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SPECIAL SESSION 68 259
Special Session 68: Analysis and Simulations of Nonlinear Systems
Wei Feng, University of North Carolina Wilmington, USAZhaosheng Feng, University of Texas-Pan American, USA
The aim of the special session is to address analytic and computational aspects of nonlinear systems anddi↵erential equations arising particularly in mechanics and mathematical biology. The session will focuson recent advances in mathematical analysis and simulations of various systems of mixed types (includingPDE models, delay equations, and impulsive di↵erential equations), equally with applications to mechanics,population dynamics, wave phenomena, formation of patterns and other physical contexts.
Pullback random attractor for the FitzHugh-Nagumo system on unbounded domains
Abiti AdiliNMT, [email protected] Bi Xiang
We study the asymptotic behavior of solutions of theFitzHugh-Nagumo system defined on unbounded do-mains with non-autonomous deterministic as well asstochastic terms. We first prove the pullback asymp-totic compactness of solutions and then establishthe existence of a unique pullback random attractorfor the system. We further show that the randomattractor is periodic when the non-autonomous de-terministic terms are periodic.
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Some computational challenges in analyzingglobal dynamics of certain nonlinear discretedynamical systems
Sukanya BasuGrand Valley State University, [email protected]
We present some computational challenges involvedin analyzing global behavior of solutions to certainclasses of nonlinear discrete dynamical systems whichhave applications in mathematical biology and ecol-ogy. We also suggest some innovative geometricand computer-based approaches to overcome thesechallenges.
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Mathematical modeling with reaction-di↵usion-advection systems and its applica-tion in biology
Kanadpriya BasuUniversity of South Carolina, [email protected] Liu
In this talk, I will discuss about the computationalchallenges to solve reaction di↵usion advection equa-tions and how to overcome these di�culties by de-signing e↵ective and robust numerical techniques.
Moreover, I will show some interesting applicationsof reaction di↵usion models arising from complexbiological systems.
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A fast finite di↵erence method for fractionaldi↵usion equations
Treena BasuUniversity of South Carolina, [email protected] Wang
Fractional di↵usion equations model phenomena ex-hibiting anomalous di↵usion that can not be modeledaccurately by the second-order di↵usion equations.Because of the nonlocal property of fractional dif-ferential operators, the numerical methods have fullcoe�cient matrices which require storage of O(N2)and computational cost ofO(N3) where N is thenumber of grid points. In this talk, I will discuss afast finite di↵erence method for fractional di↵usionequations, which only requires storage ofO(N) andcomputational cost of O(N log2 N) for a one dimen-sional fractional di↵usion equation or O(N logN)for a two dimensional fractional di↵usion equationwhile retaining the same accuracy and approxima-tion property as the regular finite di↵erence method.Numerical experiments are presented to show theutility of the method.
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Local and global well-posedness for KdV-systems
Jerry BonaUniversity of Illinois at Chicago, [email protected]
Systems of Korteweg-de Vries type arise in variousapplications. We study here a class of such systemsand obtain local and global well-posedness resultstogether with some associated examples of blow-upin finite time.
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260 9th AIMS CONFERENCE – ABSTRACTS
Hopf bifurcation analysis of a predator-preydystem with discrete and distributed delays
Canan Celik KaraaslanliBahcesehir University, [email protected]
In this paper, a ratio dependent predator-prey sys-tem with both discrete and distributed delays isinvestigated. First we consider the stability of thepositive equilibrium and the existence of local Hopfbifurcations. Then, by choosing the delay time ⌧ as abifurcation parameter, we show that Hopf bifurcationcan occur as the delay time ⌧ passes some criticalvalues. Using normal form theory and central man-ifold argument, we also establish the direction andthe stability of Hopf bifurcation. Finally, we performsome numerical simulations to support theoreticalresults.
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Initial-boundary-value problem of systems ofnonlinear dispersive equations
Hongqiu ChenUniversity of Memphis, [email protected] L. Bona
Considered here is the system8>>>>><
>>>>>:
ut + ux � uxxt + P (u, v)x = 0, x 2 (0, L), t > 0,
vt + vx � vxxt +Q(u, v)x = 0, x 2 (0, L), t > 0,
u(0, t) = f1(t), v(0, t) = g1(t), t > 0,
u(L, t) = f2(t), v(L, t) = g2(t), t > 0,
u(x, 0) = u0(x), v(x, 0) = v0(x) x 2 (0, L)
of coupled BBM-equations posed on bounded do-main for x in the bounded domain [0, L], whereu = u(x, t), v = v(x, t) are functions defined on(0, L)⇥R+, P (u, v) = Au2+Buv+Cv2 and Q(u, v) =Du2+Euv+Fv2 in which A,B, · · · , F are fixed realnumbers. We investigate conditions on both bound-ary and initial data to guarantee the system to bewell-posed.
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Lyapunov stability of elliptic periodic solu-tions of nonlinear damped equations
Jifeng ChuHohai University, Peoples Rep of [email protected]
Based on the third order approximation developedby Ortega, we present the formula of the first twistcoe�cient for a nonlinear damped equation. Asexamples, we consider the Lyapunov stability of asuperlinear damped di↵erential equation and a sin-gular damped di↵erential equations.
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Existence and global attractivity of positiveperiodic solution to a Lotka-Volterra model
Zengji DuXuzhou Normal University, Peoples Rep of [email protected]
In this talk, a Volterra model with mutual inter-ference and Holling III type functional response isinvestigated, some su�cient conditions which guar-antee the existence and global attractivity of posi-tive periodic solution for the system are obtained,two examples are given to verify the results by us-ing MatLab. We improve the main results of thereferences.
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First integral of Du�ng-van der Pol oscillatorsystem
Zhaosheng FengUniversity of Texas-Pan American, [email protected]
In this talk, we restrict our attention to a nonlin-ear Du�ng–van der Pol–type oscillator system bymeans of the first-integral method. This system hasphysical relevance as a model in certain flow-inducedstructural vibration problems, which includes thevan der Pol oscillator and the damped Du�ng oscil-lator etc as particular cases. Through applying thefirst-integral method and the Lie symmetry method,we present a general first integral formula of theDu�ng–van der Pol–type oscillator system.
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Exact solutions of the Burgers-Huxley equa-tion
Tian JingThe University of Texas-Pan American, [email protected] Feng
In this talk, we apply the Lie symmetry reductionmethod to solve exact solutions of partial di↵erentialequations such as the Burgers-Huxley equation andthe generalized Fisher equation. Through analyzingthe symmetry condition and the resultant determin-ing system, we construct classical and non-classicalinfinitesimal generators, and obtain exact solutionsaccordingly.Joint with Zhaosheng Feng
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SPECIAL SESSION 68 261
The nonlinear Schrodinger equation createdby the vibrations of an elastic plate and itsdimensional expansion
Shuya KanagawaTokyo City University, [email protected] T. Nohara
In the course of studying the theory of plates theclassical, Kirchho↵ plate theory due to Timoshenko(1970) and Ugural (1981) in which transverse nor-mal and shear stresses are neglected to study bend-ing, buckling, and natural vibrations of rectangularplates, was first established. The first order sheardeformation plate theory extends the kinematics ofthe classical, Kirchho↵ plate theory by relaxing thenormality restriction and allowing for arbitrary butconstant rotation of transverse normals and finiteelement models are developed for the precise anal-ysis of the plate characteristics in real problem. Ingeneral, the Schrodinger equation governs the spa-tial and temporal evolution of the amplitude of awavepacket propagating transversely in any disper-sive, lossless medium. The Schrodinger equationgoverns an envelope created by a wavepacket byNohara(2003). The nonlinear Schrodinger equationgoverns the non-linearity of the envelope. Manystudies of a wavepacket has been carried out in wa-ter waves, fiber-optic communication systems, andsome other area as well.
In this paper, we, first, survey the two-dimensional governing equation that describes thepropagation of a wavepacket on an elastic plate us-ing the method of multiple scales. Then we expandthe governing equation to the multi-dimensional casenot only in the sense of mathematical science butalso engineering. For example the governing equa-tion for 3-dimensional coordinate (x, y, z) shows thevibrational dynamics inside the plate. On the otherhand 2-dimensionl wavenumber (k1, k2) means thevertical wavenumber k1 and horizontal wavenumberk2 on the plate, respectively.
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On the paradoxes of enrichment and pesti-cides
Y. Charles LiUniversity of Missouri, [email protected] Feng, Yipeng Yang
For the paradox of enrichment (which arose frommodels of population dynamics), we present a reso-lution (joint with Frank Feng). For the paradox ofpesticides (which arose from experiments), we studymathematical models and present a theory for theresolution (joint with Yipeng Yang).
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Intravenous glucose tolerance test model andits global stability
Jiaxu LiUniversity of Louisville, [email protected] Wang, Andrea De Gaetano, PasqualePalumbo, Simona Panunzi
Diabetes mellitus has become a prevalent diseasein the world. Diagnostic protocol for the onset ofdiabetes mellitus is the initial step in the treatments.The intravenous glucose tolerance test (IVGTT) hasbeen considered as the most accurate method todetermine the insulin sensitivity and glucose e↵ec-tiveness. It is well known that there exists a timedelay in insulin secretion stimulated by the elevatedglucose concentration level. However, the range ofthe length of the delay in the existing IVGTT mod-els are not fully discussed and thus in many casesthe time delay may be assigned to a value out of itsreasonable range. In addition, several attempts hadbeen made to determine when the unique equilibriumpoint is globally asymptotically stable. However, allthese conditions are delay-independent. In this talk,we review the existing IVGTT models, discuss therange of the time delay and provide easy-to-checkdelay-dependent conditions for the global asymptoticstability of the equilibrium point for a simplified butpractical IVGTT model through Liapunov functionapproach. Estimates of the upper bound of the delayfor global stability are given in corollaries. In addi-tion, the numerical simulation in this work is fullyincorporated with functional initial conditions, whichis natural and more appropriate in delay di↵erentialequation systems.
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Qishao LuBeihang Univ., Peoples Rep of [email protected] Shi
Physiological experiments have confirmed the ex-istence of synchronous oscillations of neurons in dif-ferent areas of the brain. Synchronization of neuronsplays a significant role in neural information pro-cessing. Clinical evidences also point out that thesynchronization of individual neurons plays a keyrole in some pathological conditions like Parkinson’sdisease, essential tremor, and epilepsies. Accordingto experimental results, we constructed a small worldneuronal network to study the e↵ects of the couplingscheme, the intrinsic property of individual neuronsand the network topology on burst synchronizationand the rhythm dynamics of the network. The re-sults are of fundamental significance to understandinformation activities in nervous systems.
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262 9th AIMS CONFERENCE – ABSTRACTS
Analytical solutions for the BoitiLeon-Pempinelli equation
Zhuosheng LuBeijing University of Posts and Telecommunications,Peoples Rep of [email protected]
We apply the classical Lie symmetry reductionmethod and an constructive algorithm to solve exactsolutions of the BoitiLeonPempinelli equation. In-finitesimal generators, Backlund transformation andvarious types of analytical solutions of the BoitiLeon-Pempinelli equation are obtained.
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Hyper-spherical harmonics and jumps in fi-nancial markets
Indranil SenguptaUniversity of Texas- El Paso, [email protected] C. Mariani
The technique of expanding functions in terms ofspherical harmonics turns out to be very useful insolving integral equations. This motivates us toapply the same technique in mathematical finance.The governing equation in mathematical finance ismostly a parabolic partial di↵erential equation. Wepresent a technique of solving Heat equation andBlack-Scholes equation using the method of spheri-cal harmonics. We also discuss the method relatedto the Black-Scholes equation in annular domain andsome generalized Black-Scholes equations. Finallywe solve some integro-di↵erential equation arising infinancial models with jumps by using the method ofspherical and hyper-spherical harmonics.
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A mathematical model with time-varying an-tiretroviral therapy for the treatment of HIV
Nicoleta TarfuleaPurdue University Calumet, [email protected]
We present a mathematical model to investigatetheoretically and numerically the e↵ect of immunee↵ectors in modeling HIV pathogenesis. Addition-ally, by introducing drug therapy, we assess the e↵ectof treatments consisting of a combination of severalantiretroviral drugs. A periodic model of bang-bangtype and a pharmacokinetic model are employed toestimate the drug e�ciencies. Nevertheless, even inthe presence of drug therapy, ongoing viral replica-tion can lead to the emergence of drug-resistant virusvariances. Thus, by including two viral strains, wild-type and drug-resistant, we show that the inclusion
of the CTL compartment produces a higher reboundfor an individuals healthy helper T-cell compartmentthan does drug therapy alone. We investigate nu-merically how time-varying drug e�cacy due to drugdosing regimen and/or suboptimal adherence a↵ectsthe antiviral response and the emergence of drugresistance.
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Investigation of the long-time evolution of lo-calized solutions of a dispersive wave system
Michail TodorovTechnical University of Sofia, [email protected] I. Christov
We consider the long-time evolution of the solutionof a system with dispersion and nonlinearity which isthe progenitor of the di↵erent Boussinesq equations.As initial condition we use a wave system comprisedby the superposition of two analytical soliton solu-tions. We use a strongly implicit di↵erence schemewith internal iterations which allows us to follow theevolution of the solution at very long times. We fo-cus on the dynamical behavior of traveling localizedsolutions developing from critical initial data. Thesystem is rendered into a seven-diagonal band-matrixform and solved e↵ectively by specialized solver withpivoting which is stable to round-o↵ errors even for2.500.000 points of spatial resolution. The main soli-tary waves appear virtually non-deformed from theinteraction, but additional oscillations are excited atthe trailing edge of each one of them. We extract theperturbations and track their evolution for very longtimes when they tend to adopt a self-similar shape:their amplitudes decrease with the time while thelength scales increase. We test a hypothesis aboutthe dependence on time of the amplitude and thesupport of Airy-function shaped coherent structureswhich gives a very good quantitative agreement withthe numerically obtained solutions. This investiga-tion is supported by the Scientific Foundation of theBulgarian Ministry of Education, Youth, and Scienceunder Grant DDVU02/71.
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Entire and blow-up solutions: from semilinearto fully nonlinear elliptic equations
Antonio VitoloUniversity of Salerno, [email protected]
Semilinear elliptic equations arise in many fieldsof the applied sciences, in particular in mechanics,engineering and life sciences. We discuss existence,uniqueness and qualitative properties of solutions inthe whole space as well as in a bounded domain,possibly explosive on the boundary, for second orderelliptic operators with a superlinear growth in the
SPECIAL SESSION 68 263
solution, both in the case of a linear main term andof a fully nonlinear one.
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Models and applications for minority healthStudies
Xiaohui WangUniversity of Texas-Pan American, [email protected]
Minority health status is of big concern to our society.In many cases, researchers seek to understand the ef-fects of predictors (such as demographic measures,social-economic measures, insurance etc.) on healthor intervention outcomes. Generalized linear modelsare used in our studies to explore such relationships.Application examples will be illustrated.
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Qualitative properties of positive solutions ofa class of boundary blow-up problems
Lei WeiJiangsu Normal University, Peoples Rep of [email protected]
We will consider boundary blow-up problems onan unbounded domain. For the problems, we willstudy the existence, uniqueness, blow-up rate of pos-itive solutions. Our results extend some results ofProfessor Julian Lopez Gomez’s papers.
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The e↵ects of singular lines in nonlinear waveequations
Lijun ZhangZhejiang Sci-Tech University, Peoples Rep of [email protected]
In this talk, the dynamical system approach to studytraveling wave solutions of nonlinear wave equationsis introduced. We pay more careful attentions tothe travelling wave solutions of the nonlinear waveequations with nonlinear dispersive terms which cor-responding traveling wave system may have singularlines and thus will result in a variety of singularwaves. The singular traveling waves of the gener-alized nonlinear Klein-Gordon model equation arestudied as an example to demonstrate the e↵ects ofsingular lines in nonlinear wave equations.
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Energy identity for a class of approximate bi-harmonic maps into sphere in dimension four
Shenzhou ZhengSchool of Science, Beijing Jiaotong University, Peo-ples Rep of [email protected] Wang
We considered in dimension four weakly conver-gent sequences of approximate biharmonic maps intosphere with bi-tension fields bounded in Lp for somep > 1. We prove an energy identity that accounts forthe loss of Hessian energies by the sum of Hessian en-ergies over finitely many nontrivial biharmonic mapson R4.
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