scholarly journals On construction of a field of forces along given trajectories in the presence of random perturbations

2021 ◽  
Vol 101 (1) ◽  
pp. 98-103
Author(s):  
M.I. Tleubergenov ◽  
◽  
G.K. Vassilina ◽  
G.A. Tuzelbaeva ◽  
◽  
...  
Keyword(s):  
Differential Equations ◽  
Force Field ◽  
Integral Manifold ◽  
Sufficient Conditions ◽  
Second Order ◽  
Generalized Coordinates ◽  
Integral Manifolds ◽  
The Cauchy Problem ◽  
Stochastic Equivalence ◽  
The Right

In this paper, a force field is constructed along a given integral manifold in the presence of random perturbing forces. In this case, two types of integral manifolds are considered separately: 1) trajectories that depend on generalized coordinates and do not depend on generalized velocities, and 2) trajectories that depend on both generalized coordinates and generalized velocities. The construction of the force field is carried out in the class of second-order stochastic Ito differential equations. It is assumed that the functions in the right-hand sides of the equation must be continuous in time and satisfy the Lipschitz condition in generalized coordinates and generalized velocities. Also this functions satisfy the condition for linear growth in generalized coordinates and generalized velocities.These assumptions ensure the existence and uniqueness up to stochastic equivalence of the solution to the Cauchy problem of the constructed equations in the phase space, which is a strictly Markov process continuous with probability 1. To solve the two posed problems, stochastic differential equations of perturbed motion with respect to the integral manifold are constructed. Moreover, in the case when the trajectories depend on generalized coordinates and do not depend on generalized velocities, the second order equations of perturbed motion are constructed, and in the case when the trajectories depend on both generalized coordinates and generalized velocities, the first order equations of perturbed motion are constructed. And further, in both cases by Erugin’s method necessary and sufficient conditions for solving the posed problems are derived.

scholarly journals An implementation of the Chebyshev series method for the approximate analytical solution of second-order ordinary differential equations

2019 ◽  
pp. 97-103
Author(s):  
О.Б. Арушанян ◽  
С.Ф. Залеткин
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Canonical System ◽  
Second Order ◽  
Partial Sums ◽  
Problem Solution ◽  
Chebyshev Series ◽  
Canonical Systems ◽  
The Cauchy Problem ◽  
The Right

Описан один метод по применению рядов Чебышёва для интегрирования канонических систем обыкновенных дифференциальных уравнений второго порядка. Этот метод основан на аппроксимации решения задачи Коши, его первой и второй производных частичными суммами смещенных рядов Чебышёва. Коэффициенты рядов вычисляются итерационным способом с применением соотношений, связывающих коэффициенты Чебышёва решения задачи Коши, а также коэффициенты Чебышёва первой производной решения с коэффициентами Чебышёва правой части системы. Неотъемлемым элементом вычислительной схемы является использование формулы численного интегрирования Маркова для вычисления коэффициентов Чебышёва правой части системы. В статье не только сообщаются результаты, полученные численными расчетами, но и делается упор на высокоточном аналитическом представлении решения в виде частичной суммы ряда на промежутке интегрирования. A method used to apply the Chebyshev series for solving canonical systems of second order ordinary differential equations is described. This method is based on the approximation of the Cauchy problem solution and its first and second derivatives by partial sums of shifted Chebyshev series. The coefficients of these series are determined iteratively using the relations relating the Chebyshev coefficients of the solution and its first derivative with the Chebyshev coefficients found for the right-hand side of the canonical system by application of Markov's quadrature formula. The obtained numerical results are discussed and the high-precision analytical representations of the solution are proposed in the form of partial sums of Chebyshev series on a given integration segment.

scholarly journals New Theorems for Oscillations to Differential Equations with Mixed Delays

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 367
Author(s):  
Shyam Sundar Santra ◽  
Debasish Majumder ◽  
Rupak Bhattacharjee ◽  
Omar Bazighifan ◽  
Khaled Mohamed Khedher ◽  
...  
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Oscillatory Behavior ◽  
Second Order ◽  
Necessary And Sufficient Conditions ◽  
Mixed Delays ◽  
Second Order Differential Equations ◽  
Neutral Term ◽  
The Right ◽  
Necessary And Sufficient

The oscillation of differential equations plays an important role in many applications in physics, biology and engineering. The symmetry helps to deciding the right way to study oscillatory behavior of solutions of this equations. The purpose of this article is to establish new oscillatory properties which describe both the necessary and sufficient conditions for a class of nonlinear second-order differential equations with neutral term and mixed delays of the form p(ι)w′(ι)α′+r(ι)uβ(ν(ι))=0,ι≥ι0 where w(ι)=u(ι)+q(ι)u(ζ(ι)). Furthermore, examining the validity of the proposed criteria has been demonstrated via particular examples.

scholarly journals Second-Order Non-Canonical Neutral Differential Equations with Mixed Type: Oscillatory Behavior

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 318
Author(s):  
Osama Moaaz ◽  
Amany Nabih ◽  
Hammad Alotaibi ◽  
Y. S. Hamed
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Mixed Type ◽  
Relevant Literature ◽  
Sufficient Conditions ◽  
Oscillatory Behavior ◽  
Second Order ◽  
Neutral Differential Equations ◽  
Delay Differential ◽  
Oscillation Of Solutions

In this paper, we establish new sufficient conditions for the oscillation of solutions of a class of second-order delay differential equations with a mixed neutral term, which are under the non-canonical condition. The results obtained complement and simplify some known results in the relevant literature. Example illustrating the results is included.

scholarly journals Second-order impulsive differential systems with mixed and several delays

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Shyam Sundar Santra ◽  
Apurba Ghosh ◽  
Omar Bazighifan ◽  
Khaled Mohamed Khedher ◽  
Taher A. Nofal
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Oscillation Theory ◽  
Second Order ◽  
Neutral Delay ◽  
Differential Systems ◽  
Impulsive Differential Systems ◽  
Several Delays ◽  
Necessary And Sufficient

AbstractIn this work, we present new necessary and sufficient conditions for the oscillation of a class of second-order neutral delay impulsive differential equations. Our oscillation results complement, simplify and improve recent results on oscillation theory of this type of nonlinear neutral impulsive differential equations that appear in the literature. An example is provided to illustrate the value of the main results.

scholarly journals New Conditions for the Oscillation of Second-Order Differential Equations with Sublinear Neutral Terms

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1159
Author(s):  
Shyam Sundar Santra ◽  
Omar Bazighifan ◽  
Mihai Postolache
Keyword(s):  
Fluid Dynamics ◽  
Neural Networks ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Qualitative Behavior ◽  
Neutral Differential Equations ◽  
Second Order Differential Equations ◽  
Delay Differential

In continuous applications in electrodynamics, neural networks, quantum mechanics, electromagnetism, and the field of time symmetric, fluid dynamics, neutral differential equations appear when modeling many problems and phenomena. Therefore, it is interesting to study the qualitative behavior of solutions of such equations. In this study, we obtained some new sufficient conditions for oscillations to the solutions of a second-order delay differential equations with sub-linear neutral terms. The results obtained improve and complement the relevant results in the literature. Finally, we show an example to validate the main results, and an open problem is included.

scholarly journals On the solvability of the modified Cauchy problem for linear systems of generalized ordinary differential equations with singularities

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Malkhaz Ashordia ◽  
Inga Gabisonia ◽  
Mzia Talakhadze
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Systems ◽  
Unique Solvability ◽  
Sufficient Conditions ◽  
The Cauchy Problem ◽  
Generalized Ordinary Differential Equations

AbstractEffective sufficient conditions are given for the unique solvability of the Cauchy problem for linear systems of generalized ordinary differential equations with singularities.

scholarly journals New Results on Qualitative Behavior of Second Order Nonlinear Neutral Impulsive Differential Systems with Canonical and Non-Canonical Conditions

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 934
Author(s):  
Shyam Sundar Santra ◽  
Khaled Mohamed Khedher ◽  
Kamsing Nonlaopon ◽  
Hijaz Ahmad
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Oscillatory Behavior ◽  
Second Order ◽  
Discrete Equation ◽  
Qualitative Behavior ◽  
Differential Systems ◽  
Impulsive Differential Systems

The oscillation of impulsive differential equations plays an important role in many applications in physics, biology and engineering. The symmetry helps to deciding the right way to study oscillatory behavior of solutions of impulsive differential equations. In this work, several sufficient conditions are established for oscillatory or asymptotic behavior of second-order neutral impulsive differential systems for various ranges of the bounded neutral coefficient under the canonical and non-canonical conditions. Here, one can see that if the differential equations is oscillatory (or converges to zero asymptotically), then the discrete equation of similar type do not disturb the oscillatory or asymptotic behavior of the impulsive system, when impulse satisfies the discrete equation. Further, some illustrative examples showing applicability of the new results are included.

scholarly journals Periodic solutions for periodic second-order differential equations with variable potentials

2018 ◽  
Vol 24 (2) ◽  
pp. 127-137
Author(s):  
Jaume Llibre ◽  
Ammar Makhlouf
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Second Order Differential Equations ◽  
Second Order Differential Equation ◽  
Existence Of Periodic Solutions

Abstract We provide sufficient conditions for the existence of periodic solutions of the second-order differential equation with variable potentials {-(px^{\prime})^{\prime}(t)-r(t)p(t)x^{\prime}(t)+q(t)x(t)=f(t,x(t))} , where the functions {p(t)>0} , {q(t)} , {r(t)} and {f(t,x)} are {\mathcal{C}^{2}} and T-periodic in the variable t.

scholarly journals Boundedness of Solutions for Second Order Differential Equations

2020 ◽  
Vol 12 (4) ◽  
pp. 58
Author(s):  
Daniel C. Biles
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Linear Differential Equations ◽  
Boundedness Of Solutions ◽  
Second Order Differential Equations ◽  
All Solutions ◽  
Non Linear ◽  
Linear Differential

We present new theorems which specify sufficient conditions for the boundedness of all solutions for second order non-linear differential equations. Unboundedness of solutions is also considered.

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