Asymptotic reducibility of a nonlinear system of differential-functional equations to a linear system of ordinary differential equations

1977 ◽  
Vol 28 (5) ◽  
pp. 458-467
Author(s):  
M. S. Bortei ◽  
V. I. Fodchuk
Keyword(s):  
Nonlinear System ◽  
Linear System ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Functional Equations ◽  
Asymptotic Reducibility

scholarly journals The sufficient conditions for polystability of solutions of~nonlinear systems of ordinary differential equations

2018 ◽  
Vol 20 (3) ◽  
pp. 304-317
Author(s):  
Pavel A. Shamanaev ◽  
Olga S. Yazovtseva
Keyword(s):  
Banach Space ◽  
Nonlinear System ◽  
Nonlinear Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Approximation ◽  
Critical Case ◽  
Sufficient Conditions ◽  
Asymptotic Equivalence ◽  
Theorem Proof

The article states the sufficient polystability conditions for part of variables for nonlinear systems of ordinary differential equations with a sufficiently smooth right-hand side. The obtained theorem proof is based on the establishment of a local componentwise Brauer asymptotic equivalence. An operator in the Banach space that connects the solutions of the nonlinear system and its linear approximation is constructed. This operator satisfies the conditions of the Schauder principle, therefore, it has at least one fixed point. Further, using the estimates of the non-zero elements of the fundamental matrix, conditions that ensure the transition of the properties of polystability are obtained, if the trivial solution of the linear approximation system to solutions of a nonlinear system that is locally componentwise asymptotically equivalent to its linear approximation. There are given examples, that illustrate the application of proven sufficient conditions to the study of polystability of zero solutions of nonlinear systems of ordinary differential equations, including in the critical case, and also in the presence of positive eigenvalues.

scholarly journals Rhythm Drivers, Physical Properties, Mechanisms of Formation

2021 ◽  
Vol 248 ◽  
pp. 01007
Author(s):  
Mikhail Mazurov
Keyword(s):  
Mathematical Model ◽  
Partial Differential Equations ◽  
Nonlinear System ◽  
Differential Equations ◽  
Physical Properties ◽  
Ordinary Differential Equations ◽  
Numerical Study ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Mechanisms Of Formation

A mathematical model of the pacemaker is presented in the form of a nonlinear system of ordinary differential equations and in the form of a system of partial differential equations for distributed pacemakers. For the numerical study of the properties of the pacemaker, a modified axiomatic Wiener-Rosenbluth method was used using the properties of uniform almost periodic functions. Physical foundations, mechanisms of formation, properties of point and distributed pacemakers are described in detail.

scholarly journals On solvability of a nonlinear system of equations for the Fourier-Chebyshev coefficients in the problem of solving ordinary differential equations using Chebyshev series

2017 ◽  
pp. 169-174
Author(s):  
О.Б. Арушанян ◽  
С.Ф. Залеткин
Keyword(s):  
Nonlinear System ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Analytical Method ◽  
System Of Equations ◽  
Chebyshev Series ◽  
Nonlinear System Of Equations ◽  
Theoretical Substantiation ◽  
Solvability Theorem ◽  
Numerical Analytical Method

Сформулирована и доказана теорема о разрешимости нелинейной системы уравнений относительно приближенных значений коэффициентов Фурье-Чебышёва. Теорема является теоретическим обоснованием ранее предложенного численно-аналитического метода интегрирования обыкновенных дифференциальных уравнений с использованием рядов Чебышёва. A solvability theorem for a nonlinear system of equations with respect to approximate values of Fourier-Chebyshev coefficients is formulated and proved. This theorem is a theoretical substantiation for the previously proposed numerical-analytical method of solving ordinary differential equations using Chebyshev series.

scholarly journals Linearization of Two Second-Order Ordinary Differential Equations via Fiber Preserving Point Transformations

2011 ◽  
Vol 2011 ◽  
pp. 1-21 ◽  
Author(s):  
Sakka Sookmee ◽  
Sergey V. Meleshko
Keyword(s):  
Linear System ◽  
General Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Necessary Conditions ◽  
Second Order ◽  
Point Transformations ◽  
Constant Coefficients ◽  
Linearizable System

The necessary form of a linearizable system of two second-order ordinary differential equations y1″=f1(x,y1,y2,y1′,y2′), y2″=f2(x,y1,y2,y1′,y2′) is obtained. Some other necessary conditions were also found. The main problem studied in the paper is to obtain criteria for a system to be equivalent to a linear system with constant coefficients under fiber preserving transformations. A linear system with constant coefficients is chosen because of its simplicity in finding the general solution. Examples demonstrating the procedure of using the linearization theorems are presented.

scholarly journals On Mixed Convection Squeezing Flow of Nanofluids

Energies ◽  
2020 ◽  
Vol 13 (12) ◽  
pp. 3138 ◽  
Author(s):  
Sheikh Irfan Ullah Khan ◽  
Ebraheem Alzahrani ◽  
Umar Khan ◽  
Noreena Zeb ◽  
Anwar Zeb
Keyword(s):  
Nonlinear System ◽  
Differential Equations ◽  
Mixed Convection ◽  
Ordinary Differential Equations ◽  
Volume Fraction ◽  
Squeezing Flow ◽  
Physical Parameters ◽  
Gamma Alumina ◽  
Highly Nonlinear ◽  
The Impact

In this article, the impact of effective Prandtl number model on 3D incompressible flow in a rotating channel is proposed under the influence of mixed convection. The coupled nonlinear system of partial differential equations is decomposed into a highly nonlinear system of ordinary differential equations with aid of suitable similarity transforms. Then, the solution of a nonlinear system of ordinary differential equations is obtained numerically by using Runge–Kutta–Fehlberg (RKF) method. Furthermore, the surface drag force C f and the rate of heat transfer N u are portrayed numerically. The effects of different emerging physical parameters such as Hartmann number (M), Reynold’s number (Re), squeezing parameter ( β ), mixed convection parameter λ , and volume fraction ( φ ) are also incorporated graphically for γ — alumina. Due to the higher viscosity and thermal conductivity ethylene-based nanofluids, it is observed to be an effective common base fluid as compared to water. These observations portrayed the temperature of gamma-alumina ethylene-based nanofluids rising on gamma-alumina water based nanofluids.

scholarly journals A New Approximate Analytical Method for ODEs

2014 ◽  
Vol 10 (1) ◽  
pp. 19-30
Author(s):  
Hossein Aminikhah
Keyword(s):  
Exact Solution ◽  
Nonlinear System ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Analytical Method ◽  
New Method ◽  
Good Stability ◽  
Linear And Nonlinear ◽  
Stability And Accuracy ◽  
Theoretical Considerations

Abstract In this paper, we propose a new algorithm for solving ordinary differential equations. We show the superiority of this algorithm by applying the new method for some famous ODEs. Theoretical considerations are discussed. The first He's polynomials have used to reach the exact solution of these problems. This method which has good stability and accuracy properties is useful in deal with linear and nonlinear system of ordinary differential equations.

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