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Статті в журналах з теми "Free boundary model"

Автор: Grafiati

Опубліковано: 2 листопада 2024

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1

Lions, P. L., and N. Masmoudi. "On a free boundary barotropic model." Annales de l'Institut Henri Poincare (C) Non Linear Analysis 16, no. 3 (May 1999): 373–410. http://dx.doi.org/10.1016/s0294-1449(99)80018-3.

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2

Yi,, Tong Y., and Parviz E. Nikravesh. "Extraction of Free-Free Modes from Constrained Vibration Data for Flexible Multibody Models." Journal of Vibration and Acoustics 123, no. 3 (February 1, 2001): 383–89. http://dx.doi.org/10.1115/1.1375814.

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This paper presents a method for identifying the free-free modes of a structure by utilizing the vibration data of the same structure with boundary conditions. In modal formulations for flexible body dynamics, modal data are primary known quantities that are obtained either experimentally or analytically. The vibration measurements may be obtained for a flexible body that is constrained differently than its boundary conditions in a multibody system. For a flexible body model in a multibody system, depending upon the formulation used, we may need a set of free-free modal data or a set of constrained modal data. If a finite element model of the flexible body is available, its vibration data can be obtained analytically under any desired boundary conditions. However, if a finite element model is not available, the vibration data may be determined experimentally. Since experimentally measured vibration data are obtained for a flexible body supported by some form of boundary conditions, we may need to determine its free-free vibration data. The aim of this study is to extract, based on experimentally obtained vibration data, the necessary free-free frequencies and the associated modes for flexible bodies to be used in multibody formulations. The available vibration data may be obtained for a structure supported either by springs or by fixed boundary conditions. Furthermore, the available modes may be either a complete set, having as many modes as the number of degrees of freedom of the associated FE model, or an incomplete set.

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3

Liu, Yunfeng, Zhiming Guo, Mohammad El Smaily, and Lin Wang. "A Wolbachia infection model with free boundary." Journal of Biological Dynamics 14, no. 1 (January 1, 2020): 515–42. http://dx.doi.org/10.1080/17513758.2020.1784474.

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4

Kim, Kwang Ik, Zhigui Lin, and Qunying Zhang. "An SIR epidemic model with free boundary." Nonlinear Analysis: Real World Applications 14, no. 5 (October 2013): 1992–2001. http://dx.doi.org/10.1016/j.nonrwa.2013.02.003.

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5

Baker, Ruth E., Andrew Parker, and Matthew J. Simpson. "A free boundary model of epithelial dynamics." Journal of Theoretical Biology 481 (November 2019): 61–74. http://dx.doi.org/10.1016/j.jtbi.2018.12.025.

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6

Elmurodov, Alimardon, Abduraxmon Norov, N. Yuldasheva, Sanjarbek Yuldashev, and Mavjuda Sadullayeva. "Free boundary problem for predator-prey model." E3S Web of Conferences 401 (2023): 04062. http://dx.doi.org/10.1051/e3sconf/202340104062.

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In this article, we study the behavior of two species evolving in a domain with a free boundary. This system mimics the spread of invasive or new predator species, in which free boundaries represent the expanding fronts of predator species and are described by the Stefan condition. A priori estimates for the required functions are established. These estimates are used to prove the existence and uniqueness of the solution.

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7

Petrova, A. G., and V. V. Pukhnachev. "Free Boundary Problem in a Polymer Solution Model." Russian Journal of Mathematical Physics 28, no. 1 (January 2021): 96–103. http://dx.doi.org/10.1134/s1061920821010106.

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8

Takhirov, J. O., and A. Norov. "On a predator-prey model with free boundary." Uzbek Mathematical Journal 2019, no. 4 (December 2, 2019): 162–68. http://dx.doi.org/10.29229/uzmj.2019-4-17.

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9

Tambyah, Tamara A., Ryan J. Murphy, Pascal R. Buenzli, and Matthew J. Simpson. "A free boundary mechanobiological model of epithelial tissues." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, no. 2243 (November 2020): 20200528. http://dx.doi.org/10.1098/rspa.2020.0528.

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In this study, we couple intracellular signalling and cell-based mechanical properties to develop a novel free boundary mechanobiological model of epithelial tissue dynamics. Mechanobiological coupling is introduced at the cell level in a discrete modelling framework, and new reaction–diffusion equations are derived to describe tissue-level outcomes. The free boundary evolves as a result of the underlying biological mechanisms included in the discrete model. To demonstrate the accuracy of the continuum model, we compare numerical solutions of the discrete and continuum models for two different signalling pathways. First, we study the Rac–Rho pathway where cell- and tissue-level mechanics are directly related to intracellular signalling. Second, we study an activator–inhibitor system which gives rise to spatial and temporal patterning related to Turing patterns. In all cases, the continuum model and free boundary condition accurately reflect the cell-level processes included in the discrete model.

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10

Saavedra, Patricia, and L. Ridgway Scott. "Variational formulation of a model free-boundary problem." Mathematics of Computation 57, no. 196 (1991): 451. http://dx.doi.org/10.1090/s0025-5718-1991-1094958-0.

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11

Funaki, T. "Free boundary problem from stochastic lattice gas model." Annales de l'Institut Henri Poincare (B) Probability and Statistics 35, no. 5 (September 1999): 573–603. http://dx.doi.org/10.1016/s0246-0203(99)00107-7.

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12

Friedman, Avner. "Free boundary problems in biology." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, no. 2050 (September 13, 2015): 20140368. http://dx.doi.org/10.1098/rsta.2014.0368.

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In this paper, I review several free boundary problems that arise in the mathematical modelling of biological processes. The biological topics are quite diverse: cancer, wound healing, biofilms, granulomas and atherosclerosis. For each of these topics, I describe the biological background and the mathematical model, and then proceed to state mathematical results, including existence and uniqueness theorems, stability and asymptotic limits, and the behaviour of the free boundary. I also suggest, for each of the topics, open mathematical problems.

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13

BERON-VERA, F. J., and P. RIPA. "Free boundary effects on baroclinic instability." Journal of Fluid Mechanics 352 (December 10, 1997): 245–64. http://dx.doi.org/10.1017/s0022112097007222.

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The effects of a free boundary on the stability of a baroclinic parallel flow are investigated using a reduced-gravity model. The basic state has uniform density stratification and a parallel flow with uniform vertical shear in thermal-wind balance with the horizontal buoyancy gradient. A finite value of the velocity at the free (lower) boundary requires the interface to have a uniform slope in the direction transversal to that of the flow. Normal-mode perturbations with arbitrary vertical structure are studied in the limit of small Rossby number. This solution is restricted to neither a horizontal lower boundary nor a weak stratification in the basic state.In the limit of a very weak stratification and bottom slope there is a large separation between the first two deformation radii and hence short or long perturbations may be identified:(a) The short-perturbation limit corresponds to the well-known Eady problem in which case the layer bottom is effectively rigid and its slope in the basic state is immaterial.(b) In the long-perturbation limit the bottom is free to deform and the unstable wave solutions, which appear for any value of the Richardson number Ri, are sensible to its slope in the basic state. In fact, a sloped bottom is found to stabilize the basic flow.At stronger stratifications there is no distinction between short and long perturbations, and the bottom always behaves as a free boundary. Unstable wave solutions are found for Ri→∞ (unlike the case of long perturbations). The increase in stratification is found to stabilize the basic flow. At the maximum stratification compatible with static stability, the perturbation has a vanishing growth rate at all wavenumbers.Results in the long-perturbation limit corroborate those predicted by an approximate layer model that restricts the buoyancy perturbations to have a linear vertical structure. The approximate model is less successful in the short-perturbation limit since the constraint to a linear density profile does not allow the correct representation of the exponential trapping of the exact eigensolutions. With strong stratification, only the growth rate of long enough perturbations superimposed on basic states with gently sloped lower boundaries behaves similarly to that of the exact model. However, the stabilizing tendency on the basic flow as the stratification reaches its maximum is also found in the approximate model. Its partial success in this case is also attributed to the limited vertical structure allowed by the model.

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14

Volino, R. J. "A New Model for Free-Stream Turbulence Effects on Boundary Layers." Journal of Turbomachinery 120, no. 3 (July 1, 1998): 613–20. http://dx.doi.org/10.1115/1.2841760.

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A model has been developed to incorporate more of the physics of free-stream turbulence effects into boundary layer calculations. The transport in the boundary layer is modeled using three terms: (1) the molecular viscosity, ν; (2) the turbulent eddy viscosity, εT, as used in existing turbulence models; and (3) a new free-stream-induced eddy viscosity, εf. The three terms are added to give an effective total viscosity. The free-stream-induced viscosity is modeled algebraically with guidance from experimental data. It scales on the rms fluctuating velocity in the free stream, the distance from the wall, and the boundary layer thickness. The model assumes a direct tie between boundary layer and free-stream fluctuations, and a distinctly different mechanism than the diffusion of turbulence from the free-stream to the boundary layer assumed in existing higher order turbulence models. The new model can be used in combination with any existing turbulence model. It is tested here in conjunction with a simple mixing length model and a parabolic boundary layer solver. Comparisons to experimental data are presented for flows with free-stream turbulence intensities ranging from 1 to 8 percent and for both zero and nonzero streamwise pressure gradient cases. Comparisons are good. Enhanced heat transfer in higher turbulence cases is correctly predicted. The effect of the free-stream turbulence on mean velocity and temperature profiles is also well predicted. In turbulent flow, the log region in the inner part of the boundary layer is preserved, while the wake is suppressed. The new model provides a simple and effective improvement for boundary layer prediction.

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15

Yin, Hai Bin, Yu Feng Li, Yan Zhao, and Pei Juan Cui. "Theoretical and Experimental Investigation on Modal Characteristics of Rotating Flexible Beam under Dynamic Boundary Conditions." Applied Mechanics and Materials 518 (February 2014): 33–40. http://dx.doi.org/10.4028/www.scientific.net/amm.518.33.

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In order to exactly formulate the dynamics and solve the flexible vibration in the studies on the flexible structure, it is inevitable to study the modal characteristics of the flexible structure; while the accuracy of the modal characteristics relies on the choice of its boundary conditions. This paper investigates the modal characteristics of a flexible beam considering dynamic boundary conditions. Firstly, the first three mode shape functions and frequencies of a sliding-mass beam model are obtained through theoretical solutions. The concept of dynamic boundary conditions is brought forward by the study of special boundary conditions. Secondly, a rotating-free beam model is proposed to describe the dynamic boundary conditions of the rotating flexible beam. The first three frequencies and the mode shape functions of the rotating-free beam model are obtained. Finally, a series of samples and experiments proves the validity of the proposal on the rotating-free beam model.

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16

Liao, Jie. "Phase Field Model for Solidification with Boundary Interface Interaction." International Frontier Science Letters 9 (August 2016): 1–8. http://dx.doi.org/10.18052/www.scipress.com/ifsl.9.1.

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By incorporation the surface free energy in the free energy functional, a phase field model for solidification with boundary interface intersection is developed. In this model, the bulk equation is appropriately modified to account for the presence of heat diffusion inside the diffuse interface, and a relaxation boundary condition for the phase field variable is introduced to balance the interface energy and boundary surface energy in the multiphase contact region. The asymptotic analysis is applied on the phase field model to yield the free interface problem with dynamic contact point condition.

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17

Zheng, Jiayue, and Shangbin Cui. "Analysis of a tumor-model free boundary problem with a nonlinear boundary condition." Journal of Mathematical Analysis and Applications 478, no. 2 (October 2019): 806–24. http://dx.doi.org/10.1016/j.jmaa.2019.05.056.

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18

Goncharov, A. A. "Nonconservative boundary-element model of wedge free-wheel mechanisms." Journal of Machinery Manufacture and Reliability 37, no. 2 (April 2008): 123–29. http://dx.doi.org/10.3103/s1052618808020052.

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19

Bunting, Gary, Yihong Du, and Krzysztof Krakowski. "Spreading speed revisited: Analysis of a free boundary model." Networks & Heterogeneous Media 7, no. 4 (2012): 583–603. http://dx.doi.org/10.3934/nhm.2012.7.583.

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20

Lin, Zhigui. "A free boundary problem for a predator–prey model." Nonlinearity 20, no. 8 (June 22, 2007): 1883–92. http://dx.doi.org/10.1088/0951-7715/20/8/004.

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21

Huang, Yaodan, Zhengce Zhang, and Bei Hu. "Bifurcation for a free-boundary tumor model with angiogenesis." Nonlinear Analysis: Real World Applications 35 (June 2017): 483–502. http://dx.doi.org/10.1016/j.nonrwa.2016.12.003.

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22

Lorenzi, Luca, and Alessandra Lunardi. "Stability in a two-dimensional free boundary combustion model." Nonlinear Analysis: Theory, Methods & Applications 53, no. 2 (April 2003): 227–76. http://dx.doi.org/10.1016/s0362-546x(02)00289-4.

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23

Lorenzi, Luca, and Alessandra Lunardi. "“Stability in a two-dimensional free boundary combustion model”." Nonlinear Analysis: Theory, Methods & Applications 53, no. 6 (June 2003): 859–60. http://dx.doi.org/10.1016/s0362-546x(03)00076-2.

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24

Huyse, Luc, Mansa C. Singh, and Marc A. Maes. "A static drilling riser model using free boundary conditions." Ocean Engineering 24, no. 5 (May 1997): 431–44. http://dx.doi.org/10.1016/s0029-8018(96)00021-2.

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25

González, María del Mar, and Maria Pia Gualdani. "Asymptotics for a free boundary model in price formation." Nonlinear Analysis: Theory, Methods & Applications 74, no. 10 (July 2011): 3269–94. http://dx.doi.org/10.1016/j.na.2011.02.005.

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26

Friedman, Avner, and King-Yeung Lam. "Analysis of a free-boundary tumor model with angiogenesis." Journal of Differential Equations 259, no. 12 (December 2015): 7636–61. http://dx.doi.org/10.1016/j.jde.2015.08.032.

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27

Atkinson, C. "A Free Boundary Problem of a Real Option Model." SIAM Journal on Applied Mathematics 69, no. 6 (January 2009): 1793–804. http://dx.doi.org/10.1137/080723089.

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28

Gosset, Jérôme. "Multidimensional time-dependent free boundary model of ion extraction." Physics of Plasmas 6, no. 1 (January 1999): 385–94. http://dx.doi.org/10.1063/1.873292.

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29

Li, Heng, Yongzhi Xu, and Jian-Rong Zhou. "A free boundary problem arising from DCIS mathematical model." Mathematical Methods in the Applied Sciences 40, no. 10 (November 18, 2016): 3566–79. http://dx.doi.org/10.1002/mma.4246.

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30

Kukavica, Igor, and Amjad Tuffaha. "A free boundary inviscid model of flow-structure interaction." Journal of Differential Equations 413 (December 2024): 851–912. http://dx.doi.org/10.1016/j.jde.2024.08.045.

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31

Casabán, María Consuelo, Rafael Company, Vera N. Egorova, and Lucas Jódar. "Qualitative Numerical Analysis of a Free-Boundary Diffusive Logistic Model." Mathematics 11, no. 6 (March 8, 2023): 1296. http://dx.doi.org/10.3390/math11061296.

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Анотація:

A two-dimensional free-boundary diffusive logistic model with radial symmetry is considered. This model is used in various fields to describe the dynamics of spreading in different media: fire propagation, spreading of population or biological invasions. Due to the radial symmetry, the free boundary can be treated by a front-fixing approach resulting in a fixed-domain non-linear problem, which is solved by an explicit finite difference method. Qualitative numerical analysis establishes the stability, positivity and monotonicity conditions. Special attention is paid to the spreading–vanishing dichotomy and a numerical algorithm for the spreading–vanishing boundary is proposed. Theoretical statements are illustrated by numerical tests.

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32

Lepoittevin, Grégoire, and Gerald Kress. "Finite element model updating of vibrating structures under free–free boundary conditions for modal damping prediction." Mechanical Systems and Signal Processing 25, no. 6 (August 2011): 2203–18. http://dx.doi.org/10.1016/j.ymssp.2011.01.019.

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33

Lyashenko, Yu A., and Andriy Gusak. "DIGM - Entropy Balance and Free Energy Release Rate." Defect and Diffusion Forum 249 (January 2006): 81–90. http://dx.doi.org/10.4028/www.scientific.net/ddf.249.81.

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A model of alloying in the three-layer thin-film system at the low temperature is constructed. Solid solution formation takes place as a result of the diffusion-induced grain boundary migration (DIGM). The unknown parameters are determined from the set of the equations for: (1) grain boundary diffusion along the moving planar phase boundary; (2) the entropy balance in the region of the phase transformation moving with constant velocity; (3) the maximum rate of the free energy release. We consider the model system with complete solubility of the components. The main parameters are self-consistently determined using thermodynamic and kinetic description in the frame of the regular solution model. The model allows determining the concentration distribution along the planar moving phase boundary, its velocity, the thickness of the forming solid solution layer and the limiting average concentration in this layer.

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34

Saito, Hirokazu. "Compressible Fluid Model of Korteweg Type with Free Boundary Condition: Model Problem." Funkcialaj Ekvacioj 62, no. 3 (2019): 337–86. http://dx.doi.org/10.1619/fesi.62.337.

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35

Li, Mei. "The spreading fronts in a mutualistic model with delay." International Journal of Biomathematics 09, no. 06 (August 2, 2016): 1650080. http://dx.doi.org/10.1142/s1793524516500807.

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This paper is concerned with a system of semilinear parabolic equations with two free boundaries describing the spreading fronts of the invasive species in a mutualistic ecological model. The local existence and uniqueness of a classical solution are obtained and the asymptotic behavior of the free boundary problem is studied. Our results indicate that two free boundaries tend monotonically to finite or infinite limits at the same time, and the free boundary problem admits a global slow solution with unbounded free boundaries if the intra-specific competitions are strong, while if the intra-specific competitions are weak, there exist the blowup solution and global fast solution.

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36

Wang, Mingxin, and Jingfu Zhao. "A Free Boundary Problem for the Predator–Prey Model with Double Free Boundaries." Journal of Dynamics and Differential Equations 29, no. 3 (October 15, 2015): 957–79. http://dx.doi.org/10.1007/s10884-015-9503-5.

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37

Waite, Michael L., and Boualem Khouider. "Boundary Layer Dynamics in a Simple Model for Convectively Coupled Gravity Waves." Journal of the Atmospheric Sciences 66, no. 9 (September 1, 2009): 2780–95. http://dx.doi.org/10.1175/2009jas2871.1.

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Abstract A simplified model of intermediate complexity for convectively coupled gravity waves that incorporates the bulk dynamics of the atmospheric boundary layer is developed and analyzed. The model comprises equations for velocity, potential temperature, and moist entropy in the boundary layer as well as equations for the free tropospheric barotropic (vertically uniform) velocity and first two baroclinic modes of vertical structure. It is based on the multicloud model of Khouider and Majda coupled to the bulk boundary layer–shallow cumulus model of Stevens. The original multicloud model has a purely thermodynamic boundary layer and no barotropic velocity mode. Here, boundary layer horizontal velocity divergence is matched with barotropic convergence in the free troposphere and yields environmental downdrafts. Both environmental and convective downdrafts act to transport dry midtropospheric air into the boundary layer. Basic states in radiative–convective equilibrium are found and are shown to be consistent with observations of boundary layer and free troposphere climatology. The linear stability of these basic states, in the case without rotation, is then analyzed for a variety of tropospheric regimes. The inclusion of boundary layer dynamics—specifically, environmental downdrafts and entrainment of free tropospheric air—enhances the instability of both the synoptic-scale moist gravity waves and nonpropagating congestus modes in the multicloud model. The congestus mode has a preferred synoptic-scale wavelength, which is absent when a purely thermodynamic boundary layer is employed. The weak destabilization of a fast mesoscale wave, with a phase speed of 26 m s−1 and coupling to deep convection, is also discussed.

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38

KOJIMA, TAKEO. "THE SU(n)-INVARIANT MASSIVE THIRRING MODEL WITH BOUNDARY REFLECTION." International Journal of Modern Physics A 16, no. 15 (June 20, 2001): 2665–89. http://dx.doi.org/10.1142/s0217751x01003706.

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We study the SU (n)-invariant massive Thirring model with boundary reflection. Our approach is based on the free field approach. We construct the free field realizations of the boundary state and its dual. For an application of these realizations, we present integral representations for the form factors of the local operators.

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39

Zheng, Jiayue, and Shangbin Cui. "Bifurcation analysis of a tumor-model free boundary problem with a nonlinear boundary condition." Discrete & Continuous Dynamical Systems - B 22, no. 11 (2017): 0. http://dx.doi.org/10.3934/dcdsb.2020103.

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40

Mikhasev, G., E. Avdeichik, and D. Prikazchikov. "Free vibrations of nonlocally elastic rods." Mathematics and Mechanics of Solids 24, no. 5 (July 13, 2018): 1279–93. http://dx.doi.org/10.1177/1081286518785942.

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Several of the Eringen’s nonlocal stress models, including two-phase and purely nonlocal integral models, along with the simplified differential model, are studied in the case of free longitudinal vibrations of a nanorod, for various types of boundary conditions. Assuming the exponential attenuation kernel in the nonlocal integral models, the integro-differential equation corresponding to the two-phase nonlocal model is reduced to a fourth-order differential equation with additional boundary conditions, taking into account nonlocal effects in the neighbourhood of the rod ends. Exact analytical and asymptotic solutions of boundary-value problems are constructed. Formulas for natural frequencies and associated modes found in the framework of the purely nonlocal model and its ‘equivalent’ differential analogue are also compared. A detailed analysis of solutions suggests that the purely nonlocal and differential models lead to ill-posed problems.

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41

Rodin, Alexey, Leonid Klinger, and Boris S. Bokstein. "Solute Diffusion in Grain Boundaries – Outside the Scope of Fisher Model." Defect and Diffusion Forum 289-292 (April 2009): 711–18. http://dx.doi.org/10.4028/www.scientific.net/ddf.289-292.711.

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Thermodynamic and kinetic models are developed for grain boundary segregation and diffusion with regards to the possible complexes formation in grain boundary. The equilibrium state for ideal solutions can be described by the equilibrium constants b and K. The first corresponds to the pure segregation, i.e. the exchange A and B atoms between grain boundary and the bulk. The second represents the equilibrium of the reaction of the complexes formation in grain boundary. The segregation isotherm and diffusion profiles are calculated. It is shown that both b and K equilibrium constants define completely dependence of the total grain boundary concentration of B atoms on the bulk concentration and distribution of B atoms between two states: free (pure exchange) and tied (in the complexes). Segregation in both forms (free B atoms and tied B atoms) decreases grain boundary diffusivity in comparison with the absence of segregation.

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42

SHEN, LIAN, GEORGE S. TRIANTAFYLLOU, and DICK K. P. YUE. "Turbulent diffusion near a free surface." Journal of Fluid Mechanics 407 (March 25, 2000): 145–66. http://dx.doi.org/10.1017/s0022112099007466.

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We study numerically and analytically the turbulent diffusion characteristics in a low-Froude-number turbulent shear flow beneath a free surface. In the numerical study, the Navier–Stokes equations are solved directly subject to viscous boundary conditions at the free surface. From an ensemble of such simulations, we find that a boundary layer develops at the free surface characterized by a fast reduction in the value of the eddy viscosity. As the free surface is approached, the magnitude of the mean shear initially increases over the boundary (outer) layer, reaches a maximum and then drops to zero inside a much thinner inner layer. To understand and model this behaviour, we derive an analytical similarity solution for the mean flow. This solution predicts well the shape and the time-scaling behaviour of the mean flow obtained in the direct simulations. The theoretical solution is then used to derive scaling relations for the thickness of the inner and outer layers. Based on this similarity solution, we propose a free-surface function model for large-eddy simulations of free-surface turbulence. This new model correctly accounts for the variations of the Smagorinsky coefficient over the free-surface boundary layer and is validated in both a priori and a posteriori tests.

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43

TEIXEIRA, M. A. C., and S. E. BELCHER. "Dissipation of shear-free turbulence near boundaries." Journal of Fluid Mechanics 422 (November 3, 2000): 167–91. http://dx.doi.org/10.1017/s002211200000149x.

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The rapid-distortion model of Hunt & Graham (1978) for the initial distortion of turbulence by a flat boundary is extended to account fully for viscous processes. Two types of boundary are considered: a solid wall and a free surface. The model is shown to be formally valid provided two conditions are satisfied. The first condition is that time is short compared with the decorrelation time of the energy-containing eddies, so that nonlinear processes can be neglected. The second condition is that the viscous layer near the boundary, where tangential motions adjust to the boundary condition, is thin compared with the scales of the smallest eddies. The viscous layer can then be treated using thin-boundary-layer methods. Given these conditions, the distorted turbulence near the boundary is related to the undistorted turbulence, and thence profiles of turbulence dissipation rate near the two types of boundary are calculated and shown to agree extremely well with profiles obtained by Perot & Moin (1993) by direct numerical simulation. The dissipation rates are higher near a solid wall than in the bulk of the flow because the no-slip boundary condition leads to large velocity gradients across the viscous layer. In contrast, the weaker constraint of no stress at a free surface leads to the dissipation rate close to a free surface actually being smaller than in the bulk of the flow. This explains why tangential velocity fluctuations parallel to a free surface are so large. In addition we show that it is the adjustment of the large energy-containing eddies across the viscous layer that controls the dissipation rate, which explains why rapid-distortion theory can give quantitatively accurate values for the dissipation rate. We also find that the dissipation rate obtained from the model evaluated at the time when the model is expected to fail actually yields useful estimates of the dissipation obtained from the direct numerical simulation at times when the nonlinear processes are significant. We conclude that the main role of nonlinear processes is to arrest growth by linear processes of the viscous layer after about one large-eddy turnover time.

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44

Fedotov, A. F., and Yu S. Chumakov. "Multi-Equation Turbulence Model for a Free Convection Boundary Layer." Heat Transfer Research 33, no. 1-2 (2002): 6. http://dx.doi.org/10.1615/heattransres.v33.i1-2.40.

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Willis, Lisa, Karen M. Page, David S. Broomhead, and Eileen J. Cox. "Discrete free-boundary reaction-diffusion model of diatom pore occlusions." Plant Ecology and Evolution 143, no. 3 (November 30, 2010): 297–306. http://dx.doi.org/10.5091/plecevo.2010.415.

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46

Xu, Yongzhi. "A free boundary problem model of ductal carcinoma in situ." Discrete and Continuous Dynamical Systems - Series B 4, no. 1 (November 2003): 337–48. http://dx.doi.org/10.3934/dcdsb.2004.4.337.

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47

Cao, Jia-Feng, Wan-Tong Li, and Fei-Ying Yang. "Dynamics of a nonlocal SIS epidemic model with free boundary." Discrete & Continuous Dynamical Systems - B 22, no. 2 (2017): 247–66. http://dx.doi.org/10.3934/dcdsb.2017013.

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48

Liu, Yunfeng, Zhiming Guo, Mohammad El Smaily, and Lin Wang. "A Leslie-Gower predator-prey model with a free boundary." Discrete & Continuous Dynamical Systems - S 12, no. 7 (2019): 2063–84. http://dx.doi.org/10.3934/dcdss.2019133.

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49

Li, Chenglin. "A diffusive Holling–Tanner prey–predator model with free boundary." International Journal of Biomathematics 11, no. 05 (July 2018): 1850066. http://dx.doi.org/10.1142/s1793524518500663.

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This paper is concerned with a diffusive Holling–Tanner prey–predator model in a bounded domain with Dirichlet boundary condition and a free boundary. The global existence of the unique solution is proved. Moreover, the criteria governing spreading–vanishing are derived by mainly using the comparison principle. The results show that if the length of the occupying line is bigger than a threshold value (spreading barrier), then the spreading of predators will make an achievement, and, if the length of the occupying line is smaller than this spreading barrier and the spreading coefficient is relatively small depending on initial size of predators, then the predators will fail in establishing themselves and eventually die out.

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50

González, Adriana María, Juan Carlos Reginato, and Domingo Alberto Tarzia. "A FREE-BOUNDARY MODEL FOR ANAEROBIOSIS IN SATURATED SOIL AGGREGATES." Soil Science 173, no. 11 (November 2008): 758–67. http://dx.doi.org/10.1097/ss.0b013e31818d4178.

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