| Sun et al. 1] reduced the set of coupled rst order nonlinear partial diierential equations determining the wavelength-dependent, time-varying doped-ber ampli-er gain into a single ordinary diierential equation (ODE). In this paper we further simplify the ODE, greatly enhancing its utility as an analysis and design tool. We nd that the gain dynamics are completely speciied by the total number ...

The selection of the spatial basis functions is very important for model reduction of the nonlinear partial differential equations (PDEs) under time/space separation framework, which will significantly affect the accuracy and efficiency of the modeling. Using the spatial basis functions expansions and the Galerkin method, the finite-dimensional ordinary differential equation (ODE) systems can b...

Abstract: In this paper, the nonlinear Schrödinger equation with power law nonlinearity is studied. The first integral method, the Riccati sub-ODE method are efficient methods to construct the exact solutions of nonlinear partial differential equations.By means of these methods, the periodic and solitary wave solutions of the nonlinear Schrödinger equation with power law nonlinearity are obtained.

The problem of state-observation is addressed for nonlinear systems that can bemodeled by an ODE–PDE series association. The ODE subsystem assumes a triangular structure while the PDE element is of heat diffusion type. The aim is to accurately estimate online the state vector of the ODE subsystem and the distributed state of the PDE element. One major difficulty is that the state observation mu...

in this paper we propose a method for solving some well-known classes of lane-emden type equations which are nonlinear ordinary differential equations on the semi-innite domain. the proposed approach is based on an unsupervised combined articial neural networks (ucann) method. firstly, the trial solutions of the differential equations are written in the form of feed-forward neural networks co...

This paper proposes a methodology for the synthesis of nonlinear finite-dimensional feedback controllers for incompressible Newtonian fluid flows described by two-dimensional Navier Stokes equations. Combination of Galerkin’s method with approximate inertial manifolds is employed for the derivation of low-order ordinary Ž . differential equation ODE systems that accurately describe the dominant...

Ordinary differential equations (ODEs) are widely used in biomedical research and other scientific areas to model complex dynamic systems. It is an important statistical problem to estimate parameters in ODEs from noisy observations. In this article we propose a method for estimating the time-varying coefficients in an ODE. Our method is a variation of the nonlinear least squares where penalize...

We consider a linear 2 × 2 matrix ODE with two coalescing regular singularities. This coalescence is restricted with an isomonodromy condition with respect to the distance between the merging singularities in a way consistent with the ODE. In particular, a zero-distance limit for the ODE exists. The monodromy group of the limiting ODE is calculated in terms of the original one. This coalescing ...