
Georgy V. Mosolov ◽  
Ekaterina A. Sabayda ◽  
◽  
...  
Keyword(s):  
Mathematical Modeling ◽  
Mathematical Model ◽  
Roughness Coefficient ◽  
Computational Domain ◽  
Channel Deformations ◽  
Geological Surveys ◽  
Kuban River ◽  
The Stability ◽  
Protective Structures

In order to determine the possibility of using the method of mathematical modeling for making long-term forecasts of channel deformations of trunk line underwater crossing (TLUC) through water obstacles, a methodology for performing and analyzing the results of mathematical modeling of channel deformations in the TLUC zone across the Kuban River is considered. Within the framework of the work, the following tasks were solved: 1) the format and composition of the initial data necessary for mathematical modeling were determined; 2) the procedure for assigning the boundaries of the computational domain of the model was considered, the computational domain was broken down into the computational grid, the zoning of the computational domain was performed by the value of the roughness coefficient; 3) the analysis of the results of modeling the water flow was carried out without taking the bottom deformations into account, as well as modeling the bottom deformations, the specifics of the verification and calibration calculations were determined to build a reliable mathematical model; 4) considered the possibility of using the method of mathematical modeling to check the stability of the bottom in the area of TLUC in the presence of man-made dumping or protective structure. It has been established that modeling the flow hydraulics and structure of currents, making short-term forecasts of local high-altitude reshaping of the bottom, determining the tendencies of erosion and accumulation of sediments upstream and downstream of protective structures are applicable for predicting channel deformations in the zone of the TLUC. In all these cases, it is mandatory to have materials from engineering-hydro-meteorological and engineering-geological surveys in an amount sufficient to compile a reliable mathematical model.

On the displacements of the contours of the optic-electronic space images. Mathematical modeling of displacement

2017 ◽  
Vol 992 (4) ◽  
pp. 32-38 ◽  
Author(s):  
E.G. Voronin
Keyword(s):  
Mathematical Modeling ◽  
Remote Sensing ◽  
Mathematical Model ◽  
Point Of View ◽  
Design Features ◽  
Space Images ◽  
Electronic Imaging ◽  
Measuring Accuracy ◽  
Physical Laws

The article opens a cycle of three consecutive publications dedicated to the phenomenon of the displacement of the same points in overlapping scans obtained adjacent CCD matrices with opto-electronic imagery. This phenomenon was noticed by other authors, but the proposed explanation for the origin of displacements and the resulting estimates are insufficient, and developed their solutions seem controversial from the point of view of recovery of the measuring accuracy of opticalelectronic space images, determined by the physical laws of their formation. In the first article the mathematical modeling of the expected displacements based on the design features of a scanning opto-electronic imaging equipment. It is shown that actual bias cannot be forecast, because they include additional terms, which may be gross, systematic and random values. The proposed algorithm for computing the most probable values of the additional displacement and ways to address some of the systematic components of these displacements in a mathematical model of optical-electronic remote sensing.

scholarly journals Fractional-Order Colour Image Processing

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 457
Author(s):  
Manuel Henriques ◽  
Duarte Valério ◽  
Paulo Gordo ◽  
Rui Melicio
Keyword(s):  
Image Processing ◽  
Edge Detection ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Grey Scale ◽  
Colour Image ◽  
Colour Image Processing ◽  
Image Processing Algorithms ◽  
Processing Algorithms

Many image processing algorithms make use of derivatives. In such cases, fractional derivatives allow an extra degree of freedom, which can be used to obtain better results in applications such as edge detection. Published literature concentrates on grey-scale images; in this paper, algorithms of six fractional detectors for colour images are implemented, and their performance is illustrated. The algorithms are: Canny, Sobel, Roberts, Laplacian of Gaussian, CRONE, and fractional derivative.

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