scholarly journals Compatibility aspects of the method of phase synchronization for decoupling linear second-order differential equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Willy Sarlet ◽  
Tom Mestdag
Keyword(s):  
General Theory ◽  
Differential Equations ◽  
Linear Transformation ◽  
Phase Synchronization ◽  
Second Order ◽  
Order System ◽  
Linear Transformations ◽  
Second Order Differential Equations ◽  
Geometric Context ◽  
The Given

<p style='text-indent:20px;'>The so-called method of phase synchronization has been advocated in a number of papers as a way of decoupling a system of linear second-order differential equations by a linear transformation of coordinates and velocities. This is a rather unusual approach because velocity-dependent transformations in general do not preserve the second-order character of differential equations. Moreover, at least in the case of linear transformations, such a velocity-dependent one defines by itself a second-order system, which need not have anything to do, in principle, with the given system or its reformulation. This aspect, and the related questions of compatibility it raises, seem to have been overlooked in the existing literature. The purpose of this paper is to clarify this issue and to suggest topics for further research in conjunction with the general theory of decoupling in a differential geometric context.</p>

scholarly journals New collocation integrator for solving dynamic problems

2020 ◽  
pp. 131-140
Author(s):  
V.A. Avdyshev ◽  
Keyword(s):  
General Theory ◽  
Differential Equations ◽  
Second Order ◽  
Dynamic Problems ◽  
Second Order Differential Equations ◽  
Mixed Systems

A new collocation integrator on Lobatto spacings is proposed for numerically solving mixed systems of first and second order differential equations of dynamic problems. The general theory of collocation integrators is described, from which the basic formulas of the new integrator are derived.

scholarly journals Applications of measure of noncompactness for the solvability of an infinite system of second order differential equations in some integrated sequence spaces

2021 ◽  
Vol 40 (2) ◽  
pp. 573-592
Author(s):  
Rituparna Das ◽  
Niraj Sapkota
Keyword(s):  
Differential Equations ◽  
Measure Of Noncompactness ◽  
Infinite System ◽  
Sufficient Conditions ◽  
Second Order ◽  
Sequence Spaces ◽  
Second Order Differential Equations ◽  
Banach Sequence Spaces ◽  
Condensing Operator ◽  
The Given

The aim of this paper is to study the infinite system of second order differential equations along with the given boundary conditions for its solvability in some integrated sequence spaces. The result is achieved with the analytical tool namely the measure of noncompactness along with the idea of Meir-Keeler condensing operator and provides the realization of the sufficient conditions for the existence results in these Banach Sequence spaces. We also illustrate the results with examples.

8.—On a Class of Series Expansions in the Theory of Emden's Equation

1973 ◽  
Vol 71 (2) ◽  
pp. 95-110 ◽  
Author(s):  
Einar Hille
Keyword(s):  
General Theory ◽  
Differential Equations ◽  
Series Expansion ◽  
Laurent Series ◽  
Second Order ◽  
Series Expansions ◽  
Section 8 ◽  
Second Order Differential Equations ◽  
Bona Fide ◽  
Image Position

SynopsisThis paper deals with the nature of movable singularities of solutions of Emden's equationat which the solution becomes infinite. If m = 1 + 2/p with p > 1 an integer, then the solution becomes infinite at a given point x = c asBy the general theory of P. Painlevé on movable poles of solutions of non-linear second order differential equations this ‘pseudo-pole’ cannot actually be a pole of order p. Instead of a bona fide Laurent series at x = c we obtain a series expansion of the formwhere Pn(t) is a polynomial in t of degree at most [n/(2p + 2)]. The object of this paper is to derive these series and to prove convergence for p = 2. In this case deg [P6m] is strictly equal to m. For other values of p, see Section 8, Addenda.

Banach Lie Algebroids

2011 ◽  
Vol 57 (2) ◽  
pp. 409-416
Author(s):  
Mihai Anastasiei
Keyword(s):  
Differential Equations ◽  
Smooth Manifold ◽  
Vector Bundles ◽  
Second Order ◽  
Lie Algebroids ◽  
Second Order Differential Equations

Banach Lie AlgebroidsFirst, we extend the notion of second order differential equations (SODE) on a smooth manifold to anchored Banach vector bundles. Then we define the Banach Lie algebroids as Lie algebroids structures modeled on anchored Banach vector bundles and prove that they form a category.

scholarly journals New oscillation criteria for second-order neutral differential equations with distributed deviating arguments

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Osama Moaaz ◽  
Choonkil Park ◽  
Elmetwally M. Elabbasy ◽  
Waed Muhsin
Keyword(s):  
Differential Equations ◽  
Second Order ◽  
The Other ◽  
Riccati Transformation ◽  
Transformation Technique ◽  
Deviating Arguments ◽  
Neutral Differential Equations ◽  
Second Order Differential Equations ◽  
Continuous Delay ◽  
New Criteria

AbstractIn this work, we create new oscillation conditions for solutions of second-order differential equations with continuous delay. The new criteria were created based on Riccati transformation technique and comparison principles. Furthermore, we obtain iterative criteria that can be applied even when the other criteria fail. The results obtained in this paper improve and extend the relevant previous results as illustrated by examples.

scholarly journals Fixed point theorems via Generalized WF-Contractions with applications

SeMA Journal ◽  
2021 ◽  
Author(s):  
Rosana Rodríguez-López ◽  
Rakesh Tiwari
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Second Order ◽  
Point Boundary ◽  
New Class ◽  
Second Order Differential Equations ◽  
Fixed Point Result

AbstractThe aim of this paper is to introduce a new class of mixed contractions which allow to revise and generalize some results obtained in [6] by R. Gubran, W. M. Alfaqih and M. Imdad. We also provide an example corresponding to this class of mappings and show how the new fixed point result relates to the above-mentioned result in [6]. Further, we present an application to the solvability of a two-point boundary value problem for second order differential equations.

scholarly journals Oscillation of Second-Order Differential Equations with Multiple and Mixed Delays under a Canonical Operator

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1323
Author(s):  
Shyam Sundar Santra ◽  
Rami Ahmad El-Nabulsi ◽  
Khaled Mohamed Khedher
Keyword(s):  
Differential Equations ◽  
Necessary Conditions ◽  
Second Order ◽  
Multiple Delays ◽  
Mixed Delays ◽  
Neutral Differential Equations ◽  
Canonical Operator ◽  
Second Order Differential Equations ◽  
The Future ◽  
Sufficient And Necessary Conditions

In this work, we obtained new sufficient and necessary conditions for the oscillation of second-order differential equations with mixed and multiple delays under a canonical operator. Our methods could be applicable to find the sufficient and necessary conditions for any neutral differential equations. Furthermore, we proved the validity of the obtained results via particular examples. At the end of the paper, we provide the future scope of this study.

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