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The Lie-group Shooting Method for radial Symmetric Solutions of the Yamabe Equation

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Computer Modeling in Engineering & Sciences 2015年第4期104卷 329-351页

作者： S.Abbasbandy r.a.van gorder M.Hajiketabi Department of Mathematics Imam Khomeini International UniversityGhazvin 34149Iran Mathematical Institute University of OxfordAndrew Wiles BuildingRadcliffe Observatory QuarterWoodstock RoadOxfordOX26GGUnited Kingdom

We transform the Yamabe equation on a ball of arbitrary dimension greater than two into a nonlinear singularly boundary value problem on the unit interval[0;1].Then we apply Lie-group shooting method(LGSM)to search a missing initial condition of slope through a weighting factor r 2(0;1).The best r is determined by matching the right-end boundary *** the initial slope is available we can apply the group preserving scheme(GPS)to calculate the solution,which is highly *** LGSM we obtain precise radial symmetric solutions of the Yamabe *** results are useful in demonstrating the utility of Lie-group based numerical approaches to solving nonlinear differential equations.

关键词： Yamabe equation nonlinear singularly boundary value problem group preserving scheme Lie-group shooting method

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Applied Mathematics and Mechanics(English Edition) 2011年第11期32卷 1447-1456页

作者： J.HAUSSErMANN K.VAJrAVELU r.a.van gorder Department of Mathematics University of Central Florida P.O.Box 161364OrlandoFlorida 32816-1364USA

The Lin-reissner-Tsien equation describes unsteady transonic flows under the transonic approximation. In the present paper, the equation is reduced to an ordinary differential equation via a similarity transformation. The resulting equation is then solved analytically and even exactly in some cases. Numerical simulations are provided for the cases in which there is no exact solution. Travelling wave solutions are also obtained.

关键词： Lin-reissner-Tsien equation self-similar solution transonic approximation,nonlinear partial differential equation

Travelling Waves for a Density Dependent Diffusion Nagumo Equation over the real Line

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Communications in Theoretical Physics 2012年第7期58卷 5-11页

作者： robert A. van gorder Department of Mathematics University of Central FloridaP.O.Box 161364OrlandoFL 32816-1364USA

We consider the density dependent diffusion Nagumo equation, where the diffusion coefficient is a simple power function. This equation is used in modelling electrical pulse propagation in nerve axons and in population genetics (amongst other areas). In the present paper, the δ-expansion method is applied to a travelling wave reduction of the problem, so that we may obtain globally valid perturbation solutions (in the sense that the perturbation solutions are valid over the entire infinite domain, not just locally; hence the results are a generalization of the local solutions considered recently in the literature). The resulting boundary value problem is solved on the real line subject to conditions at z →±∞. Whenever a perturbative method is applied, it is important to discuss the accuracy and convergence properties of the resulting perturbation expansions. We compare our results with those of two different numerical methods (designed for initial and boundary value problems, respectively) and deduce that the perturbation expansions agree with the numerical results after a reasonable number of iterations. Finally, we are able to discuss the influence of the wave speed c and the asymptotic concentration value α on the obtained solutions. Upon recasting the density dependent diffusion Nagumo equation as a two-dimensional dynamical system, we are also able to discuss the influence of the nonlinear density dependence (which is governed by a power-law parameter m) on oscillations of the travelling wave solutions.

关键词： density dependent diffusion equation nonlinear partial differential equation perturbation method δ-expansion method

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

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

Sterilization Efficacy of Demolizer Technology for Onsite Treatment of Sharps and Other regulated Medical Waste against Staphylococcus aureus, Escherichia coli, Candida albicans,Mycobacterium phleiand Bacillus subtilisSpores

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Open Journal of Medical Microbiology 2012年第3期2卷 77-83页

作者： James Marsden Jasdeep Saini Maria Teresa Ortega Diane gorder Pamela Hatesohl Department of Animal Science and Industry Kansas State University Manhattan USA

This study was performed to evaluate the efficacy of the Demolizer? technology for the on-site sterilization of low vo-lumes of regulated medical waste. The objective was to demonstrate a minimum of 6 log10 reduction of the dry heat sterilization process applied by the Demolizer? II system for the representative organisms, Staphylococcus aureus, Escherichia coli, Candida albicans, Mycobacterium phlei and Bacillus subtilis spores (formerly Bacillus subtilis) on simulated medical waste consistent with numerous regulatory standards for medical waste treatment. The system cycle was heat treatment at a minimum temperature of 350?F and held at or above this temperature for a minimum of 90 minutes. Upon completion of treatment, there was no evidence of growth in the bacterial species after treatment. Given the minimum detection level of 4 CFU/ml, the Demolizer? II system demonstrated a minimum sterilization efficacy of 6.6 log10 for both S. aureus and E. coli as representative gram-positive and gram-negative bacteria species. Candida albicans (6.7 log10 CFU/ml), Mycobacterium phlei (9.0 log10 CFU/ml) and Bacillus subtilis (6.3 log10 CFU/ml) were completely eliminated after sterilizing representative medical waste in the Demolizer? II system for 90 minutes at a minimum temperature of 350?F. Also, the Demolizer? II exceeded typical recognized standards for medical waste treatment of a 6 log10 reduction of Mycobacteria and a 4 log10 reduction of the appropriate Bacillus endospore.

关键词： regulated Medical Waste Onsite Treatment Sharps Demolizer Technology