On solutions of a second-order quasilinear differential system representable by Fourier series with slowly varying parameters

1998 ◽  
Vol 50 (5) ◽  
pp. 741-753 ◽  
Author(s):  
A. V. Kostin ◽  
S. A. Shchegolev
Keyword(s):  
Fourier Series ◽  
Differential System ◽  
Second Order ◽  
Varying Parameters

An Asymptotical Stability in Variational Sense for Second Order Systems

2010 ◽  
Vol 10 (1) ◽  
Author(s):  
Dariusz Idczak ◽  
Stanisław Walczak
Keyword(s):  
Sobolev Space ◽  
Differential System ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Second Order ◽  
Functional Parameter ◽  
Asymptotical Stability ◽  
Parameter Control ◽  
Initial Condition ◽  
Second Order Systems

AbstractIn this paper, a new, variational concept of asymptotical stability of zero solution to an ordinary differential system of the second order, considered in Sobolev space, is presented. Sufficient conditions for an asymptotical stability in a variational sense with respect to initial condition and functional parameter (control) are given. Relation to the classical asymptotical stability is illustrated.

scholarly journals Second Approximate Solution of Duffing Equation with Strong Nonlinearity by Homotopy Perturbation Method

1970 ◽  
Vol 30 ◽  
pp. 59-75
Author(s):  
M Alhaz Uddin ◽  
M Abdus Sattar
Keyword(s):  
Approximate Solution ◽  
Nonlinear Systems ◽  
Perturbation Method ◽  
Differential System ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Second Order ◽  
Strong Nonlinearity ◽  
Analytical Technique ◽  
Homotopy Perturbation

 In this paper, the second order approximate solution of a general second order nonlinear ordinary differential system, modeling damped oscillatory process is considered. The new analytical technique based on the work of He’s homotopy perturbation method is developed to find the periodic solution of a second order ordinary nonlinear differential system with damping effects. Usually the second or higher order approximate solutions are able to give better results than the first order approximate solutions. The results show that the analytical approximate solutions obtained by homotopy perturbation method are uniformly valid on the whole solutions domain and they are suitable not only for strongly nonlinear systems, but also for weakly nonlinear systems. Another advantage of this new analytical technique is that it also works for strongly damped, weakly damped and undamped systems. Figures are provided to show the comparison between the analytical and the numerical solutions. Keywords: Homotopy perturbation method; damped oscillation; nonlinear equation; strong nonlinearity. GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 30 (2010) 59-75  DOI: http://dx.doi.org/10.3329/ganit.v30i0.8504

Basic Fourier series

1979 ◽  
Vol 369 (1736) ◽  
pp. 115-136 ◽  
Keyword(s):  
Time Series ◽  
Fourier Series ◽  
Stochastic Processes ◽  
Numerical Approximation ◽  
Basic Number ◽  
Second Order ◽  
Orthogonal Functions ◽  
Sturm Liouville

The concept of basic number is applied to the development of a simple analogue of the Sturm–Liouville system of the second order. This is then employed to deduce a family of q -orthogonal functions, which leads to a generalization of the Fourier and Fourier–Bessel expansions. The numerical approximation of basic integrals is discussed and some aspects of the evaluation of C a (q; x) are mentioned. A few of the zeros of this function are listed, and, in conclusion, an indication is given of the possibility of applying the analysis presented in this paper to thé study of stochastic processes and time-series.

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