scholarly journals Бинарные соответствия и обратная задача химической кинетики

2018 ◽  
Author(s):  
L.I. Kononenko ◽  
A.E. Gutman
Keyword(s):  
Inverse Problem ◽  
Differential Equations ◽  
Chemical Kinetics ◽  
Unique Solvability ◽  
Auxiliary Problem ◽  
Problem Definition ◽  
Finite Sets ◽  
The Stability ◽  
Definition Of ◽  
The Right

We show how binary correspondences can be used for simple formalization of the notion of problem, definition of the basic components of problems, their properties, and constructions. In particular, formalization of the following notions is presented: condition, data, unknowns, and solutions of a problem, solvability and unique solvability, inverse problem, composition and restriction of problems, isomorphism between problems. We also consider topological problems and the related notions of stability and correctness. A connection is indicated between the stability and continuity of a uniquely solvable topological problem. The definition of parametrized set is given. The notions are introduced of parametrized problem, the problem of reconstruction of an object by the values of parameters, as well as the notions of locally free set of parameters and stability with respect to a set of parameters. As an illustration, we consider a singularly perturbed system of ordinary differential equations which describe a process in chemical kinetics and burning. Direct and inverse problems are stated for such a system. We extend the class of problems under study by considering polynomials of arbitrary degree as the right-hand sides of the differential equations. It is shown how the inverse problem of chemical kinetics can be corrected and made more practical by means of the composition with a simple auxiliary problem which represents the relation between functions and finite sets of numerical characteristics being measured. For the corrected inverse problem, formulas for the solution are presented and the conditions of unique solvability are indicated. Within the study of solvability, a criterion is established for linear independence of functions in terms of finite sets of their values. With the help of the criterion, realizability is clarified of the condition for unique solvability of the inverse problem of chemical kinetics.

scholarly journals On the Lyapunov functionals method in the stability problem of Volterra integro-differential equations

2018 ◽  
Vol 20 (3) ◽  
pp. 260-272
Author(s):  
A. S. Andreev ◽  
O. A. Peregudova
Keyword(s):  
Differential Equations ◽  
Stability Problem ◽  
Infinite Delay ◽  
Zero Solution ◽  
Model Systems ◽  
Lyapunov Functionals ◽  
Structure Equations ◽  
Positive Limit Set ◽  
The Stability ◽  
The Right

In this paper, we consider the problem of applying the method of Lyapunov functionals to investigate the stability of non-linear integro-differential equations, the right-hand side of which is the sum of the components of the instantaneous action and also ones with a finite and infinite delay. The relevance of the problem is the widespread use of such complicated in structure equations in modeling the controllers using integral regulators for mechanical systems, as well as biological, physical and other processes. We develop the Lyapunov functionals method in the direction of revealing the limiting properties of solutions by means of Lyapunov functionals with a semi-definite derivative. We proved the theorems on the quasi-invariance of a positive limit set of bounded solution as well as ones on the asymptotic stability of the zero solution including a uniform one. The results are achieved by constructing a new structure of the topological dynamics of the equations under study. The theorems proved are applied in solving the stability problem of two model systems which are generalizations of a number of known models of natural science and technology.

scholarly journals Application of M-Matrices for the Study of Mathematical Models of Living Systems

2018 ◽  
Vol 13 (1) ◽  
pp. 208-237 ◽  
Author(s):  
N.V. Pertsev ◽  
B.Yu. Pichugin ◽  
A.N. Pichugina
Keyword(s):  
Differential Equations ◽  
Stability Problem ◽  
Distributed Delay ◽  
Linearization Method ◽  
Living Systems ◽  
High Dimensional ◽  
Linear Differential ◽  
The Stability ◽  
Several Delays ◽  
The Right

Some results are presented of application of M-matrices to the study the stability problem of the equilibriums of differential equations used in models of living systems. The models studied are described by differential equations with several delays, including distributed delay, and by high-dimensional systems of differential equations. To study the stability of the equilibriums the linearization method is used. Emerging systems of linear differential equations have a specific structure of the right-hand parts, which allows to effectively use the properties of M-matrices. As examples, the results of studies of models arising in immunology, epidemiology and ecology are presented.

scholarly journals Unique solvability of boundary value problem for functional differential equations with involution

2021 ◽  
Vol 103 (3) ◽  
pp. 68-75
Author(s):  
K.Zh. Nazarova ◽  
◽  
K.I. Usmanov ◽  
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Unique Solvability ◽  
Boundary Value ◽  
Fredholm Type ◽  
Integral Sign ◽  
Integral Differential Equations ◽  
The Right ◽  
Integral Differential Equation

In this paper we consider a boundary value problem for systems of Fredholm type integral-differential equations with involutive transformation, containing derivative of the required function on the right-hand side under the integral sign. Applying properties of an involutive transformation, original boundary value problem is reduced to a boundary value problem for systems of integral-differential equations, containing derivative of the required function on the right side under the integral sign. Assuming existence of resolvent of the integral equation with respect to the kernel K˜2(t, s) (this is the kernel of the integral equation that contains the derivative of the desired function) and using properties of the resolvent, integral-differential equation with a derivative on the right-hand side is reduced to a Fredholm type integral-differential equation, in which there is no derivative of the desired function on the right side of the equation. Further, the obtained boundary value problem is solved by the parametrization method created by Professor D. Dzhumabaev. Based on this method, the problem is reduced to solving a special Cauchy problem with respect to the introduced new functions and to solving systems of linear algebraic equations with respect to the introduced parameters. An algorithm to find a solution is proposed. As is known, in contrast to the Cauchy problem for ordinary differential equations, the special Cauchy problem for systems of integral-differential equations is not always solvable. Necessary conditions for unique solvability of the special Cauchy problem were established. By using results obtained by Professor D. Dzhumabaev, necessary and sufficient conditions for the unique solvability of the original problem were established.

scholarly journals Mixed forced, parametric, and self-oscillations with nonideal energy source and lagging forces

2021 ◽  
Vol 29 (5) ◽  
pp. 739-750
Author(s):  
Alishir Alifov ◽  
Keyword(s):  
Energy Source ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Point Of View ◽  
Ease Of Use ◽  
Direct Linearization ◽  
Plane Change ◽  
The Stability ◽  
The Right

The purpose of this study is to determine the effect of retarded forces in elasticity and damping on the dynamics of mixed forced, parametric, and self-oscillations in a system with limited excitation. A mechanical frictional self-oscillating system driven by a limited-power engine was used as a model. Methods. In this work, to solve the nonlinear differential equations of motion of the system under consideration, the method of direct linearization is used, which differs from the known methods of nonlinear mechanics in ease of use and very low labor and time costs. This is especially important from the point of view of calculations when designing real devices. Results. The characteristic of the friction force that causes self-oscillations, represented by a general polynomial function, is linearized using the method of direct linearization of nonlinearities. Using the same method, solutions of the differential equations of motion of the system are constructed, equations are obtained for determining the nonstationary values of the amplitude, phase of oscillations and the speed of the energy source. Stationary motions are considered, as well as their stability by means of the Routh–Hurwitz criteria. Performed calculations obtained information about the effect of delays on the dynamics of the system. Conclusion. Calculations have shown that delays shift the amplitude curves to the right and left, up and down on the amplitude–frequency plane, change their shape, and affect the stability of motion.

AN EFFICIENT NUMERICAL METHOD FOR SOLVING AN INVERSE WAVE PROBLEM

2013 ◽  
Vol 10 (03) ◽  
pp. 1350009 ◽  
Author(s):  
REZA POURGHOLI ◽  
AMIN ESFAHANI
Keyword(s):  
Inverse Problem ◽  
Numerical Method ◽  
Auxiliary Problem ◽  
Linear Wave ◽  
Singular Value Decomposition Method ◽  
The Matrix ◽  
Uniqueness Of The Solution ◽  
The Stability ◽  
Stability And Accuracy ◽  
Value Decomposition

In this paper, we will first study the existence and uniqueness of the solution of a one-dimensional inverse problem for an inhomogeneous linear wave equation with initial and boundary conditions via an auxiliary problem. Then a stable numerical method consisting of zeroth-, first-, and second-order Tikhonov regularization to the matrix form of Duhamel's principle for solving this inverse problem is presented. The stability and accuracy of the scheme presented is evaluated by comparison with the Singular Value Decomposition method. Some numerical experiments confirm the utility of this algorithm as the results are in good agreement with the exact data.

scholarly journals Tomographic Inverse Problem with Estimating Missing Projections

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Masashi Kimura ◽  
Yusaku Yamaguchi ◽  
Omar M. Abou Al-Ola ◽  
Tetsuya Yoshinaga
Keyword(s):  
Inverse Problem ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Artifact Reduction ◽  
Metal Artifact Reduction ◽  
System Of Differential Equations ◽  
Initial Value ◽  
Degraded Image ◽  
Novel Method ◽  
The Stability

Image reconstruction in computed tomography can be treated as an inverse problem, namely, obtaining pixel values of a tomographic image from measured projections. However, a seriously degraded image with artifacts is produced when a certain part of the projections is inaccurate or missing. A novel method for simultaneously obtaining a reconstructed image and an estimated projection by solving an initial-value problem of differential equations is proposed. A system of differential equations is constructed on the basis of optimizing a cost function of unknown variables for an image and a projection. Three systems described by nonlinear differential equations are constructed, and the stability of a set of equilibria corresponding to an optimized solution for each system is proved by using the Lyapunov stability theorem. To validate the theoretical result given by the proposed method, metal artifact reduction was numerically performed.

scholarly journals A Comprehensive Mathematical Model for SARS-CoV-2 in Caputo Derivative

2021 ◽  
Vol 5 (4) ◽  
pp. 271
Author(s):  
Yu Gu ◽  
Muhammad Altaf Khan ◽  
Y. S. Hamed ◽  
Bassem F. Felemban
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Caputo Derivative ◽  
Real Data ◽  
Model Parameters ◽  
Globally Asymptotically Stable ◽  
The Real ◽  
Free Case ◽  
The Stability ◽  
Definition Of

In the present work, we study the COVID-19 infection through a new mathematical model using the Caputo derivative. The model has all the possible interactions that are responsible for the spread of disease in the community. We first formulate the model in classical differential equations and then extend it into fractional differential equations using the definition of the Caputo derivative. We explore in detail the stability results for the model of the disease-free case when R0<1. We show that the model is stable locally when R0<1. We give the result that the model is globally asymptotically stable whenever R0≤1. Further, to estimate the model parameters, we consider the real data of the fourth wave from Pakistan and provide a reasonable fitting to the data. We estimate the basic reproduction number for the proposed data to be R0=1.0779. Moreover, using the real parameters, we present the numerical solution by first giving a reliable scheme that can numerically handle the solution of the model. In our simulation, we give the graphical results for some sensitive parameters that have a large impact on disease elimination. Our results show that taking into consideration all the possible interactions can describe COVID-19 infection.

scholarly journals Safety peace in the Holy Quran : السلام الأمني في القرآن الكريم

2017 ◽  
Vol 1 (1) ◽  
Author(s):  
Samira Ahmed Mostafa Majdoubah -   Fadlan bin Mohammed Othma
Keyword(s):  
The World ◽  
Holy Quran ◽  
Peace And Security ◽  
The Holy Quran ◽  
The Stability ◽  
Definition Of ◽  
The Right

Praise be to God who blessed the Muslims of security in the world and the afterlife and prayers and peace upon our Prophet Mohammad (p.b.u.h). Security and peace are of the great blessings that God bestowed on the Islamic nation individuals and groups. They are playing a very important role in the achievement and innovation. They are essential need for the continuation of life and the habitation of land, and without it doesn’t have the stability and happiness, and this is done only by referring to the teaching of Koran and the Sunnah which guide man to the right proper way, and are making him happy and flood it safety and security upon him, and give psychological serenity and heart satisfaction feeling of comfortably to overcome the difficulties of life and its problems. This study was based on inductive analytical subject of (peaceful security in the Quran), the researcher states through the study the definition of peace and security in the language and in the terminology of peace, words and their meanings as set out in the context of the Qur'an with an indication to the importance of achieving the security of peace among the members of the community by referring to the Quran and teachings of the Prophet, peace be upon him

On the Stability and Boundedness of Differential Systems

1962 ◽  
Vol 58 (3) ◽  
pp. 492-496 ◽  
Author(s):  
V. Lakshmikantham
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Real Banach Space ◽  
Closed Linear Operator ◽  
Differential Systems ◽  
The Real ◽  
The Stability ◽  
Definition Of ◽  
Image Position ◽  
The Right

Consider the differential systemswhere A(t), g(t, y) and g(t, y) are operators acting in the real Banach space E, A(t) is an unbounded, closed, linear operator for each t in 0 ≤ t < ∞ and x0, y0 belong to the domain of definition of the operator A (t0). Let ‖x‖ denote the norm of an element x ε: E and R(λ, t) the resolvent of A(t). Here and in the following the prime denotes the right-hand derivative.

scholarly journals On the Lyapunov functionals method in the stability problem of Volterra integro-differential equations

2018 ◽  
Vol 20 (3) ◽  
pp. 260-272
Author(s):  
Aleksandr S. Andreev ◽  
Olga A. Peregudova
Keyword(s):  
Differential Equations ◽  
Stability Problem ◽  
Infinite Delay ◽  
Zero Solution ◽  
Model Systems ◽  
Lyapunov Functionals ◽  
Structure Equations ◽  
Positive Limit Set ◽  
The Stability ◽  
The Right

In this paper, we consider the problem of applying the method of Lyapunov functionals to investigate the stability of non-linear integro-differential equations, the right-hand side of which is the sum of the components of the instantaneous action and also ones with a finite and infinite delay. The relevance of the problem is the widespread use of such complicated in structure equations in modeling the controllers using integral regulators for mechanical systems, as well as biological, physical and other processes. We develop the Lyapunov functionals method in the direction of revealing the limiting properties of solutions by means of Lyapunov functionals with a semi-definite derivative. We proved the theorems on the quasi-invariance of a positive limit set of bounded solution as well as ones on the asymptotic stability of the zero solution including a uniform one. The results are achieved by constructing a new structure of the topological dynamics of the equations under study. The theorems proved are applied in solving the stability problem of two model systems which are generalizations of a number of known models of natural science and technology.

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