scholarly journals A remark on renormalization group theoretical perturbation in a class of ordinary differential equations

2020 ◽  
Author(s):  
Atsuo Kuniba
Keyword(s):  
Renormalization Group ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Inversion Formula ◽  
Elementary Proof ◽  
Functional Relation ◽  
Perturbation Series ◽  
Van Der Pol ◽  
Oscillator Type ◽  
Secular Terms

Abstract We revisit the renormalization group (RG) theoretical perturbation theory on oscillator type second order ordinary differential equations. For a class of potentials, we show a simple functional relation among secular coefficients of the harmonics in the naive perturbation series. It leads to an inversion formula between bare and renormalized amplitudes and an elementary proof of the absence of secular terms in all order of the RG series. The result covers non-autonomous as well as autonomous cases and refines earlier studies including the classic examples as Van der Pol, Mathieu, Duffing and Rayleigh equations.

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.

On Integrating Factors and a Perturbation Method

1995 ◽  
Author(s):  
W. T. van Horssen
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Ordinary Differential Equations ◽  
First Integrals ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Order Ordinary Differential Equation ◽  
First Order ◽  
Integrating Factors ◽  
Complete Solutions

Abstract In this paper the fundamental concept (due to Euler, 1734) of how to make a first order ordinary differential equation exact by means of integrating factors, is extended to n-th order (n ≥ 2) ordinary differential equations and to systems of first order ordinary differential equations. For new classes of differential equations first integrals or complete solutions can be constructed. Also a perturbation method based on integrating factors can be developed. To show how this perturbation method works the method is applied to the well-known Van der Pol equation.

scholarly journals Approximate transformations for van der Pol-type equations

2006 ◽  
Vol 2006 ◽  
pp. 1-11 ◽  
Author(s):  
R. K. Gazizov ◽  
C. M. Khalique
Keyword(s):  
Differential Equations ◽  
Small Parameter ◽  
Ordinary Differential Equations ◽  
Transformation Groups ◽  
Van Der Pol ◽  
Approximate Symmetries ◽  
Approximate Integration ◽  
Successive Reduction

Classification of van der Pol-type equations with respect to admitted approximate transformation groups transforming a small parameter is given. It is shown that approximate symmetries transforming the small parameter as well as the usual approximate symmetries can be used for approximate integration (e.g., by method of successive reduction of order) of ordinary differential equations with a small parameter.

scholarly journals Simplifying Coefficients in Differential Equations Associated with Higher Order Bernoulli Numbers of the Second Kind

2017 ◽  
Author(s):  
Feng Qi ◽  
Da-Wei Niu ◽  
Bai-Ni Guo
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Inversion Formula ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Higher Order ◽  
Bell Polynomials

In the paper, by virtue of the Faà di Bruno formula, some properties of the Bell polynomials of the second kind, and an inversion formula for the Stirling numbers of the first and second kinds, the authors establish meaningfully and significantly two identities which simplify coefficients in a family of ordinary differential equations associated with higher order Bernoulli numbers of the second kind.

scholarly journals Simplifying coefficients in a family of nonlinear ordinary differential equations

2019 ◽  
Vol 22 (2) ◽  
pp. 293-297 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Inversion Formula ◽  
Stirling Numbers ◽  
Combinatorial Analysis ◽  
Bell Polynomials ◽  
Nonlinear Ordinary Differential Equations ◽  
Faà Di Bruno Formula

By virtue of the Faá di Bruno formula, properties of the Stirling numbers and the Bell polynomials of the second kind, the binomial inversion formula, and other techniques in combinatorial analysis, the author finds a simple, meaningful, and signicant expression for coefficients in a family of nonlinear ordinary differential equations.

scholarly journals Simple forms for coefficients in two families of ordinary differential equations

2018 ◽  
Vol 6 (1) ◽  
pp. 7
Author(s):  
Feng Qi
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Inversion Formula ◽  
Stirling Numbers ◽  
Bell Polynomials ◽  
Faà Di Bruno Formula

In the paper, by virtue of the Faá di Bruno formula, properties of the Bell polynomials of the second kind, and the inversion formula for the Stirling numbers of the first and second kinds, the author finds simple, meaningful, and significant forms for coefficients in two families of ordinary differential equations.

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