scholarly journals Unexpected Solutions of the Nehari Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
A. Gritsans ◽  
F. Sadyrbaev
Keyword(s):  
Differential Equation ◽  
Multiple Solutions ◽  
Integral Functional ◽  
Characteristic Number ◽  
Nonlinear Ordinary Differential Equation ◽  
Oscillatory Solutions ◽  
Nehari Problem ◽  
Characteristic Numbers ◽  
Regular Shape ◽  
Asymmetric Shape

The Nehari characteristic numbers λn(a,b) are the minimal values of an integral functional associated with a boundary value problem (BVP) for nonlinear ordinary differential equation. In case of multiple solutions of the BVP, the problem of identifying of minimizers arises. It was observed earlier that for nonoscillatory (positive) solutions of BVP those with asymmetric shape can provide the minimal value to a functional. At the same time, an even solution with regular shape is not a minimizer. We show by constructing the example that the same phenomenon can be observed in the Nehari problem for the fifth characteristic number λn(a,b) which is associated with oscillatory solutions of BVP (namely, with those having exactly four zeros in (a,b)).

Design of evolutionary cubic spline intelligent solver for nonlinear Painlevé-I transcendent

2021 ◽  
Author(s):  
Sharafat Ali ◽  
Iftikhar Ahmad ◽  
Muhammad Asif Zahoor Raja ◽  
Siraj ul Islam Ahmad ◽  
Muhammad Shoaib
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Genetic Algorithms ◽  
Statistical Analysis ◽  
Quadratic Programming ◽  
Sequential Quadratic Programming ◽  
Process Variation ◽  
Nonlinear Ordinary Differential Equation ◽  
Second Derivative ◽  
Search Technique

In this research paper, an innovative bio-inspired algorithm based on evolutionary cubic splines method (CSM) has been utilized to estimate the numerical results of nonlinear ordinary differential equation Painlevé-I. The computational mechanism is used to support the proposed technique CSM and optimize the obtained results with global search technique genetic algorithms (GAs) hybridized with sequential quadratic programming (SQP) for quick refinement. Painlevé-I is solved by the proposed technique CSM-GASQP. In this process, variation of splines is implemented for various scenarios. The CSM-GASQP produces an interpolated function that is continuous upto its second derivative. Also, splines proved to be stable than a single polynomial fitted to all points, and reduce wiggles between the tabulated points. This method provides a reliable and excellent procedure for adaptation of unknown coefficients of splines by searching globally exploiting the performance of GA-SQP algorithms. The convergence, exactness and accuracy of the proposed scheme are examined through the statistical analysis for the several independent runs.

Approximated Solutions for Axisymmetric Wrinkled Inflated Membranes

2015 ◽  
Vol 82 (11) ◽  
Author(s):  
Riccardo Barsotti
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Finite Element ◽  
Nonlinear Ordinary Differential Equation ◽  
Experimental Results ◽  
Limit Case ◽  
Reasonable Approximation ◽  
Constituent Material ◽  
Actual Solution ◽  
Wrinkled Membrane

The axisymmetric inflation problem for a wrinkled membrane is solved by means of a simple nonlinear ordinary differential equation. The solution is illustrated in full details. Both the free and constrained cases are addressed, in the limit case where the membrane is fully wrinkled. In the constrained inflation problem, no slippage is allowed between the membrane and the constraining surfaces. It is shown that an actual membrane can in no way reach the fully wrinkled configuration during free inflation, regardless of the membrane's initial configuration and constituent material. The fully wrinkled solution is compared to some finite element results obtained by means of an expressly developed iterative–incremental procedure. When the values of the inflating pressure and length of the meridian lie within a suitable applicability range, the fully wrinkled solution may represent a reasonable approximation of the actual solution. A comparison with some numerical and experimental results available in the literature is illustrated.

scholarly journals Approximate solution of nonlinear ordinary differential equation using ZZ decomposition method

2020 ◽  
Vol 4 (1) ◽  
pp. 448-455
Author(s):  
Mulugeta Andualem ◽  
◽  
Atinafu Asfaw ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Approximate Solution ◽  
Initial Value Problems ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Nonlinear Ordinary Differential Equation ◽  
Initial Value ◽  
Transform Method ◽  
Adomian Decomposition

Nonlinear initial value problems are somewhat difficult to solve analytically as well as numerically related to linear initial value problems as their variety of natures. Because of this, so many scientists still searching for new methods to solve such nonlinear initial value problems. However there are many methods to solve it. In this article we have discussed about the approximate solution of nonlinear first order ordinary differential equation using ZZ decomposition method. This method is a combination of the natural transform method and Adomian decomposition method.

Dynamic Models for Magnetorheological Dampers and its Feedback Linearization

2009 ◽  
Vol 79-82 ◽  
pp. 1205-1208 ◽  
Author(s):  
Cheng Zhang ◽  
Lin Xiang Wang
Keyword(s):  
Differential Equation ◽  
Phase Transition ◽  
Ordinary Differential Equation ◽  
Feedback Linearization ◽  
Dynamic Models ◽  
Nonlinear Ordinary Differential Equation ◽  
Transition Theory ◽  
Magnetorheological Dampers ◽  
Differential Model ◽  
Mr Dampers

In the current paper, the hysteretic dynamics of magnetorheological dampers is modeled by a differential model. The differential model is constructed on the basis of a phenomenological phase transition theory. The model is expressed as a second order nonlinear ordinary differential equation with bifurcations embedded in. Due to the differential nature of the model, the hysteretic dynamics of the MR dampers can be linearized and controlled by introducing a feedback linearization strategy.

scholarly journals The Kramers problem for SDEs driven by small, accelerated Lévy noise with exponentially light jumps

2020 ◽  
pp. 2150019
Author(s):  
André de Oliveira Gomes ◽  
Michael A. Högele
Keyword(s):  
Differential Equation ◽  
Stochastic Integral ◽  
Stable State ◽  
Nonlinear Ordinary Differential Equation ◽  
Large Deviations Principle ◽  
Perturbed System ◽  
State Formula ◽  
Escape Problem ◽  
Kramers Problem ◽  
Weak Convergence Approach

We establish Freidlin–Wentzell results for a nonlinear ordinary differential equation starting close to the stable state [Formula: see text], say, subject to a perturbation by a stochastic integral which is driven by an [Formula: see text]-small and [Formula: see text]-accelerated Lévy process with exponentially light jumps. For this purpose, we derive a large deviations principle for the stochastically perturbed system using the weak convergence approach developed by Budhiraja, Dupuis, Maroulas and collaborators in recent years. In the sequel, we solve the associated asymptotic first escape problem from the bounded neighborhood of [Formula: see text] in the limit as [Formula: see text] which is also known as the Kramers problem in the literature.

Investigation of the Oscillatory Behavior of Electrostatically-Actuated Microbeams

2010 ◽  
Author(s):  
Mahdi Mojahedi ◽  
Mahdi Moghimi Zand ◽  
Mohammad Taghi Ahmadian
Keyword(s):  
Differential Equation ◽  
Homotopy Perturbation Method ◽  
Axial Stress ◽  
Nonlinear Partial Differential Equation ◽  
Nonlinear Ordinary Differential Equation ◽  
Oscillatory Behavior ◽  
Electrostatically Actuated ◽  
Homotopy Analysis ◽  
Analytic Expressions ◽  
Good Agreement

Vibrations of electrostatically-actuated microbeams are investigated. Effects of electrostatic actuation, axial stress and midplane stretching are considered in the model. Galerkin’s decomposition method is utilized to convert the governing nonlinear partial differential equation to a nonlinear ordinary differential equation. Homotopy perturbation method (i.e. a special and simpler case of homotopy analysis method) is utilized to find analytic expressions for natural frequencies of predeformed microbeam. Effects of increasing the voltage, midplane stretching, axial force and higher modes contribution on natural frequency are also studied. The anayltical results are in good agreement with the numerical results in the literature.

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