ALGORITHM OF USING HURWITZ CONDITIONS TO STUDY STABILITY OF STEADY STATES OF CONTINUOUS FERMENTATION PROCESSES WITH NONLINEAR KINETICS

2017 ◽  
pp. 125-135
Author(s):  
Aleksey Georgievich Borodkin ◽  
Elena Lvovna Gordeeva ◽  
Yuliya Lvovna Gordeeva ◽  
Yuriy Alekseevich Komissarov
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Continuous Fermentation ◽  
Raw Materials ◽  
Sufficient Conditions ◽  
Steady States ◽  
Microbiological Synthesis ◽  
Hurwitz Matrix ◽  
First Approximation ◽  
The Mathematical Model

The authors of the article produce the algorithm to evaluate the stability of the steady states for microbiological synthesis using mathematical models for kinetics based on the unstructured approach. The object under study is a continuous biotechnological process aiming to obtain a target product, in addition to biomass. The estimation of stability is performed using Hurwitz matrix. In this process the use of raw materials containing components for additional producing of the primary substrate are possible. The algorithm includes five steps: formulation of the mathematical model that takes into account biological and technological limits; definition of steady states (one or several) for which it is necessary to evaluate conditions of stability and to calculate process parameters for the adopted steady states resulted from solving a system of nonlinear algebraic equations of the mathematical model; formulation of the equations of the first approximation (the system of first-order linear differential equations with constant coefficients) and evaluation of the coefficients corresponding to the adopted steady states; formation of Hurwitz matrix and calculation of its items; calculation of necessary and sufficient conditions for stability using Hurwitz matrix. Two methods of calculating Hurwitz matrix have been shown. The first method is implemented by transforming the system of four differential equations of the first approximation into one fourth-order ordinary differential equation. In the course of transformation there were determined calculated ratios for coefficients of the matrix. In the second method the coefficients are calculated by solving vector equation for the system of first approximation with the introduction of eigen values. The algorithm has been used for evaluating steady states of microbiological synthesis for lactic acid production from raw materials containing the component reproducing the main substrate. The estimation has been carried out for two steady states, one of which is optimal. For numerical values of the constants (from the literature) both steady states were stable.

scholarly journals ASYMPTOTIC EXPANSIONS OF SOLUTIONS FOR HIV EVOLUTION MODEL

2017 ◽  
Vol 19 (3) ◽  
pp. 5-11
Author(s):  
A.A. Archibasov
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Asymptotic Expansions ◽  
Boundary Function ◽  
Evolution Model ◽  
Singularly Perturbed ◽  
First Approximation ◽  
The Mathematical Model ◽  
Hiv Evolution

In the paper the mathematical model of HIV evolution is considered. This model is a singularly perturbed partial integro-differential equations system. Based on the Tikhonov—Vasilieva method of boundary function the first approximation of the system solutions is realized.

Glycemia monitoring

1998 ◽  
Vol 2 ◽  
pp. 23-30
Author(s):  
Igor Basov ◽  
Donatas Švitra
Keyword(s):  
Diabetes Mellitus ◽  
Mathematical Model ◽  
Numerical Methods ◽  
Differential Equations ◽  
Blood Sugar ◽  
Self Regulation ◽  
Physiological System ◽  
Non Linear ◽  
The Mathematical Model

Here a system of two non-linear difference-differential equations, which is mathematical model of self-regulation of the sugar level in blood, is investigated. The analysis carried out by qualitative and numerical methods allows us to conclude that the mathematical model explains the functioning of the physiological system "insulin-blood sugar" in both normal and pathological cases, i.e. diabetes mellitus and hyperinsulinism.

scholarly journals Computational Study of the Coupled Mechanism of Thermophoretic Transportation and Mixed Convection Flow around the Surface of a Sphere

Molecules ◽  
2020 ◽  
Vol 25 (11) ◽  
pp. 2694
Author(s):  
Amir Abbas ◽  
Muhammad Ashraf ◽  
Yu-Ming Chu ◽  
Saqib Zia ◽  
Ilyas Khan ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Mixed Convection ◽  
Computational Study ◽  
Convection Flow ◽  
Mixed Convection Flow ◽  
Partial Differential ◽  
Primitive Form ◽  
The Mathematical Model

The main goal of the current work was to study the coupled mechanism of thermophoretic transportation and mixed convection flow around the surface of the sphere. To analyze the characteristics of heat and fluid flow in the presence of thermophoretic transportation, a mathematical model in terms of non-linear coupled partial differential equations obeying the laws of conservation was formulated. Moreover, the mathematical model of the proposed phenomena was approximated by implementing the finite difference scheme and boundary value problem of fourth order code BVP4C built-in scheme. The novelty point of this paper is that the primitive variable formulation is introduced to transform the system of partial differential equations into a primitive form to make the line of the algorithm smooth. Secondly, the term thermophoretic transportation in the mass equation is introduced in the mass equation and thus the effect of thermophoretic transportation can be calculated at different positions of the sphere. Basically, in this study, some favorite positions around the sphere were located, where the velocity field, temperature distribution, mass concentration, skin friction, and rate of heat transfer can be calculated simultaneously without any separation in flow around the surface of the sphere.

scholarly journals Mathematical Model of Spatial Motion of the Controlled Parachute-Tether System of the Wind Kite Type

2021 ◽  
Vol 24 (4) ◽  
pp. 17-24
Author(s):  
V.M. Churkin ◽  
T.Yu. Churkina ◽  
A.M. Girin
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Solid Body ◽  
Equations Of Motion ◽  
Vertical Plane ◽  
Spatial Motion ◽  
Mathematical Task ◽  
Spatial Movement ◽  
Elastic Thread ◽  
The Mathematical Model

Mathematical modeling is created for the mathematical task of spatial motion of the controlled parachute-tether system of the “wind kite” type. The mathematical model parachute-tether system consists of a model of the main parachute and a model of the braking parachute. The parachutes are connected by the tether. The model of the main parachute is supposed to be the solid body. This solid body has two planes of symmetry. The braking parachute is the solid body with axial symmetry. The tether model is an absolutely flexible elastic thread. The tether is connected by ideal hinges with the main parachute and braking parachute. The control of the main parachute is carried out by changing the length of the control slings. Changing the length causes deformation of the dome. This is the reason for the change in its aerodynamics. Maneuvering of the main parachute occurs in the vertical plane, when the length of the control slings changes simultaneously. Maneuvering of the main parachute in space is carried out when the length of the control slings changes, when the slings are given a travel difference. The system of dynamic and kinematic equations is designed for calculating the controlled spatial movement of the main parachute, braking parachute and tether. The option exists when the mass of the tether and the forces applied to the tether cannot be neglected. The motion of the tether is represented by the equations of motion of an absolutely flexible elastic thread in projections on the axis of a natural trihedron. The mathematical model is represented by a system of ordinary differential equations and partial differential equations. The problem is solved using various numerical methods. The solution is possible with the help of an integrated numerical and analytical approach as well.

scholarly journals Sensitivity analysis and practical identifiability of the mathematical model for partial differential equations

2021 ◽  
Vol 2092 (1) ◽  
pp. 012012
Author(s):  
O Krivorotko ◽  
D Andornaya
Keyword(s):  
Mathematical Model ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Sensitivity Matrix ◽  
Partial Differential ◽  
Identifiability Analysis ◽  
Practical Identifiability ◽  
Eigenvalue Method ◽  
Simulation Results ◽  
The Mathematical Model

Abstract A sensitivity-based identifiability analysis of mathematical model for partial differential equations is carried out using an orthogonal method and an eigenvalue method. These methods are used to study the properties of the sensitivity matrix and the effects of changes in the model coefficients on the simulation results. Practical identifiability is investigated to determine whether the coefficients can be reconstructed with noisy experimental data. The analysis is performed using correlation matrix method with allowance for Gaussian noise in the measurements. The results of numerical calculations to obtain identifiable sets of parameters for the mathematical model arising in social networks are presented and discussed.

scholarly journals OPTIMAL ORGANIZATIONAL HIERARCHIES IN FIRMS

2007 ◽  
Vol 8 (2) ◽  
pp. 79-99 ◽  
Author(s):  
Sergey Mishin
Keyword(s):  
Mathematical Model ◽  
Sufficient Conditions ◽  
Cost Functions ◽  
Minimal Cost ◽  
Organizational Hierarchy ◽  
Modern Economy ◽  
Organizational Hierarchies ◽  
The Mathematical Model

One of the main problems in modern economy is to construct an efficient organizational hierarchy allowing to control the firm with minimal cost. This paper describes the mathematical model of optimal hierarchies in firms. Optimal hierarchies for several classes of cost functions are obtained. Particularly, sufficient conditions for tree optimality, 2hier‐archy (any manager has two immediate subordinates) optimality and two‐tier hierarchy optimality are defined.

scholarly journals Mathematical Model with Feedback Loop in Company Control

2021 ◽  
Vol 114 ◽  
pp. 01020
Author(s):  
Arkadiy Plotnikov ◽  
Tatyana Goryacheva ◽  
Flyura Kazakova ◽  
Ekaterina Zakharchenko
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Feedback Loop ◽  
Structural Model ◽  
Production Costs ◽  
Operator Form ◽  
Single Industry ◽  
Dynamic Relationships ◽  
The Mathematical Model ◽  
Further Development

This article is devoted to the further development of feedback loop models. They are used in the management of a single-industry firm. The development of the studying consists in the mathematical modeling of transients and assessing their impact on economic indicators. To solve this problem, the following was done in the article. A structural model of the functioning of a single-industry firm (SIF) is presented. It is built in accordance with the theory of automatic control (TAC) and includes elements: a management unit, a production unit, a sales unit, information nodes, as well as a feedback loop (FL). Equations and relationships describing the logic of the functioning of the company as a production system are presented. They allowed us to derive dynamic relationships and differential equations that reflect feedback loops on revenue and production costs. The system of expressions in operator form is presented, which describes the contour of the FL SIF. It has the form of a system of differential equations. It forms the basis of the mathematical model of SIF in the control system. This model made it possible to obtain a graphical interpretation of transients with closed and open FL based on the use of the Mathcad editor. Transients are fluctuations in sales volumes and production costs in the presence of disturbing influences. This is the scientific result and determines the novelty of the article.

scholarly journals Research of kinetics of change in fractional composition of wood raw materials being shredded

2020 ◽  
Author(s):  
Ю.Н. Власов ◽  
Е.В. Нестерова ◽  
Е.Г. Хитров
Keyword(s):  
Experimental Data ◽  
Mathematical Model ◽  
Differential Equations ◽  
Fractional Composition ◽  
Raw Materials ◽  
Calculated Data ◽  
General Transformation ◽  
Differential Model ◽  
Integral Differential Equations ◽  
Kinetics Of

В технике при изучении кинетики измельчения материалов пользуются математическими моделями в виде интегро-дифференциальных уравнений, решение которых трудоемко и не всегда приводит к получению наглядных результатов. Цель настоящей статьи разработать математическую модель, раскрывающую кинетику изменения фракционного состава измельчаемых древесных материалов, позволяющую на практике проводить оценку фракционного состава обрабатываемого сырья во времени. Методы исследования математический анализ, численные методы решения дифференциальных уравнений и обработки расчетных данных. Измельчение рассмотрено как многостадийный процесс, при котором фракции материала (узкие классы) под воздействием рабочего органа машины-измельчителя претерпевают превращения, происходящие как последовательно, так и параллельно, причем скорости превращений и доли вновь образованных узких классов материала определяются исходными размерами измельчаемых фракций и параметрами рабочего органа измельчителя. Предложена система дифференциальных уравнений, описывающая в общем превращения узких классов при измельчении, причем коэффициенты уравнений позволяют учесть произвольный вид функций скоростей измельчения фракций и выхода продуктов измельчения. Предложенная система является альтернативой интегро-дифференциальному уравнению балансовой модели измельчения. Выполнена оценка значений параметров математической модели на примере измельчения коры. По результатам сопоставления результатов моделирования с экспериментальными данными, полученными предыдущими исследователями, установлено, что предложенная дифференциальная модель изменения фракционного состава материала при принятых предпосылках к расчету ее параметров качественно и количественно описывает экспериментальных данные с высокой точностью. In techniques at study of kinetics of shredding of materials use mathematical models in the form of the integral-differential equations, which solution is laborious and not always leads to reception of evident results. The purpose of this article is to develop a mathematical model, which reveals the kinetics of change in fractional composition of wood materials being shredded, allowing in practice to evaluate the fractional composition of the processed raw materials in time. Methods of research include mathematical analysis, numerical methods for solving differential equations and processing of calculated data. Shredding is considered as multistage process at which fractions of a material (narrow classes) under the influence of a working body of the shredder machine undergo transformations occurring both consistently and in parallel, and rates of transformations and a share of again formed narrow classes of the material are defined by initial sizes of shredded fractions and parameters of the working body. The system of the differential equations describing in the general transformation of narrow classes at grinding is offered, and factors of the equations allow to consider any kind of functions of speeds of grinding of fractions and the output of shredding products. The proposed system is an alternative to the integral-differential equation of the balance shredding model. The estimation of values of parameters of the mathematical model on an example of bark shredding is carried out. By results of comparison of results of modeling with the experimental data received by previous researchers it is established that the offered differential model of change of fractional composition of the material at the accepted preconditions to calculation of its parameters qualitatively and quantitatively describes the experimental data with high accuracy.

SELF-SIMILAR PROBLEMS FOR MODELING THE SURFACE CHEMICAL REACTIONS WITH THE GRAVITATION

1998 ◽  
Vol 3 (1) ◽  
pp. 25-32
Author(s):  
Jânis Cepîtis ◽  
Harijs Kalis
Keyword(s):  
Boundary Layer ◽  
Mathematical Model ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Chemical Reactions ◽  
Glass Fibre ◽  
Surface Chemical ◽  
Self Similar ◽  
Surface Chemical Reactions ◽  
The Mathematical Model

The mathematical model of a chemical reaction which takes place on the surface of the uniformly moving vertically imbedded glass fibre material is considered. The effect of gravitation is taken into account. Boussinesq's and boundary layer fittings allow to derive boundary value problems for self‐similar systems of ordinary differential equations.

Modeling Colorectal Cancer

2013 ◽  
pp. 53-75 ◽  
Author(s):  
Svetoslav Nikolov ◽  
Mukhtar Ullah ◽  
Momchil Nenov ◽  
Julio Vera Gonzalez ◽  
Peter Raasch ◽  
...  
Keyword(s):  
Mathematical Modeling ◽  
Colorectal Cancer ◽  
Mathematical Model ◽  
Systems Biology ◽  
Numerical Simulations ◽  
Time Delays ◽  
Sufficient Conditions ◽  
Extended Version ◽  
Necessary And Sufficient ◽  
The Mathematical Model

Mathematical modeling is increasingly used to improve our understanding of colorectal cancer. In the first part of this chapter, the authors give a review of systems biology approaches to investigate colorectal cancer. In the second part, the mathematical model proposed by Johnston et al. (2007) is expanded to include time delays and analysed for its stability. For both models, the original and the extended version, the authors obtain the necessary and sufficient conditions for stability. This is confirmed by numerical simulations. Thus, some new mathematical and biological results are obtained.

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