scholarly journals Asymptotic method for solution of identification problem of the nonlinear dynamic systems

Filomat ◽  
2018 ◽  
Vol 32 (3) ◽  
pp. 1025-1033 ◽  
Author(s):  
F.A. Aliev ◽  
N.A. Ismailov ◽  
A.A. Namazov ◽  
N.A. Safarova ◽  
M.F. Rajabov ◽  
...  
Keyword(s):  
Differential Equations ◽  
Small Parameter ◽  
Least Squares Method ◽  
Identification Problem ◽  
Euler Method ◽  
Vector Parameter ◽  
Constant Vector ◽  
Unknown Constant ◽  
Linear Differential ◽  
The Right

A dynamic system, when the motion of the object is described by the system of nonlinear ordinary differential equations, is considered. The right part of the system involves the phase coordinates as a unknown constant vector-parameter and a small number. The statistical data are taken from practice: the initial and final values of the object coordinates. Using the method of quasilinearization the given equation is reduced to the system of linear differential equations, where the coefficients of the coordinate and unknown parameter, also of the perturbations depend on a small parameter linearly. Then, by using the least-squares method the unknown constant vector-parameter is searched in the form of power series on a small parameter and for the coefficients of zero and the first orders the analytical formulas are given. The fundamental matrices both in a zero and in the first approach are constructed approximately, by means of the ordinary Euler method. On an example the determination of the coefficient of hydraulic resistance (CHR) in the lift in the oil extraction by gas lift method is illustrated, as the obtained results in the first approaching coincides with well-known results on order of 10-2.

scholarly journals II. On the solution of linear differential equations

1848 ◽  
Vol 138 ◽  
pp. 31-54 ◽  
Keyword(s):  
Differential Equations ◽  
Differential Expression ◽  
Linear Differential Equations ◽  
Right Hand ◽  
Independent Variable ◽  
Linear Differential ◽  
The Right

If the operation of differentiation with regard to the independent variable x be denoted by the symbol D, and if ϕ (D) represent any function of D composed of integral powers positive or negative, or both positive and negative, it may easily be shown, that ϕ (D){ψ x. u } = ψ x. ϕ (D) u + ψ' x. ϕ' (D) u + ½ψ" x. ϕ" (D) u + 1/2.3 ψ"' x. ϕ"' (D) u + . . . (1.) and that ϕx .ψ(D) u = ψ(D){ ϕx. u } - ψ'(D){ ϕ'x. u } + ½ψ"(D){ ϕ"x. u } - 1/2.3ψ"'(D){ ϕ"'x. u } + . . (2.) and these general theorems are expressions of the laws under which the operations of differentiation, direct and inverse, combine with those operations which are de­noted by factors, functions of the independent variable. It will be perceived that the right-hand side of each of these equations is a linear differential expression; and whenever an expression assumes or can be made to assume either of these forms, its solution is determined; for the equations ϕ (D){ψ x. u } = P and ϕx . ψ(D) u = P are respectively equivalent to u = (ψ x ) -1 { ϕ (D)} -1 P and u = {ψ(D)} -1 (( ϕx ) -1 P).

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 Efficient quantum algorithm for dissipative nonlinear differential equations

2021 ◽  
Vol 118 (35) ◽  
pp. e2026805118
Author(s):  
Jin-Peng Liu ◽  
Herman Øie Kolden ◽  
Hari K. Krovi ◽  
Nuno F. Loureiro ◽  
Konstantina Trivisa ◽  
...  
Keyword(s):  
Differential Equations ◽  
Quantum Algorithms ◽  
Nonlinear Differential Equations ◽  
Quantum Algorithm ◽  
Euler Method ◽  
Linear Differential Equations ◽  
Dimensional System ◽  
Worst Case ◽  
Potential Applications ◽  
Linear Differential

Nonlinear differential equations model diverse phenomena but are notoriously difficult to solve. While there has been extensive previous work on efficient quantum algorithms for linear differential equations, the linearity of quantum mechanics has limited analogous progress for the nonlinear case. Despite this obstacle, we develop a quantum algorithm for dissipative quadratic n-dimensional ordinary differential equations. Assuming R<1, where R is a parameter characterizing the ratio of the nonlinearity and forcing to the linear dissipation, this algorithm has complexity T2q poly(log⁡T,log⁡n,log⁡1/ϵ)/ϵ, where T is the evolution time, ϵ is the allowed error, and q measures decay of the solution. This is an exponential improvement over the best previous quantum algorithms, whose complexity is exponential in T. While exponential decay precludes efficiency, driven equations can avoid this issue despite the presence of dissipation. Our algorithm uses the method of Carleman linearization, for which we give a convergence theorem. This method maps a system of nonlinear differential equations to an infinite-dimensional system of linear differential equations, which we discretize, truncate, and solve using the forward Euler method and the quantum linear system algorithm. We also provide a lower bound on the worst-case complexity of quantum algorithms for general quadratic differential equations, showing that the problem is intractable for R≥2. Finally, we discuss potential applications, showing that the R<1 condition can be satisfied in realistic epidemiological models and giving numerical evidence that the method may describe a model of fluid dynamics even for larger values of R.

scholarly journals Error of small parameter methods in solving shortened equations of a synchronized oscillator

Radiotekhnika ◽  
2021 ◽  
pp. 113-117
Author(s):  
V.V. Rapin
Keyword(s):  
Differential Equations ◽  
Small Parameter ◽  
Relative Error ◽  
Nonlinear Differential Equations ◽  
Parameter Method ◽  
Frequency Detuning ◽  
Small Parameter Method ◽  
First Approximation ◽  
Linear Differential ◽  
Zero Frequency

The paper considers the use of recently appeared analytical methods for solving shortened equations of a synchronized oscillator. These are a quasi-small parameter method and a combined small parameter method. Both methods use the classic small parameter method. A peculiarity of their application is that in this case they are used for solving nonlinear differential equations that do not contain a small parameter. The difference between the above methods is in obtaining the equations of the first approximation. In the quasi-small parameter method, they are linear differential equations obtained by linearizing the original nonlinear differential equations in the area of the zero frequency detuning. In the combined small parameter method, the equations of the first approximation are obtained by approximating the original nonlinear differential equations. Of course, a number of transformations of these equations were made for this. The approximation made it possible to obtain better representation of the original nonlinear differential equations by means of linear differential equations. This representation provided a smaller error, which in both cases was presented as a discrepancy. The discrepancy does not allow obtaining a relative error and investigating its peculiarity. A study of the relative error of the quasi-small parameter method shows that this error is a continuous function of the frequency detuning with a zero value for a zero frequency detuning. A function representing relative error has a gap at zero frequency detuning for the combined small parameter method. However, this kind of gap can be eliminated by additional function definition.

CONTINUOUS DEPENDENCE ON A PARAMETER OF THE SOLUTIONS OF IMPULSIVE DIFFERENTIAL EQUATIONS IN A BANACH SPACE

1993 ◽  
Vol 03 (04) ◽  
pp. 477-483
Author(s):  
D.D. BAINOV ◽  
S.I. KOSTADINOV ◽  
NGUYEN VAN MINH ◽  
P.P. ZABREIKO
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Differential Equations ◽  
Small Parameter ◽  
Continuous Dependence ◽  
Impulsive Differential Equation ◽  
Impulsive Differential Equations ◽  
Right Hand ◽  
Dependence On A Parameter ◽  
The Right

Continuous dependence of the solutions of an impulsive differential equation on a small parameter is proved under the assumption that the right-hand side of the equation and the impulse operators satisfy conditions of Lipschitz type.

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