Nonlinear problems related to the Thomas-Fermi equation

2004 ◽  
pp. 673-770
Author(s):  
Philippe Bénilan ◽  
Haïm Brezis
Keyword(s):  
Nonlinear Problems ◽  
Thomas Fermi

An optimal iteration method with application to the Thomas-Fermi equation

Open Physics ◽  
2011 ◽  
Vol 9 (3) ◽  
Author(s):  
Vasile Marinca ◽  
Nicolae Herişanu
Keyword(s):  
Approximate Solution ◽  
Approximate Method ◽  
Iteration Method ◽  
Nonlinear Problems ◽  
Analytical Approximate Solution ◽  
Thomas Fermi ◽  
Strongly Nonlinear ◽  
Good Agreement

AbstractThe aim of this paper is to introduce a new approximate method, namely the Optimal Parametric Iteration Method (OPIM) to provide an analytical approximate solution to Thomas-Fermi equation. This new iteration approach provides us with a convenient way to optimally control the convergence of the approximate solution. A good agreement between the obtained solution and some well-known results has been demonstrated. The proposed technique can be easily applied to handle other strongly nonlinear problems.

Nonlinear problems related to the Thomas-Fermi equation

2003 ◽  
Vol 3 (4) ◽  
pp. 673-770 ◽  
Author(s):  
Philippe Bénilan † ◽  
Haïm Brezis
Keyword(s):  
Nonlinear Problems ◽  
Thomas Fermi

scholarly journals Nonlinearities Distribution Homotopy Perturbation Method Applied to Solve Nonlinear Problems: Thomas-Fermi Equation as a Case Study

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
U. Filobello-Nino ◽  
H. Vazquez-Leal ◽  
K. Boubaker ◽  
A. Sarmiento-Reyes ◽  
A. Perez-Sesma ◽  
...  
Keyword(s):  
Approximate Solution ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Problems ◽  
Thomas Fermi ◽  
Homotopy Perturbation

We propose an approximate solution of T-F equation, obtained by using the nonlinearities distribution homotopy perturbation method (NDHPM). Besides, we show a table of comparison, between this proposed approximate solution and a numerical of T-F, by establishing the accuracy of the results.

A Comprehensive Investigation of Ensembles of Gaussian-Based and Gradient-Based Transformed Reliability Analyses: When and How to Use Them

2016 ◽  
Author(s):  
Po Ting Lin ◽  
Wei-Hao Lu ◽  
Shu-Ping Lin
Keyword(s):  
Design Optimization ◽  
Design Space ◽  
Optimization Problems ◽  
Nonlinear Problems ◽  
Kernel Functions ◽  
Sensitivity Analyses ◽  
Normal Space ◽  
Highly Nonlinear ◽  
Standard Normal ◽  
Gradient Based

In the past few years, researchers have begun to investigate the existence of arbitrary uncertainties in the design optimization problems. Most traditional reliability-based design optimization (RBDO) methods transform the design space to the standard normal space for reliability analysis but may not work well when the random variables are arbitrarily distributed. It is because that the transformation to the standard normal space cannot be determined or the distribution type is unknown. The methods of Ensemble of Gaussian-based Reliability Analyses (EoGRA) and Ensemble of Gradient-based Transformed Reliability Analyses (EGTRA) have been developed to estimate the joint probability density function using the ensemble of kernel functions. EoGRA performs a series of Gaussian-based kernel reliability analyses and merged them together to compute the reliability of the design point. EGTRA transforms the design space to the single-variate design space toward the constraint gradient, where the kernel reliability analyses become much less costly. In this paper, a series of comprehensive investigations were performed to study the similarities and differences between EoGRA and EGTRA. The results showed that EGTRA performs accurate and effective reliability analyses for both linear and nonlinear problems. When the constraints are highly nonlinear, EGTRA may have little problem but still can be effective in terms of starting from deterministic optimal points. On the other hands, the sensitivity analyses of EoGRA may be ineffective when the random distribution is completely inside the feasible space or infeasible space. However, EoGRA can find acceptable design points when starting from deterministic optimal points. Moreover, EoGRA is capable of delivering estimated failure probability of each constraint during the optimization processes, which may be convenient for some applications.

scholarly journals Real-time automatic uncertainty estimation of coseismic single rectangular fault model using GNSS data

2021 ◽  
Vol 73 (1) ◽  
Author(s):  
Keitaro Ohno ◽  
Yusaku Ohta ◽  
Satoshi Kawamoto ◽  
Satoshi Abe ◽  
Ryota Hino ◽  
...  
Keyword(s):  
Real Time ◽  
Disaster Response ◽  
Order Approximation ◽  
Nonlinear Problems ◽  
Large Earthquake ◽  
Fault Model ◽  
Deformation Monitoring ◽  
Model Parameters ◽  
The Sea Of Japan ◽  
Plate Interface

AbstractRapid estimation of the coseismic fault model for medium-to-large-sized earthquakes is key for disaster response. To estimate the coseismic fault model for large earthquakes, the Geospatial Information Authority of Japan and Tohoku University have jointly developed a real-time GEONET analysis system for rapid deformation monitoring (REGARD). REGARD can estimate the single rectangular fault model and slip distribution along the assumed plate interface. The single rectangular fault model is useful as a first-order approximation of a medium-to-large earthquake. However, in its estimation, it is difficult to obtain accurate results for model parameters due to the strong effect of initial values. To solve this problem, this study proposes a new method to estimate the coseismic fault model and model uncertainties in real time based on the Bayesian inversion approach using the Markov Chain Monte Carlo (MCMC) method. The MCMC approach is computationally expensive and hyperparameters should be defined in advance via trial and error. The sampling efficiency was improved using a parallel tempering method, and an automatic definition method for hyperparameters was developed for real-time use. The calculation time was within 30 s for 1 × 106 samples using a typical single LINUX server, which can implement real-time analysis, similar to REGARD. The reliability of the developed method was evaluated using data from recent earthquakes (2016 Kumamoto and 2019 Yamagata-Oki earthquakes). Simulations of the earthquakes in the Sea of Japan were also conducted exhaustively. The results showed an advantage over the maximum likelihood approach with a priori information, which has initial value dependence in nonlinear problems. In terms of application to data with a small signal-to-noise ratio, the results suggest the possibility of using several conjugate fault models. There is a tradeoff between the fault area and slip amount, especially for offshore earthquakes, which means that quantification of the uncertainty enables us to evaluate the reliability of the fault model estimation results in real time.

scholarly journals Semi-Lagrangian Subgrid Reconstruction for Advection-Dominant Multiscale Problems with Rough Data

2021 ◽  
Vol 87 (2) ◽  
Author(s):  
Konrad Simon ◽  
Jörn Behrens
Keyword(s):  
Nonlinear Problems ◽  
Higher Dimensions ◽  
Two Dimensions ◽  
Multiscale Methods ◽  
Galerkin Methods ◽  
Standard Methods ◽  
Multiscale Problems ◽  
Eulerian Method ◽  
Lower Order Terms ◽  
New Framework

AbstractWe introduce a new framework of numerical multiscale methods for advection-dominated problems motivated by climate sciences. Current numerical multiscale methods (MsFEM) work well on stationary elliptic problems but have difficulties when the model involves dominant lower order terms. Our idea to overcome the associated difficulties is a semi-Lagrangian based reconstruction of subgrid variability into a multiscale basis by solving many local inverse problems. Globally the method looks like a Eulerian method with multiscale stabilized basis. We show example runs in one and two dimensions and a comparison to standard methods to support our ideas and discuss possible extensions to other types of Galerkin methods, higher dimensions and nonlinear problems.

scholarly journals Analytical Method for Geometric Nonlinear Problems Based on Offshore Derricks

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 610
Author(s):  
Chunbao Li ◽  
Hui Cao ◽  
Mengxin Han ◽  
Pengju Qin ◽  
Xiaohui Liu
Keyword(s):  
Analytical Method ◽  
Bending Moment ◽  
Nonlinear Problems ◽  
Weather Conditions ◽  
Order Effect ◽  
Practical Calculation ◽  
Geometrically Nonlinear ◽  
Uniform Load ◽  
Safety Research ◽  
Calculation Results

The marine derrick sometimes operates under extreme weather conditions, especially wind; therefore, the buckling analysis of the components in the derrick is one of the critical contents of engineering safety research. This paper aimed to study the local stability of marine derrick and propose an analytical method for geometrically nonlinear problems. The rod in the derrick is simplified as a compression rod with simply supported ends, which is subjected to transverse uniform load. Considering the second-order effect, the differential equations were used to establish the deflection, rotation angle, and bending moment equations of the derrick rod under the lateral uniform load. This method was defined as a geometrically nonlinear analytical method. Moreover, the deflection deformation and stability of the derrick members were analyzed, and the practical calculation formula was obtained. The Ansys analysis results were compared with the calculation results in this paper.

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