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.

Hyperasymptotics for nonlinear ODEs I. A Riccati equation

2005 ◽  
Vol 461 (2060) ◽  
pp. 2503-2520 ◽  
Author(s):  
A.B Olde Daalhuis
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Riccati Equation ◽  
Asymptotic Expansions ◽  
Riemann Sheet ◽  
Nonlinear Ordinary Differential Equations ◽  
Numerical Illustration ◽  
Nonlinear Odes ◽  
Borel Transform ◽  
Sheet Structure

We illustrate how one can obtain hyperasymptotic expansions for solutions of nonlinear ordinary differential equations. The example is a Riccati equation. The main tools that we need are transseries expansions and the Riemann sheet structure of the Borel transform of the divergent asymptotic expansions. Hyperasymptotic expansions determine the solutions uniquely. A numerical illustration is included.

A Novel General and Robust Method Based on NAOP for Solving Nonlinear Ordinary Differential Equations and Partial Differential Equations by Cellular Neural Networks

2013 ◽  
Vol 135 (3) ◽  
Author(s):  
Jean Chamberlain Chedjou ◽  
Kyandoghere Kyamakya
Keyword(s):  
Neural Networks ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Cellular Neural Networks ◽  
Nonlinear Ordinary Differential Equations ◽  
Van Der Pol ◽  
Partial Differential ◽  
Nonlinear Odes ◽  
First Order

This paper develops and validates through a series of presentable examples, a comprehensive high-precision, and ultrafast computing concept for solving nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs) with cellular neural networks (CNN). The core of this concept is a straightforward scheme that we call "nonlinear adaptive optimization (NAOP),” which is used for a precise template calculation for solving nonlinear ODEs and PDEs through CNN processors. One of the key contributions of this work is to demonstrate the possibility of transforming different types of nonlinearities displayed by various classical and well-known nonlinear equations (e.g., van der Pol-, Rayleigh-, Duffing-, Rössler-, Lorenz-, and Jerk-equations, just to name a few) unto first-order CNN elementary cells, and thereby enabling the easy derivation of corresponding CNN templates. Furthermore, in the case of PDE solving, the same concept also allows a mapping unto first-order CNN cells while considering one or even more nonlinear terms of the Taylor's series expansion generally used in the transformation of a PDE in a set of coupled nonlinear ODEs. Therefore, the concept of this paper does significantly contribute to the consolidation of CNN as a universal and ultrafast solver of nonlinear ODEs and/or PDEs. This clearly enables a CNN-based, real-time, ultraprecise, and low-cost computational engineering. As proof of concept, two examples of well-known ODEs are considered namely a second-order linear ODE and a second order nonlinear ODE of the van der Pol type. For each of these ODEs, the corresponding precise CNN templates are derived and are used to deduce the expected solutions. An implementation of the concept developed is possible even on embedded digital platforms (e.g., field programmable gate array (FPGA), digital signal processor (DSP), graphics processing unit (GPU), etc.). This opens a broad range of applications. Ongoing works (as outlook) are using NAOP for deriving precise templates for a selected set of practically interesting ODEs and PDEs equation models such as Lorenz-, Rössler-, Navier Stokes-, Schrödinger-, Maxwell-, etc.

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 On the complete integrability and linearization of nonlinear ordinary differential equations. III. Coupled first-order equations

2008 ◽  
Vol 465 (2102) ◽  
pp. 585-608 ◽  
Author(s):  
V.K Chandrasekar ◽  
M Senthilvelan ◽  
M Lakshmanan
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Autonomous Systems ◽  
Three Dimensional ◽  
Complete Integrability ◽  
Nonlinear Ordinary Differential Equations ◽  
Nonlinear Odes ◽  
First Order ◽  
Open Questions ◽  
Motion Time

Continuing our study on the complete integrability of nonlinear ordinary differential equations (ODEs), in this paper we consider the integrability of a system of coupled first-order nonlinear ODEs of both autonomous and non-autonomous types. For this purpose, we modify the original Prelle–Singer (PS) procedure so as to apply it to both autonomous and non-autonomous systems of coupled first-order ODEs. We briefly explain the method of finding integrals of motion (time-independent as well as time-dependent integrals) for two and three coupled first-order ODEs by extending the PS method. From this we try to answer some of the open questions in the original PS method. We also identify integrable cases for the two-dimensional Lotka–Volterra system and three-dimensional Rössler system as well as other examples including non-autonomous systems in a straightforward way using this procedure. Finally, we develop a linearization procedure for coupled first-order ODEs.

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.

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