variation of constants
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2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Faming Guo ◽  
Ricai Luo ◽  
Xiaolan Qin ◽  
Yunfei Yi

In this paper, we study the problem of exponential stability for the Hopfield neural network with time-varying delays. Different from the existing results, we establish new stability criteria by employing the method of variation of constants and Gronwall’s integral inequality. Finally, we give several examples to show the effectiveness and applicability of the obtained criterion.


2021 ◽  
Author(s):  
Fritz Schwarz

The subject of this article are linear and quasilinear differential equations of second order that may be decomposed into a first-order component with guaranteed solution procedure for obtaining closed-form solutions. These are homogeneous or inhomogeneous linear components, special Riccati components, Bernoulli, Clairaut or d’Alembert components. Procedures are described how they may be determined and how solutions of the originally given second order equation may be obtained from them. This makes it possible to solve new classes of differential equations and opens up a new area of research. Applying decomposition to linear inhomogeneous equations a simple procedure for determining a special solution follows. It is not based on the method of variation of constants of Lagrange, and consequently does not require the knowledge of a fundamental system. Algorithms based on these results are implemented in the computer algebra system ALLTYPES which is available on the website www.alltypes.de.


2021 ◽  
Vol 25 (Spec. issue 1) ◽  
pp. 65-75
Author(s):  
Ali Sirma ◽  
Resat Kosker ◽  
Muzaffer Akat

In this study, we propose a numerical scheme for stochastic oscillators with additive noise obtained by the method of variation of constants formula using generalized numerical integrators. For both of the displacement and the velocity components, we show that the scheme has an order of 3/2 in one step convergence and a first order in overall convergence. Theoretical statements are supported by numerical experiments.


2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yonghui Xia ◽  
Hai Huang ◽  
Kit Ian Kou

<p style='text-indent:20px;'>Quaternion-valued differential equations (QDEs) is a new kind of differential equations. In this paper, an algorithm was presented for solving linear nonhomogeneous quaternionic-valued differential equations. The variation of constants formula was established for the nonhomogeneous quaternionic-valued differential equations. Moreover, several examples showed the feasibility of our algorithm. Finally, some open problems end this paper.</p>


2020 ◽  
Vol 18 (12) ◽  
pp. 18-29
Author(s):  
Layali Y. Salih AL-Mashhadani ◽  
Ahmed H. Abdullah

Time variation of constants of nature is still a question of debate among astronomers, physicists, geologists, and palaeontologists. But are the fundamental physical constants really varying in space or time and how changing these parameters may occur?. Paul Dirac was interested in this question in the large number hypothesis (LNH). He arrived by coincidence at the revolutionary hypothesis that the gravitational constant G should be varied inversely with the cosmic time t. LNH sparked off many ideas and arguments about the possibility of time or space variations of the fundamental constants of nature. In this work, we review details and arguments regarding the time and space variation of dimensional and dimensionless constants based on a detailed comparison for the recorded literature over about one and a half-century.


2020 ◽  
Vol 99 (3) ◽  
pp. 62-74
Author(s):  
M. Akat ◽  
◽  
R. Kosker ◽  
A. Sirma ◽  
◽  
...  

In this paper, a numerical approach is proposed based on the variation-of-constants formula for the numerical discretization Langevin-type equations. Linear and non-linear cases are treated separately. The proofs of convergence have been provided for the linear case, and the numerical implementation has been executed for the non-linear case. The order one convergence for the numerical scheme has been shown both theoretically and numerically. The stability of the numerical scheme has been shown numerically and depicted graphically.


2020 ◽  
pp. 1-43
Author(s):  
Pierre Magal ◽  
Ousmane Seydi

Abstract In this paper, we extend to the non-Hille–Yosida case a variation of constants formula for a nonautonomous and nonhomogeneous Cauchy problems first obtained by Gühring and Räbiger. By using this variation of constants formula, we derive a necessary and sufficient condition for the existence of an exponential dichotomy for the evolution family generated by the associated nonautonomous homogeneous problem. We also prove a persistence result of the exponential dichotomy for small perturbations. Finally, we illustrate our results by considering two examples. The first example is a parabolic equation with nonlocal and nonautonomous boundary conditions, and the second example is an age-structured model that is a hyperbolic equation.


Author(s):  
Zhongli You ◽  
JinRong Wang ◽  
Yong Zhou ◽  
Michal Fečkan

AbstractIn this paper, we study finite time stability for linear and nonlinear delay systems with linear impulsive conditions and linear parts defined by permutable matrices. We introduce a new concept of impulsive delayed matrix function and apply the variation of constants method to seek a representation of solution of linear impulsive delay systems, which can be well used to deal with finite time stability. We establish sufficient conditions for the finite time stability results by using the properties of impulsive delayed matrix exponential and Gronwall’s integral inequalities. Finally, we give numerical examples to demonstrate the validity of theoretical results and present some possible advantage by comparing the current work with the previous literature.


2018 ◽  
Vol 19 (3) ◽  
pp. 525
Author(s):  
José Vanterler da Costa Sousa ◽  
Edmundo Capelas de Oliveira

In this paper, we present and prove a new truncated V-fractional Taylor's formula using the truncated V-fractional variation of constants formula. In this sense, we present the truncated V-fractional Taylor's remainder by means of V-fractional integral, essential for analyzing and comparing the error, when approaching functions by polynomials. From these new results, some applications were made involving some inequalities, specifically, we generalize the Cauchy-Schwartz inequality.


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