scholarly journals Study of stiff differential equations of mathematical description of izomerization of pentane-hexane cut process

2021 ◽  
Vol 2131 (2) ◽  
pp. 022003
Author(s):  
R I Faskhutdinova ◽  
A G Faskhutdinov ◽  
L V Enikeeva ◽  
I M Gubaydullin
Keyword(s):  
Differential Equations ◽  
Chemical Kinetics ◽  
Mathematical Description ◽  
Hexane Fraction ◽  
Chemical Transformations ◽  
System Of Differential Equations ◽  
Stiff Differential Equations ◽  
Stiff System ◽  
Catalytic Isomerization ◽  
The Given

Abstract This paper provides a brief overview of the existing definitions of a stiff system of differential equations. Further, on the example of the accepted scheme of chemical transformations of the catalytic isomerization process of the pentane-hexane fraction, the stiffness of the system of differential equations was studied. In the course of the work, a method for studying the direct kinetic problem for stiffness is presented. In the Matlab software, the results of solving a system of differential equations by five methods (solvers) were compared. The given method can be tried for solving other problems of chemical kinetics.

Modeling of the Process of Drying and Coalescing Nanodispersion Spreading

2021 ◽  
Vol 887 ◽  
pp. 557-563
Author(s):  
D.M. Mordasov ◽  
M.D. Mordasov
Keyword(s):  
Contact Angle ◽  
Differential Equations ◽  
Mathematical Description ◽  
Transition Function ◽  
System Of Differential Equations ◽  
Theoretical Justification ◽  
Spreading Process ◽  
Optimal Amount ◽  
First Order ◽  
Energy Processes

The spreading process of drying and coalescing nanodispersion was simulated using the method of analogies. A mathematical description of the energy processes in the proposed physical model was obtained in the form of a system of differential equations of the first order. A transition function that describes the dynamics of the change in the contact angle when the nanodispersion drop spreads was obtained as a result of solving the system of differential equations. The physical meaning of the transition function coefficients was established. Based on the analysis of the ratio of the transition function coefficients, a theoretical justification for the results of experiments on choosing the optimal amount of desiccant introduced into styrene-acrylic nanodispersion was given.

scholarly journals Minimizing of the quadratic functional on Hopfield networks

2021 ◽  
pp. 1-20
Author(s):  
Oleksandr Boichuk ◽  
Dmytro Bihun ◽  
Victor Feruk ◽  
Oleksandr Pokutnyi
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Sufficient Conditions ◽  
Quadratic Functional ◽  
Problem Solution ◽  
Linear Boundary ◽  
System Of Differential Equations ◽  
Hopfield Networks ◽  
Necessary And Sufficient ◽  
The Given

In this paper, we consider the continuous Hopfield model with a weak interaction of network neurons. This model is described by a system of differential equations with linear boundary conditions. Also, we consider the questions of finding necessary and sufficient conditions of solvability and constructive construction of solutions of the given problem, which turn into solutions of the linear generating problem, as the parameter $\varepsilon$ tends to zero. An iterative algorithm for finding solutions has been constructed. The problem of finding the extremum of the target functions on the given problem solution is considered. To minimize a functional, an accelerated method of conjugate gradients is used. Results are illustrated with examples for the case of three neurons.

scholarly journals Dynamics and Stability of Hopf Bifurcation for One Non-linear System

TEM Journal ◽  
2021 ◽  
pp. 820-824
Author(s):  
Vahidin Hadžiabdić ◽  
Midhat Mehuljić ◽  
Jasmin Bektešević ◽  
Adnan Mašić
Keyword(s):  
Hopf Bifurcation ◽  
Linear System ◽  
Differential Equations ◽  
Detailed Analysis ◽  
Limit Cycle ◽  
Local Stability ◽  
System Of Differential Equations ◽  
The One ◽  
Non Linear System ◽  
The Given

In this paper we will look at the one system of ODE and analyze it. We aim to determine the points of equilibrium; examine their character and establish the existence of a bifurcation for the corresponding parameter value. A detailed analysis of local stability was performed for all values of the given parameter. For a certain value of the parameter, the existence of supercritical Hopf bifurcation of the observed system of differential equations has been proved. Also, the existence of a limit cycle that is always stable has been proved.

scholarly journals Estimates of the Solution of the Balance Model Considering the Environmental Factor and Investment

2020 ◽  
pp. 119-127
Author(s):  
Marina Pavlova ◽  
Larisa Tolmacheva ◽  
Elena Nazarova
Keyword(s):  
Differential Equations ◽  
Nonnegative Solution ◽  
Balance Model ◽  
System Of Differential Equations ◽  
Suggested Model ◽  
Equilibrium Prices ◽  
Successful Solution ◽  
Manufactured Products ◽  
Recycling Of Wastes ◽  
The Given

The article explains the nonlinear balance model that considers disposal and recycling of wastes and investments. The suggested model is the equilibrium prices model in which the costs of harmful wastage disposal and recycling are considered. Besides, there are nonlinear interrelations between the branches of production, which allows us to predict the release of useful products, which is necessary for the economistsanalysts who are engaged in forecasting the manufactured products. For the model which is described by a system of differential equations, the conditions are created when the system of differential equations has only one solution. The paper defines the conditions under which this model is solvable and has a nonnegative solution, if at the same time the given values can be negative. For the model the methods of creating bilateral estimated solutions are adapted; the method of improving bilateral estimation is offered. Unlike the methods of searching the precise solution, the application of the method of bilateral estimation facilitates successful solution of tasks with big dimension of the processed models, without resorting to direct integration. The results of this article can be used in the solution of specific tasks of mathematics, economics, biology and other tasks with nonlinear interrelations.

scholarly journals Periodic solutions of quasilinear non-autonomous systems with impulses

1985 ◽  
Vol 31 (2) ◽  
pp. 185-197 ◽  
Author(s):  
S.G. Hristova ◽  
D.D. Bainov
Keyword(s):  
Periodic Solution ◽  
Differential Equations ◽  
Small Parameter ◽  
Periodic Solutions ◽  
Autonomous Systems ◽  
Sufficient Conditions ◽  
System Of Differential Equations ◽  
Image Position ◽  
The Given

The paper considers a system of differential equations with impulse perturbations at fixed moments in time of the formwhere x ∈ Rn, ε is a small parameter,Sufficient conditions are found for the existence of the periodic solution of the given system in the critical and non-critical cases.

scholarly journals KNIZHNIK–ZAMOLODCHIKOV-TYPE EQUATIONS FOR GAUGED WZNW MODELS

1998 ◽  
Vol 13 (08) ◽  
pp. 1345-1367 ◽  
Author(s):  
IAN I. KOGAN ◽  
ALEX LEWIS ◽  
OLEG A. SOLOVIEV
Keyword(s):  
Black Hole ◽  
Differential Equations ◽  
Correlation Functions ◽  
Striking Similarity ◽  
Minimal Models ◽  
System Of Differential Equations ◽  
The Given ◽  
Zamolodchikov Equation

We study correlation functions of coset constructions by utilizing the method of gauge dressing. The given method results in a system of differential equations which are generalizations of the Knizhnik–Zamolodchikov equation. As an example, we apply this method to the minimal models and to the Witten 2D black hole. We exhibit a striking similarity between the latter and the gravitational dressing. In particular, we look for logarithmic operators in the 2D black hole.

Computing Model for Vibratory Digging-Out of Sugar Beet Roots

2014 ◽  
Vol 1 (2) ◽  
pp. 52-60
Author(s):  
V. Bulgakov ◽  
V. Adamchuk ◽  
H. Kaletnyk
Keyword(s):  
Differential Equations ◽  
Sugar Beet ◽  
Vertical Plane ◽  
Forward Motion ◽  
Admissible Velocity ◽  
Beet Root ◽  
Optimum Speed ◽  
Analytical Research ◽  
Sugar Beet Root ◽  
The Given

The new design mathematical model of the sugar beet roots vibration digging-out process with the plowshare vibration digging working part has been created. In this case the sugar beet root is simulated as a solid body , while the plowshare vibration digging working part accomplishes fl uctuations in the longitudinal - vertical plane with the given amplitude and frequency in the process of work . The aim of the current research has been to determine the dependences between the design and kinematic parameters of the sugar beet roots vibra- tion digging-out technological process from soil , which provide the ir non-damage. Methods . For the aim ac- complishment, the methods of design mathematical models constructing based on the classical laws of me- chanics are applied. The solution of the obtained differential equations is accomplished with the PC involve- ment. Results . The differential equations of the sugar beet root’s motion in course of the vibration digging-out have been comprised . They allow to determine the admissible velocity of the vibration digging working part’s forward motion depending on the angular parameters of the latter. In the result of the computational simula- tion i.e., the solution of the obtained analytical dependence by PC, the graphic dependences of the admissible velocity of plowshare v ibration digging working part’s forward motion providing the extraction of the sugar beet root from soil without the breaking-off of its tail section have been determined. Conclusions . Due to the performed analytical research , it has been established that γ = 13 ... 16 ° , β = 20 ... 30 ° should be considered as the most reasonable values of γ and β angles of the vibration digging working part providing both its forward motion optimum speed and sugar beet root digging-out from the soil without damage . On the ground of the data obtained from the analytical rese arch, the new vibration digging working parts for the sugar beet roots have been designed; also the patents of Ukraine for the inventions have been obtained for them.

Stress and Deformation

1996 ◽  
Author(s):  
Gerhard Oertel
Keyword(s):  
Differential Equations ◽  
Mathematical Description ◽  
Finite Strain ◽  
Stress Strain ◽  
Complex Problems ◽  
Vector Algebra ◽  
Stress And Deformation

Students of geology who may have only a modest background in mathematics need to become familiar with the theories of stress, strain, and other tensor quantities, so that they can follow, and apply to their own research, developments in modern, quantitative geology. This book, based on a course taught by the author at UCLA, can provide the proper introduction. Included throughout the eight chapters are 136 complex problems, advancing from vector algebra in standard and subscript notations, to the mathematical description of finite strain and its compounding and decomposition. Fully worked solutions to the problems make up the largest part of the book. With their help, students can monitor their progress, and geologists will be able to utilize subscript and matrix notations and formulate and solve tensor problems on their own. The book can be successfully used by anyone with some training in calculus and the rudiments of differential equations.

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