DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

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MODELLING THE EFFECT OF SURGICAL STRESS AND BACTERIAL GROWTH IN HUMAN CORNEA D. ROY MAHAPATRA AND R. V. N. MELNIK This paper reports a mathematical model and finite-element simulation of the dynamic piezoelectricity in human cornea including the effectofdehydrationandstressgenerated due to incision and bacterial growth. A constitutive model is proposed for the numerical characterization of cornea based on the available experimental data. The constitutive model is then employed to derive the conservation law for the dynamic piezoelectricity supplemented by the time-dependent equation for the electromagnetic field. The resulting system of partial differential equations is solved numerically with finite-element methodology. Numerical results presented here demonstrate promising applications of the developed model in aiding refractive surgery and a better understanding of regenerative processes in cornea. Copyright © 2006 D. R. Mahapatra and R. V. N. Melnik. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Cornea is one of the most delicate and active tissue systems in human and several other species. Any major change in the equilibrium stress in the sclera and cornea due to incision, excessive swelling, and bacterial growth can cause deterioration of the refractive performance of the cornea. Cornea consists of a complex architecture of the collagen fibrils dispersed in the matrix containing proteoglycans. The anisotropic structure of this composite system is distributed over the stromal layer. Computer models for surgical aid in the past had been developed, these models neglected the effect of complex tissue architecture and the resulting constitutive behavior in the long-term tissue remodelling. The role of these factors in context of surgical procedures has been brought into focus only recently and mathematical model has been proposed (see [6]). Apart from the collagen orientation-dependent anisotropy in cornea, it may be noted that the dynamic nature of the refractive property, which is very little understood in the case of cornea as compared to the sclera, is dependent on the electrical permittivity, Hindawi Publishing Corporation Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 721–731
PURELY VECTOR GROUND STATE FOR A NONLINEAR NONAUTONOMOUS SCHRÖDINGER SYSTEM LILIANE DE ALMEIDA MAIA, EUGENIO MONTEFUSCO, AND BENEDETTA PELLACCI Existence of a positive purely vector ground state solution is established, via variational methods, for a nonautonomous system of weakly coupled nonlinear Schrödinger equations. Copyright © 2006 Liliane de Almeida Maia et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction This paper is concerned with the existence of nontrivial solutions and among them, ground state solutions for systems of coupled nonlinear Schrödinger equations. Such systems arise in several branches of mathematical physics, in particular in describing the interaction between waves of different frequencies and also the interaction between orthogonally polarized components in nonlinear optical fibers (see [6, 13]). After performing suitable simplifications and variable rescaling, such models are written as systems of partial differential equations of the form iφ t + φ xx + ( |φ| 2 + b|ψ| 2) φ = 0, iψ t + ψ xx + ( |ψ| 2 + b|φ| 2) ψ = 0, (1.1) where φ and ψ represent the complex amplitudes of two-wave packets and b is a realvalued cross-phase modulation coefficient. Looking for standing wave solutions of (1.1), that is, solutions of the form φ(x,t) = e iω2 1t u(x), ψ(x,t) = e iω2 2t v(x), (1.2) and performing a rescaling of variables, one obtains that u and v satisfy the following system: −u xx + u =|u| 2 u + b|v| 2 u in R, −v xx + ω 2 v =|v| 2 v + b|u| 2 v in R, (1.3) where ω 2 = ω 2 2/ω 2 1. The existence of vector solitary waves (1.2) as solutions to (1.1) has been investigated by theoretical and numerical means, as reviewed in [13]. If b = 0, in Hindawi Publishing Corporation Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 733–742
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Proceedings of the Conference on Di
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Proceedings of the Conference on Di
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CONTENTS Preface...................
Page 8 and 9:
Inequalities for positive solutions
Page 10 and 11:
Modelling the effect of surgical st
Page 12:
Maximum principles for elliptic equ
Page 16 and 17:
A UNIFIED APPROACH TO MONOTONE METH
Page 18 and 19:
IMPULSIVE CONTROL OF THE POPULATION
Page 20 and 21:
BOUNDARY ESTIMATES FOR BLOW-UP SOLU
Page 22 and 23:
SECOND-ORDER DIFFERENTIAL GRADIENT
Page 24 and 25:
ON STRUCTURE OF FRACTIONAL SPACES G
Page 26 and 27:
ON THE STABILITY OF THE DIFFERENCE
Page 28 and 29:
TWO-WEIGHTED POINCARÉ-TYPE INTEGRA
Page 30 and 31:
PRODUCT DIFFERENCE EQUATIONS APPROX
Page 32 and 33:
QUASIDIFFUSION MODEL OF POPULATION
Page 34 and 35:
FIXED-SIGN EIGENFUNCTIONS OF TWO-PO
Page 36 and 37: MULTIPLE POSITIVE SOLUTIONS OF SUPE
Page 38 and 39: SPECTRAL STABILITY OF ELLIPTIC SELF
Page 40 and 41: HO CHARACTERISTIC EQUATIONS SURPASS
Page 42 and 43: ASYMPTOTIC STABILITY IN DISCRETE MO
Page 44 and 45: CONTINUOUS AND DISCRETE NONLINEAR I
Page 46 and 47: ON PARABOLIC SYSTEMS WITH DISCONTIN
Page 48 and 49: INEQUALITIES FOR POSITIVE SOLUTIONS
Page 50 and 51: NONOSCILLATION OF ONE OF THE COMPON
Page 52 and 53: ANNULAR JET AND COAXIAL JET FLOW JO
Page 54 and 55: JUSTIFICATION OF QUADRATURE-DIFFERE
Page 56 and 57: VLASOV-ENSKOG EQUATION WITH THERMAL
Page 58 and 59: SUBDIFFERENTIAL OPERATOR APPROACH T
Page 60 and 61: A DISCRETE EIGENFUNCTIONS METHOD FO
Page 62 and 63: ON NONLINEAR SCHRÖDINGER EQUATIONS
Page 64 and 65: NUMERICAL SOLUTION OF A TRANSMISSIO
Page 66 and 67: THE SEMILINEAR EVOLUTION EQUATION F
Page 68 and 69: ON DOUBLY PERIODIC SOLUTIONS OF QUA
Page 70 and 71: DYNAMICS OF A DISCONTINUOUS PIECEWI
Page 72 and 73: PROBABILISTIC SOLUTIONS OF THE DIRI
Page 74 and 75: MONOTONICITY RESULTS AND INEQUALITI
Page 76 and 77: WELL-POSEDNESS AND BLOW-UP OF SOLUT
Page 78 and 79: REARRANGEMENT ON THE UNIT BALL FOR
Page 80 and 81: CONVERGENCE OF SERIES OF TRANSLATIO
Page 82 and 83: ON MINIMAL (MAXIMAL) COMMON FIXED P
Page 84 and 85: BOUNDEDNESS OF SOLUTIONS OF FUNCTIO
Page 88 and 89: LIMIT CYCLES OF LIÉNARD SYSTEMS M.
Page 90 and 91: MAXIMUM PRINCIPLES AND DECAY ESTIMA
Page 92 and 93: LIPSCHITZ REGULARITY OF VISCOSITY S
Page 94 and 95: ON AN ELASTIC BEAM FULLY EQUATION W
Page 96 and 97: BOUNDARY VALUE PROBLEMS ON THE HALF
Page 98 and 99: EXISTENCE AND UNIQUENESS OF AN INTE
Page 100 and 101: STABILITY OF SOME MODELS OF CIRCULA
Page 102 and 103: LIPSCHITZ CLASSES OF A-HARMONIC FUN
Page 104 and 105: A TWO-DEGREES-OF-FREEDOM HAMILTONIA
Page 106 and 107: QUANTIZATION OF LIGHT FIELD IN PERI
Page 108 and 109: CONSERVATION LAWS IN QUANTUM SUPER
Page 110 and 111: ON SETS OF ZERO HAUSDORFF s-DIMENSI
Page 112 and 113: ON THE GLOBAL ATTRACTOR IN A CLASS
Page 114 and 115: SANDWICH PAIRS MARTIN SCHECHTER Sin
Page 116 and 117: EXISTENCE RESULTS FOR EVOLUTION INC
Page 118 and 119: AVERAGING DOMAINS: FROM EUCLIDEAN S
Page 120 and 121: CONCENTRATION-COMPACTNESS PRINCIPLE
Page 122 and 123: NONLINEAR VARIATIONAL INCLUSION PRO
Page 124 and 125: MAXIMUM PRINCIPLES FOR ELLIPTIC EQU
Page 126 and 127: GLOBAL CONVERGENT ALGORITHM FOR PAR
Page 128 and 129: SECOND-ORDER NONLINEAR OSCILLATIONS
Page 130 and 131: ANALYTIC SOLUTIONS OF UNSTEADY CRYS
Page 132 and 133: UNBOUNDEDNESS OF SOLUTIONS OF PLANA
Page 134 and 135: QUENCHING OF SOLUTIONS OF NONLINEAR
Page 136 and 137:
VARIATION-OF-PARAMETERS FORMULAE AN
Page 138:
DIFFERENCE EQUATIONS AND THE ELABOR
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DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

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