Nonlinear Ordinary Differential Equations (nonlinear + ordinary_differential_equation)

Distribution by Scientific Domains

Selected Abstracts

Analysis of velocity equation of steady flow of a viscous incompressible fluid in channel with porous walls

M. Babaelahi
Abstract Steady flow of a viscous incompressible fluid in a channel, driven by suction or injection of the fluid through the channel walls, is investigated. The velocity equation of this problem is reduced to nonlinear ordinary differential equation with two boundary conditions by appropriate transformation and convert the two-point boundary-value problem for the similarity function into an initial-value problem in which the position of the upper channel. Then obtained differential equation is solved analytically using differential transformation method and compare with He's variational iteration method and numerical solution. These methods can be easily extended to other linear and nonlinear equations and so can be found widely applicable in engineering and sciences. Copyright 2009 John Wiley & Sons, Ltd. [source]

Optimization of Operating Temperature for Continuous Immobilized Glucose Isomerase Reactor with Pseudo Linear Kinetics

N.M. Faqir
Abstract In this work, the optimal operating temperature for the enzymatic isomerization of glucose to fructose using a continuous immobilized glucose isomerase packed bed reactor is studied. This optimization problem describing the performance of such reactor is based on reversible pseudo linear kinetics and is expressed in terms of a recycle ratio. The thermal deactivation of the enzyme as well as the substrate protection during the reactor operation is considered. The formulation of the problem is expressed in terms of maximization of the productivity of fructose. This constrained nonlinear optimization problem is solved using the disjoint policy of the calculus of variations. Accordingly, this method of solution transforms the nonlinear optimization problem into a system of two coupled nonlinear ordinary differential equations (ODEs) of the initial value type, one equation for the operating temperature profile and the other one for the enzyme activity. The ODE for the operating temperature profile is dependent on the recycle ratio, operating time period, and the reactor residence time as well as the kinetics of the reaction and enzyme deactivation. The optimal initial operating temperature is selected by solving the ODEs system by maximizing the fructose productivity. This results into an unconstrained one-dimensional optimization problem with simple bounds on the operating temperature. Depending on the limits of the recycle ratio, which represents either a plug flow or a mixed flow reactor, it is found that the optimal temperature of operation is characterized by an increasing temperature profile. For higher residence time and low operating periods the residual enzyme activity in the mixed flow reactor is higher than that for the plug flow reactor, which in turn allows the mixed flow reactor to operate at lower temperature than that of the plug flow reactor. At long operating times and short residence time, the operating temperature profiles are almost the same for both reactors. This could be attributed to the effect of substrate protection on the enzyme stability, which is almost the same for both reactors. Improvement in the fructose productivity for both types of reactors is achieved when compared to the constant optimum temperature of operation. The improvement in the fructose productivity for the plug flow reactor is significant in comparison with the mixed flow reactor. [source]

Analysis of adiabatic shear bands in heat-conducting elastothermoviscoplastic materials by the meshless local Bubnov,Galerkin method

R. C. Batra
Abstract Transient finite coupled thermomechanical simple shearing deformations of a block made of an elastothermoviscoplastic material that exhibits strain and strain-rate hardening, and thermal softening are studied by using the meshless local Bubnov,Galerkin method. A local nonlinear weak formulation and a semidiscrete formulation of the problem are derived. The prescribed velocity at the top and the bottom surfaces of the block is enforced by using a set of Lagrange multipliers. A homogeneous solution of the problem is perturbed by superimposing on it a temperature bump at the center of the block, and the resulting nonlinear initial-boundary-value problem is solved numerically. We have developed an integration scheme to numerically integrate the set of coupled nonlinear ordinary differential equations. The inhomogeneous deformations of the block are found to concentrate in a narrow region of intense plastic deformation usually called a shear band. For a material exhibiting enhanced thermal softening, it is shown that as the shear stress within the region of localization collapses, an unloading elastic shear wave emanates outward from the edges of the shear band. In the absence of an analytical solution, the computed results have been compared with those obtained by the finite element and the modified smoothed particle hydrodynamics methods. Copyright 2008 John Wiley & Sons, Ltd. [source]

A Note on a Nonlinear Model of a Piezoelectric Rod

R. Gausmann Dipl.-Ing.
If piezoceramics are excited by weak electric fields a nonlinear behavior can be observed, if the excitation frequency is close to a resonance frequency of the system. To derive a theoretical model nonlinear constitutive equations are used, to describe the longitudinal oscillations of a slender piezoceramic rod near the first resonance frequency. Hamilton's principle is used to receive a variational principle for the piezoelectric rod. Introducing a Rayleigh Ritz ansatz with the eigenfunctions of the linearized system to approximate the exact solution leads to nonlinear ordinary differential equations. These equations are approximated with the method of harmonic balance. Finally it is possible to calculate the amplitudes of the displacements numerically. As a result it is shown, that the Duffing type nonlinearities found in measurements can be described with this model. [source]

Forcing Function Diagnostics for Nonlinear Dynamics

BIOMETRICS, Issue 3 2009
Giles Hooker
Summary This article investigates the problem of model diagnostics for systems described by nonlinear ordinary differential equations (ODEs). I propose modeling lack of fit as a time-varying correction to the right-hand side of a proposed differential equation. This correction can be described as being a set of additive forcing functions, estimated from data. Representing lack of fit in this manner allows us to graphically investigate model inadequacies and to suggest model improvements. I derive lack-of-fit tests based on estimated forcing functions. Model building in partially observed systems of ODEs is particularly difficult and I consider the problem of identification of forcing functions in these systems. The methods are illustrated with examples from computational neuroscience. [source]

Differential Equation Modeling of HIV Viral Fitness Experiments: Model Identification, Model Selection, and Multimodel Inference

BIOMETRICS, Issue 1 2009
Hongyu Miao
Summary Many biological processes and systems can be described by a set of differential equation (DE) models. However, literature in statistical inference for DE models is very sparse. We propose statistical estimation, model selection, and multimodel averaging methods for HIV viral fitness experiments in vitro that can be described by a set of nonlinear ordinary differential equations (ODE). The parameter identifiability of the ODE models is also addressed. We apply the proposed methods and techniques to experimental data of viral fitness for HIV-1 mutant 103N. We expect that the proposed modeling and inference approaches for the DE models can be widely used for a variety of biomedical studies. [source]

Maximum Likelihood Estimation in Dynamical Models of HIV

BIOMETRICS, Issue 4 2007
J. Guedj
Summary The study of dynamical models of HIV infection, based on a system of nonlinear ordinary differential equations (ODE), has considerably improved the knowledge of its pathogenesis. While the first models used simplified ODE systems and analyzed each patient separately, recent works dealt with inference in non-simplified models borrowing strength from the whole sample. The complexity of these models leads to great difficulties for inference and only the Bayesian approach has been attempted by now. We propose a full likelihood inference, adapting a Newton-like algorithm for these particular models. We consider a relatively complex ODE model for HIV infection and a model for the observations including the issue of detection limits. We apply this approach to the analysis of a clinical trial of antiretroviral therapy (ALBI ANRS 070) and we show that the whole algorithm works well in a simulation study. [source]