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A Linearization Approach for Rational Nonlinear Models in Mathematical Physics

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Communications in Theoretical Physics 2012年第4期57卷 530-540页

作者： Robert a.van Gorder Department of Mathematics University of Central FloridaP.O.Box 161364OrlandoFL 32816-1364USA

In this paper, a novel method for linearization of rational second order nonlinear models is discussed. In particular, we discuss an application of the 5 expansion method （created to deal with problems in Quantum Field Theory） which will enable both the linearization and perturbation expansion of such equations. Such a method allows for one to quickly obtain the order zero perturbation theory in terms of certain special functions which are governed by linear equations. Higher order perturbation theories can then be obtained in terms of such special functions. One benefit to such a method is that it may be applied even to models without small physical parameters, as the perturbation is given in terms of the degree of nonlinearity, rather than any physical parameter. As an application, we discuss a method of linearizing the six Painlev~ equations by an application of the method. In addition to highlighting the benefits of the method, we discuss certain shortcomings of the method.

关键词： perturbation method Painleve equations δ-expansion nonlinear differential equations

Solutions to a Novel Casimir Equation for the Ito System

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Communications in Theoretical Physics 2011年第11期56卷 801-804页

作者： Robert a.van Gorder Department of Mathematics University of Central Florida Orlando

We discuss two classes of solutions to a novel Casimir equation associated with the Ito system,a couplednonlinear wave *** travelling wave solutions and separable self-similar solutions are *** a numberof cases,explicit exact solutions are *** results,particularly the exact solutions,are useful in that they provideus a baseline of comparison to any numerical ***,such solutions provide us a glimpse of the behaviorof the Ito system,and hence the behavior of a type of nonlinear wave equation,for certain parameter regimes.

关键词： nonlinear wave equation Ito system travelling waves self-similar solutions

Solutions to Forced and Unforced Lin–Reissner–Tsien Equations for Transonic Gas Flows on Various Length Scales

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Communications in Theoretical Physics 2017年第3期67卷 309-316页

作者： Kyle A.Theaker Robert a.van Gorder Department of Mathematics University of Central Florida Mathematical Institute University of Oxford Andrew Wiles Building Radcliffe Observatory Quarter

The Lin–Reissner–Tsien equation is useful for studying transonic gas flows, and has appeared in both forced and unforced forms in the literature. Defining arbitrary spatial scalings, we are able to obtain a family of exact similarity solutions depending on one free parameter in addition to the model parameter holding the scalings. Numerical solutions compare favorably with the exact solutions in regions where the exact solutions are valid. Mixed wave-similarity solutions, which describe wave propagation in one variable and self-similar scaling of the entire solution, are also given,and we show that such solutions can only exist when the wave propagation is sufficiently slow. We also extend the Lin–Reissner–Tsien equation to have a forcing term, as such equations have entered the physics literature recently. We obtain both wave and self-similar solutions for the forced equations, and we are able to give conditions under which the force function allows for exact solutions. We then demonstrate how to obtain these exact solutions in both the traveling wave and self-similar cases. There results constitute new and potentially physically interesting exact solutions of the Lin–Reissner–Tsien equation and in particular suggest that the forced Lin–Reissner–Tsien equation warrants further study.

关键词： Lin-Reissner-Tsien equation similarity transform nonlinear waves exact solutions

An unexpected role for ergothioneine

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National Science Review 2015年第4期2卷 382-383页

作者： Zedu Huang Wilfred a.van der Donk Institute for Genomic Biology Department of Chemistry Howard Hughes Medical InstituteUniversity of Illinois at Urbana-Champaign

Ever since the appearance of atmospheric oxygen,organisms have needed ways to protect themselves from damage caused by oxidative ***(GSH,Fig.1A)is the dominant organothiol in eukarya to detoxify products formed from o... 详细信息

关键词： EGT MSH An unexpected role for ergothioneine

Balanced Monitoring of Flow Phenomena in Moving Mesh Methods

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Communications in Computational Physics 2010年第1期7卷 138-170页

作者： a.van Dam P.A.Zegeling Department of Mathematics Utrecht UniversityP.O.Box 80.0103508 TA UtrechtThe Netherlands

Adaptive moving mesh research usually focuses either on analytical deriva-tions for prescribed solutions or on pragmatic solvers with challenging physical appli-cations. In the latter case, the monitor functions that steer mesh adaptation are oftendefined in an ad-hoc way. In this paper we generalize our previously used moni-tor function to a balanced sum of any number of monitor components. This avoidsthe trial-and-error parameter fine-tuning that is often used in monitor functions. Thekey reason for the new balancing method is that the ratio between the maximum andaverage value of a monitor component should ideally be equal for all *** as a monitor component is a good motivating example for this. Entropy alsoturns out to be a very informative monitor component. We incorporate the monitorfunction in an adaptive moving mesh higher-order finite volume solver with HLLCfluxes, which is suitable for nonlinear hyperbolic systems of conservation laws. Whenapplied to compressible gas flow it produces very sharp results for shocks and otherdiscontinuities. Moreover, it captures small instabilities (Richtmyer-Meshkov, Kelvin-Helmholtz). Thus showing the rich nature of the example problems and the effective-ness of the new monitor balancing.

关键词： Moving mesh method conservative interpolation balanced monitor function direc-tional adaptation hydrodynamics implosion problem

Coal mine entry rating system:A case study

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International Journal of Mining Science and Technology 2021年第1期31卷 145-151页

作者： Mark a.van Dyke Ted M.Klemetti Craig Compton National Institute for Occupational Safety and Health Pittsburgh Mining Research DivisionPittsburghPA 15236USA

Coal mines are continuously seeking to determine the performance of entries with different ground control products and installation *** are many factors that impact how an entry will perform which include but are not limited to geology,overburden,bolting type and pattern,and mine *** the National Institute for Occupational Safety and Health(NIOSH),research has been instituted to examine the relationship of the parts of a coal mine entry as a system and not as individual *** study this relationship,the first step in this study was to create a numeric rating system that accurately reflects visual observations of the mine entry and is easy to *** researchers devised this rating system to improve upon previous ideas,offering increased flexibility which can be incorporated into an overall entry condition that offers different levels of confidence based on the user’s time devoted to the *** new entry rating system was implemented at three different mines over varying periods of time to evaluate the ground response to the geology,bolt installation pattern,stress changes by mining,overburden,and time dependency.

关键词： Coal Geology Rating system Support

Several Types of Similarity Solutions for the Hunter-Saxton Equation

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Communications in Theoretical Physics 2015年第6期63卷 675-681页

作者： Mathew Baxter Robert a.van Gorder Kuppalapalle Vajravelu Department of Mathematics Florida Gulf Coast University Mathematical Institute University OxfordAndrew Wiles BuildingRadcliffe Observatory Quarter Department of Mathematics University of Central Florida

We study separable and self-similar solutions to the Hunter-Saxton equation, a nonlinear wave equation which has been used to describe an instability in the director field of a nematic liquid crystal （among other applications）. Essentially, we study solutions which arise from a nonlinear inhomogeneous ordinary differential equation which is obtained by an exact similarity transform for the ttunter-Saxton equation. For each type of solution, we are able to obtain some simple exact solutions in closed-form, and more complicated solutions through an analytical approach. We find that there is a whole family of self-similar solutions, each of which depends on an arbitrary parameter. This parameteressentially controls the manner of self-similarity and can be chosen so that the self-similar solutions agree with given initial data. The simpler solutions found constitute exact soIutions to a nonlinear partial differential equation, and hence are also useful in a mathematical sense. Analytical solutions demonstrate the variety of behaviors possible within the wider family of similarity solutions. Both types of solutions cast light on self-similar phenomenon arising in the Hunter-Saxton equation.

关键词： Hunter-Saxton equation nonlinear waves self-similar solutions analytical methods exact solutions

Bleeder entry evaluation using condition mapping and numerical modeling

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International Journal of Mining Science and Technology 2021年第1期31卷 137-143页

作者： Ted M.Klemetti Mark a.van Dyke Gabriel S.Esterhuizen National Institute for Occupational Safety and Health Pittsburgh Mining Research DivisionPittsburghPA 15236USA

One of the most common critical areas of longwall mining in terms of ground stability are the gateroad and bleeder *** critical entries provide much-needed safe access for miners and allow for adequate ventilation required for dilution of hazardous airborne contaminants and must remain open during mining of a multi-panel *** paper is focused on the stability of the longwall entries subjected to a single abutment load such as bleeders,first tailgate,and last *** tailgate and last headgate are also referred to as blind headgate and tailgate.A study of a longwall district through conditions mapping,support evaluations,and numerical modeling was conducted and evaluated by researchers from the National Institute for Occupational Safety and Health(NIOSH).The condition mapping and support evaluations were performed on entries that spanned the previous five years of mining and relied on a diverse selection of supports to maintain the functionality of the *** modeling was also conducted to evaluate various support types with further investigation and comparison to the condition *** study demonstrated the importance of the abutment load decay versus distance from the gob edge,the potential for a reduction in material handling related injuries,as well as optimal usage of secondary and standing support.

关键词： Stability assessment Bleeder entry Abutment loading Condition mapping Longwall Ground control

Evaluation of seismic potential in a longwall mine with massive sandstone roof under deep overburden

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International Journal of Mining Science and Technology 2018年第1期28卷 115-119页

作者： Mark a.van Dyke Wen H.Su Joe Wickline Ground Control Branch NIOSHPittsburgh15236 PAUSA Coronado Coal LLC BeckleyWVUSA

A recent seismic event was recorded by a deep longwall mine in Virginia at 3.7 ML on the local magnitude scale and 3.4 MMS by the United States Geological Survey(USGS) in *** investigations by the National Institute for Occupational Safety and Health(NIOSH) and Coronado Coal researchers have shown that this event was associated with geological features that have also been associated with other, similar seismic events in *** mapping and geological exploration in the mining area has made it possible to forecast possible locations for future seismic *** order to use the geology as a forecaster of mining-induced seismic events and their energy potential, two primary components are *** first component is a long history of recorded seismic events with accurately plotted *** second component is a high density of geologic data within the mining *** this case, 181 events of 1.0 MLor greater were recorded by the mine's seismic network between January, 2009, and October, *** the mining area, 897 geophysical logs, 224 core holes, and 1031 fiberscope holes were examined by mine *** this information, it was found that overburden thickness, sandstone thickness, and sandstone quality contributed greatly to seismic *** the data was analyzed, a pattern became apparent indicating that the majority of seismic events occurred under specific *** forecast maps were created based on geology of previous seismic *** forecast maps have shown an accuracy of within 74%–89% when compared to the recorded 181 events that were1.0 MLor greater when considering three major geological criteria of overburden thickness of 579.12 m or greater, 6.096–12.192 m of sandstone within 15.24 m of the Pocahontas number 3 seam, and a longwall caving height of 4.572 m or less.

关键词： Seismic Mapping Deep overburden Massive sandstone