scholarly journals On the asymptotic analysis of bounded solutions to nonlinear differential equations of second order

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Cemil Tunç ◽  
Sizar Abid Mohammed
Keyword(s):  
Differential Equations ◽  
Lyapunov Functions ◽  
Nonlinear Differential Equations ◽  
Second Order ◽  
Nonlinear Ordinary Differential Equations ◽  
Bounded Solutions ◽  
Boundedness Of Solutions ◽  
Nonlinear Odes ◽  
Inequality Techniques ◽  
Lyapunov Second Method

Abstract In this paper, we consider two different models of nonlinear ordinary differential equations (ODEs) of second order. We construct two new Lyapunov functions to investigate boundedness of solutions of those nonlinear ODEs of second order. By using the Lyapunov direct or second method and inequality techniques, we prove two new theorems on the boundedness solutions of those ODEs of second order as $t \to \infty $t→∞. When we compare the conditions of the theorems of this paper with those of Meng in (J. Syst. Sci. Math. Sci. 15(1):50–57, 1995) and Sun and Meng in (Ann. Differ. Equ. 18(1):58–64 2002), we can see that our theorems have less restrictive conditions than those in (Meng in J. Syst. Sci. Math. Sci. 15(1):50–57, 1995) and Sun and Meng in (Ann. Differ. Equ. 18(1):58–64 2002) because of the two new suitable Lyapunov functions. Next, in spite of the use of the Lyapunov second method here and in (Meng in J. Syst. Sci. Math. Sci. 15(1):50–57, 1995; Sun and Meng in Ann. Differ. Equ. 18(1):58–64 2002), the proofs of the results of this paper are proceeded in a very different way from that used in the literature for the qualitative analysis of ODEs of second order. Two examples are given to show the applicability of our results. At the end, we can conclude that the results of this paper generalize and improve the results of Meng in (J. Syst. Sci. Math. Sci. 15(1):50–57, 1995), Sun and Meng in (Ann. Differ. Equ. 18(1):58–64 2002), and some other that can be found in the literature, and they have less restrictive conditions than those in these references.

scholarly journals On the complete integrability and linearization of nonlinear ordinary differential equations. IV. Coupled second-order equations

2008 ◽  
Vol 465 (2102) ◽  
pp. 609-629 ◽  
Author(s):  
V.K Chandrasekar ◽  
M Senthilvelan ◽  
M Lakshmanan
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Nonlinear Differential Equations ◽  
Second Order ◽  
Nonlinear Ordinary Differential Equations ◽  
Integrals Of Motion ◽  
Nonlinear Odes ◽  
Second Order Equations ◽  
Integrating Factors

Coupled second-order nonlinear differential equations are of fundamental importance in dynamics. In this part of our study on the integrability and linearization of nonlinear ordinary differential equations (ODEs), we focus our attention on the method of deriving a general solution for two coupled second-order nonlinear ODEs through the extended Prelle–Singer procedure. We describe a procedure to obtain integrating factors and the required number of integrals of motion so that the general solution follows straightforwardly from these integrals. Our method tackles both isotropic and non-isotropic cases in a systematic way. In addition to the above-mentioned method, we introduce a new method of transforming coupled second-order nonlinear ODEs into uncoupled ones. We illustrate the theory with potentially important examples.

scholarly journals Quasi–additive solutions of nonlinear differential equations II

1988 ◽  
Vol 38 (1) ◽  
pp. 19-21 ◽  
Author(s):  
A.S. Jones
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Nonlinear Differential Equations ◽  
Second Order ◽  
Nonlinear Ordinary Differential Equations

In a previous paper, the author sought to classify those solutions of second order nonlinear ordinary differential equations which can be expressed as sums of solutions of related equations. In that paper one sub-class of solutions was overlooked. This paper is to remedy that defect.

Hyperasymptotics for nonlinear ODEs II. The first Painlevé equation and a second-order Riccati equation

2005 ◽  
Vol 461 (2062) ◽  
pp. 3005-3021 ◽  
Author(s):  
A.B Olde Daalhuis
Keyword(s):  
Differential Equations ◽  
Riccati Equation ◽  
Second Order ◽  
Painlevé Equation ◽  
Nonlinear Ordinary Differential Equations ◽  
Positive Real ◽  
Nonlinear Odes ◽  
First Order ◽  
Sheet Structure ◽  
Painleve Equation

This paper is a sequel to Olde Daalhuis (Olde Daalhuis 2005 Proc. R. Soc. A 461 , 2503–2520) in which we constructed hyperasymptotic expansions for a simple first-order Riccati equation. In this paper we illustrate that the method also works for more complicated nonlinear ordinary differential equations, and that in those cases the Riemann sheet structure of the so-called Borel transform is much more interesting. The two examples are the first Painlevé equation and a second-order Riccati equation. The main tools that we need are transseries expansions and Stokes multipliers. Hyperasymptotic expansions determine the solutions uniquely. Some details are given about solutions that are real-valued on the positive real axis.

A boundary value problem on the whole line to second order nonlinear differential equations

2010 ◽  
Vol 17 (2) ◽  
pp. 373-390
Author(s):  
Christos G. Philos ◽  
Ioannis K. Purnaras
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Ordinary Differential Equations ◽  
Nonlinear Differential Equations ◽  
Boundary Value ◽  
Second Order ◽  
Nonlinear Ordinary Differential Equations ◽  
Specific Class ◽  
Uniqueness Of Solutions ◽  
Linear Ordinary Differential Equations

Abstract Second order nonlinear ordinary differential equations are considered, and a certain boundary value problem on the whole line is studied. Two theorems are obtained as main results. The first theorem is established by the use of the Schauder theorem and concerns the existence of solutions, while the second theorem is concerned with the existence and uniqueness of solutions and is derived by the Banach contraction principle. These two theorems are applied, in particular, to the specific class of second order nonlinear ordinary differential equations of Emden–Fowler type and to the special case of second order linear ordinary differential equations, respectively.

scholarly journals A note on boundedness of solutions to a class of non-autonomous differential equations of second order

2010 ◽  
Vol 4 (2) ◽  
pp. 361-372 ◽  
Author(s):  
Cemil Tunç
Keyword(s):  
Theoretical Analysis ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Second Order ◽  
Boundedness Of Solutions ◽  
Integral Test ◽  
Liapunov Functions ◽  
Autonomous Differential Equations

By defining some appropriate Liapunov functions, we discuss boundedness of solutions to a class of non-autonomous and nonlinear differential equations of second order. By this work, we prove some results established in the literature by Liapunov's second method instead of the integral test. We give six examples to illustrate the theoretical analysis in this work and effectiveness of the method utilized here.

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