American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Homogenization of nonlinear dissipative hyperbolic problems exhibiting arbitrarily many spatial and temporal scales
  • NHM Home
  • This Issue
  • Next Article
    Stability of non-autonomous difference equations with applications to transport and wave propagation on networks
December  2016, 11(4): 603-625. doi: 10.3934/nhm.2016011

A steady-state mathematical model for an EOS capacitor: The effect of the size exclusion

Federica Di Michele 1, , Bruno Rubino 1, and  Rosella Sampalmieri 1,

1. 

Department of Information Engineering, Computer Sciences and Mathematics, University of L'Aquila - via Vetoio, loc. Coppito, I-67100 L'Aquila, Italy, Italy, Italy

Received  December 2014 Revised  May 2016 Published  October 2016

In this paper we present a suitable mathematical model to describe the behaviour of a hybrid electrolyte-oxide-semiconductor (EOS) device, that could be considered for application to neuro-prothesis and bio-devices. In particular, we discuss the existence and uniqueness of solutions also including the effects of the size exclusion in narrow structures such as ionic channels or nanopores. The result is proved using a fixed point argument on the whole domain.
    Our results provide information about the charge distribution and the potential behaviour on the device domain, and can represent a suitable framework for the development of stable numerical tools for innovative nanodevice modelling.
Keywords: EOS capacitor, quantum models, size exclusion, steady sate solution..
Mathematics Subject Classification: Primary: 34B15, 58D30; Secondary: 82D3.
Citation: Federica Di Michele, Bruno Rubino, Rosella Sampalmieri. A steady-state mathematical model for an EOS capacitor: The effect of the size exclusion. Networks & Heterogeneous Media, 2016, 11 (4) : 603-625. doi: 10.3934/nhm.2016011
References:
[1]

U. Ascher, J. Christiansen and R. D. Russell, Collocation software for boundary-value ODEs,, ACM Transactions on Mathematical Software (TOMS), 7 (1981), 209.  doi: 10.1145/355945.355950.  Google Scholar

[2]

J. N. Y. Aziz, R. Genov, B. L. Bardakjian, M. Derchansky and P. L. Carlen, Brain-silicon interface for high-resolution in vitro neural recording,, IEEE Transactions on Biomedical Circuits and Systems, 1 (2007), 56.   Google Scholar

[3]

G. Bader and U. Ascher, A new basis implementation for a mixed order boundary value ODE solver,, SIAM J. Sci. Stat. Comput., 8 (1987), 483.  doi: 10.1137/0908047.  Google Scholar

[4]

R. Baronas, F. Ivanauskas and J. Kulys, Mathematical Modeling of Biosensors: An Introduction for Chemists and Mathematicians,, Springer Science & Business Media, (2010).  doi: 10.1007/978-90-481-3243-0_5.  Google Scholar

[5]

S. Baumgartner and C. Heitzinger, Existence and local uniqueness for 3d self-consistent multiscale models of field-effect sensors,, Commun. Math. Sci, 10 (2012), 693.  doi: 10.4310/CMS.2012.v10.n2.a13.  Google Scholar

[6]

M. Bayer, C. Uhl and P. Vogl, Theoretical study of electrolyte gate AlGaN/GaN field effect transistors,, Journal of Applied Physics, 97 (2005).  doi: 10.1063/1.1847730.  Google Scholar

[7]

S. Birner, Modeling of semiconductor nanostructures and semiconductor-electrolyte interfaces,, Ph.D thesis, (2011).   Google Scholar

[8]

S. Birner, S. Hackenbuchner, M. Sabathil, G. Zandler, J.A. Majewski, T. Andlauer, T. Zibold, R. Morschl, A. Trellakis and P. Vogl, Modeling of Semiconductor Nanostructures with nextnano3,, Acta Physica Polonica A, 110 (2006), 111.  doi: 10.12693/APhysPolA.110.111.  Google Scholar

[9]

S. Birner, C. Uhl, M. Bayer and P. Vogl, Theoretical model for the detection of charged proteins with a silicon-on-insulator sensor,, Journal of Physics: Conference Series, 107 (2008).  doi: 10.1088/1742-6596/107/1/012002.  Google Scholar

[10]

M. Burger, R. S. Eisenberg and H. W. Engl, Inverse problems related to ion channel selectivity,, SIAM Journal on Applied Mathematics, 67 (2007), 960.  doi: 10.1137/060664689.  Google Scholar

[11]

M. Burger, B. Schlake and M.-T. Wolfram, Nonlinear Poisson-Nernst-Planck equations for ion flux through confined geometries,, Nonlinearity, 25 (2012), 961.  doi: 10.1088/0951-7715/25/4/961.  Google Scholar

[12]

E. Cianci, S. Lattanzio, G. Seguini, S. Vassanelli and M. Fanciulli, Atomic layer deposited $TiO_2$ for implantable brain-chip interfacing devices,, Thin Solid Films, 520 (2012), 4745.   Google Scholar

[13]

C. De Falco, E. Gatti, A. L. Lacaita and R. Sacco, Quantum-corrected drift-diffusion models for transport in semiconductor devices,, Journal of Computational Physics, 204 (2005), 533.  doi: 10.1016/j.jcp.2004.10.029.  Google Scholar

[14]

W. Dreyer, C. Guhlke and R. Müller, Overcoming the shortcomings of the Nernst-Planck model,, Physical Chemistry Chemical Physics, 15 (2013), 7075.  doi: 10.1039/c3cp44390f.  Google Scholar

[15]

P. Fromherz, Semiconductor chips with ion channels, nerve cells and brain,, Physica E: Low-dimensional Systems and Nanostructures, 16 (2003), 24.  doi: 10.1016/S1386-9477(02)00578-7.  Google Scholar

[16]

P. Fromherz, Three levels of neuroelectronic interfacing,, Annals of the New York Academy of Sciences, 1093 (2006), 143.   Google Scholar

[17]

P. Fromherz, Joining microelectronics and microionics: Nerve cells and brain tissue on semiconductor chips,, Solid-State Electronics, 52 (2008), 1364.   Google Scholar

[18]

I. Gasser and A. Jüngel, The quantum hydrodynamic model for semiconductors in thermal equilibrium,, Zeitschrift für Angewandte Mathematik und Physik (ZAMP), 48 (1997), 45.  doi: 10.1007/PL00001469.  Google Scholar

[19]

D. Gillespie, W. Nonner and R. S. Eisenberg, Coupling Poisson-Nernst-Planck and density functional theory to calculate ion flux,, Journal of Physics: Condensed Matter, 14 (2002), 12129.  doi: 10.1088/0953-8984/14/46/317.  Google Scholar

[20]

W. M. Grill, S. E. Norman and R. V. Bellamkonda, Implanted neural interfaces: Biochallenges and engineered solutions,, Annual Review of Biomedical Engineering, 11 (2009), 1.  doi: 10.1146/annurev-bioeng-061008-124927.  Google Scholar

[21]

Y. He, I. Gamba, H.-C. Lee and K. Ren, On the modeling and simulation of reaction-transfer dynamics in semiconductor-electrolyte solar cells,, SIAM Journal on Applied Mathematics, 75 (2015), 2515.  doi: 10.1137/130935148.  Google Scholar

[22]

C. Heitzinger, R. Kennell, G. Klimeck, N. Mauser, M. McLennan and C. Ringhofer, Modeling and simulation of field-effect biosensors (BioFETs) and their deployment on the nanoHUB,, Journal of Physics: Conference Series, 107 (2008).  doi: 10.1088/1742-6596/107/1/012004.  Google Scholar

[23]

C. Heitzinger and G. Klimeck, Computational aspects of the three-dimensional feature-scale simulation of silicon-nanowire field-effect sensors for DNA detection,, Journal of Computational Electronics, 6 (2007), 387.  doi: 10.1007/s10825-006-0139-x.  Google Scholar

[24]

C. Heitzinger, N. J. Mauser and C. Ringhofer, Multiscale modeling of planar and nanowire field-effect biosensors,, SIAM Journal on Applied Mathematics, 70 (2010), 1634.  doi: 10.1137/080725027.  Google Scholar

[25]

A. Jüngel and I. V. Stelzer, Existence Analysis of Maxwell-Stefan Systems for Multicomponent Mixtures,, SIAM Journal on Mathematical Analysis, 45 (2013), 2421.  doi: 10.1137/120898164.  Google Scholar

[26]

P. A. Markowich, The Stationary Semiconductor Device Equations,, Springer Science & Business Media, (1986).  doi: 10.1007/978-3-7091-3678-2.  Google Scholar

[27]

P. A. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations,, Springer-Verlag: Berlin, (1990).  doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[28]

M. Mojarradi, D. Binkley, B. Blalock, R. Andersen, N. Ulshoefer, T. Johnson and L. Del Castillo, A miniaturized neuroprosthesis suitable for implantation into the brain,, IEEE Transactions on Neural Systems and Rehabilitation Engineering, 11 (2003), 38.  doi: 10.1109/TNSRE.2003.810431.  Google Scholar

[29]

X. Navarro, T.B Krueger, N. Lago, S. Micera, T. Stieglitz and P. Dario, A critical review of interfaces with the peripheral nervous system for the control of neuroprostheses and hybrid bionic systems,, Journal of the Peripheral Nervous System, 10 (2005), 229.  doi: 10.1111/j.1085-9489.2005.10303.x.  Google Scholar

[30]

Y. Ohno, K. Maehashi, Y. Yamashiro and K. Matsumoto, Electrolyte-gated graphene field-effect transistors for detecting pH and protein adsorption,, Nano Letters, 9 (2009), 3318.  doi: 10.1021/nl901596m.  Google Scholar

[31]

W. R. Patterson, Y. Song, C. W. Bull, I. Ozden, A. P. Deangellis, C. Lay, J. L. McKay, A. V. Nurmikko, J. D. Donoghue and B. W. Connors, A microelectrode/microelectronic hybrid device for brain implantable neuroprosthesis applications,, IEEE Transactions on Biomedical Engineering, 51 (2004), 1845.  doi: 10.1109/TBME.2004.831521.  Google Scholar

[32]

I. Peitz and P. Fromherz, Electrical interfacing of neurotransmitter receptor and field effect transistor,, The European Physical Journal E: Soft Matter and Biological Physics, 30 (2009), 223.  doi: 10.1140/epje/i2009-10461-3.  Google Scholar

[33]

R. Popovtzer, A. Natan and Y. Shacham-Diamand, Mathematical model of whole cell based bio-chip: An electrochemical biosensor for water toxicity detection,, Journal of Electroanalytical Chemistry, 602 (2007), 17.  doi: 10.1016/j.jelechem.2006.11.022.  Google Scholar

[34]

M.J. Schöning and A. Poghossian, Bio FEDs (Field-Effect Devices): State-of-the-Art and New Directions,, Electroanalysis, 18 (2006), 1893.   Google Scholar

[35]

W. M. Siu and R. S. C. Cobbold, Basic properties of the electrolyte-SiO2-Si system: Physical and theoretical aspects,, IEEE Transactions on Electron Devices, 26 (1979), 1805.   Google Scholar

[36]

A. Stett, B. Muller and P. Fromherz, Two-way silicon-neuron interface by electrical induction,, Physical Review E, 55 (1997), 1779.  doi: 10.1103/PhysRevE.55.1779.  Google Scholar

[37]

T. Tokuda, Y. L. Pan, A. Uehara, K. Kagawa, M. Nunoshita and J. Ohta, Flexible and extendible neural interface device based on cooperative multi-chip CMOS LSI architecture,, Sensors and Actuators A: Physical, 122 (2005), 88.  doi: 10.1016/j.sna.2005.03.065.  Google Scholar

[38]

R. E. G. van Hal, J. C. T. Eijkel and P. Bergveld, A general model to describe the electrostatic potential at electrolyte oxide interfaces,, Advances in Colloid and Interface Science, 69 (1996), 31.   Google Scholar

[39]

M. W. Shinwari, M. J. Deen and D. Landheer, Study of the electrolyte-insulator-semiconductor field-effect transistor (EISFET) with applications in biosensor design,, Microelectronics Reliability, 47 (2007), 2025.   Google Scholar

show all references

References:
[1]

U. Ascher, J. Christiansen and R. D. Russell, Collocation software for boundary-value ODEs,, ACM Transactions on Mathematical Software (TOMS), 7 (1981), 209.  doi: 10.1145/355945.355950.  Google Scholar

[2]

J. N. Y. Aziz, R. Genov, B. L. Bardakjian, M. Derchansky and P. L. Carlen, Brain-silicon interface for high-resolution in vitro neural recording,, IEEE Transactions on Biomedical Circuits and Systems, 1 (2007), 56.   Google Scholar

[3]

G. Bader and U. Ascher, A new basis implementation for a mixed order boundary value ODE solver,, SIAM J. Sci. Stat. Comput., 8 (1987), 483.  doi: 10.1137/0908047.  Google Scholar

[4]

R. Baronas, F. Ivanauskas and J. Kulys, Mathematical Modeling of Biosensors: An Introduction for Chemists and Mathematicians,, Springer Science & Business Media, (2010).  doi: 10.1007/978-90-481-3243-0_5.  Google Scholar

[5]

S. Baumgartner and C. Heitzinger, Existence and local uniqueness for 3d self-consistent multiscale models of field-effect sensors,, Commun. Math. Sci, 10 (2012), 693.  doi: 10.4310/CMS.2012.v10.n2.a13.  Google Scholar

[6]

M. Bayer, C. Uhl and P. Vogl, Theoretical study of electrolyte gate AlGaN/GaN field effect transistors,, Journal of Applied Physics, 97 (2005).  doi: 10.1063/1.1847730.  Google Scholar

[7]

S. Birner, Modeling of semiconductor nanostructures and semiconductor-electrolyte interfaces,, Ph.D thesis, (2011).   Google Scholar

[8]

S. Birner, S. Hackenbuchner, M. Sabathil, G. Zandler, J.A. Majewski, T. Andlauer, T. Zibold, R. Morschl, A. Trellakis and P. Vogl, Modeling of Semiconductor Nanostructures with nextnano3,, Acta Physica Polonica A, 110 (2006), 111.  doi: 10.12693/APhysPolA.110.111.  Google Scholar

[9]

S. Birner, C. Uhl, M. Bayer and P. Vogl, Theoretical model for the detection of charged proteins with a silicon-on-insulator sensor,, Journal of Physics: Conference Series, 107 (2008).  doi: 10.1088/1742-6596/107/1/012002.  Google Scholar

[10]

M. Burger, R. S. Eisenberg and H. W. Engl, Inverse problems related to ion channel selectivity,, SIAM Journal on Applied Mathematics, 67 (2007), 960.  doi: 10.1137/060664689.  Google Scholar

[11]

M. Burger, B. Schlake and M.-T. Wolfram, Nonlinear Poisson-Nernst-Planck equations for ion flux through confined geometries,, Nonlinearity, 25 (2012), 961.  doi: 10.1088/0951-7715/25/4/961.  Google Scholar

[12]

E. Cianci, S. Lattanzio, G. Seguini, S. Vassanelli and M. Fanciulli, Atomic layer deposited $TiO_2$ for implantable brain-chip interfacing devices,, Thin Solid Films, 520 (2012), 4745.   Google Scholar

[13]

C. De Falco, E. Gatti, A. L. Lacaita and R. Sacco, Quantum-corrected drift-diffusion models for transport in semiconductor devices,, Journal of Computational Physics, 204 (2005), 533.  doi: 10.1016/j.jcp.2004.10.029.  Google Scholar

[14]

W. Dreyer, C. Guhlke and R. Müller, Overcoming the shortcomings of the Nernst-Planck model,, Physical Chemistry Chemical Physics, 15 (2013), 7075.  doi: 10.1039/c3cp44390f.  Google Scholar

[15]

P. Fromherz, Semiconductor chips with ion channels, nerve cells and brain,, Physica E: Low-dimensional Systems and Nanostructures, 16 (2003), 24.  doi: 10.1016/S1386-9477(02)00578-7.  Google Scholar

[16]

P. Fromherz, Three levels of neuroelectronic interfacing,, Annals of the New York Academy of Sciences, 1093 (2006), 143.   Google Scholar

[17]

P. Fromherz, Joining microelectronics and microionics: Nerve cells and brain tissue on semiconductor chips,, Solid-State Electronics, 52 (2008), 1364.   Google Scholar

[18]

I. Gasser and A. Jüngel, The quantum hydrodynamic model for semiconductors in thermal equilibrium,, Zeitschrift für Angewandte Mathematik und Physik (ZAMP), 48 (1997), 45.  doi: 10.1007/PL00001469.  Google Scholar

[19]

D. Gillespie, W. Nonner and R. S. Eisenberg, Coupling Poisson-Nernst-Planck and density functional theory to calculate ion flux,, Journal of Physics: Condensed Matter, 14 (2002), 12129.  doi: 10.1088/0953-8984/14/46/317.  Google Scholar

[20]

W. M. Grill, S. E. Norman and R. V. Bellamkonda, Implanted neural interfaces: Biochallenges and engineered solutions,, Annual Review of Biomedical Engineering, 11 (2009), 1.  doi: 10.1146/annurev-bioeng-061008-124927.  Google Scholar

[21]

Y. He, I. Gamba, H.-C. Lee and K. Ren, On the modeling and simulation of reaction-transfer dynamics in semiconductor-electrolyte solar cells,, SIAM Journal on Applied Mathematics, 75 (2015), 2515.  doi: 10.1137/130935148.  Google Scholar

[22]

C. Heitzinger, R. Kennell, G. Klimeck, N. Mauser, M. McLennan and C. Ringhofer, Modeling and simulation of field-effect biosensors (BioFETs) and their deployment on the nanoHUB,, Journal of Physics: Conference Series, 107 (2008).  doi: 10.1088/1742-6596/107/1/012004.  Google Scholar

[23]

C. Heitzinger and G. Klimeck, Computational aspects of the three-dimensional feature-scale simulation of silicon-nanowire field-effect sensors for DNA detection,, Journal of Computational Electronics, 6 (2007), 387.  doi: 10.1007/s10825-006-0139-x.  Google Scholar

[24]

C. Heitzinger, N. J. Mauser and C. Ringhofer, Multiscale modeling of planar and nanowire field-effect biosensors,, SIAM Journal on Applied Mathematics, 70 (2010), 1634.  doi: 10.1137/080725027.  Google Scholar

[25]

A. Jüngel and I. V. Stelzer, Existence Analysis of Maxwell-Stefan Systems for Multicomponent Mixtures,, SIAM Journal on Mathematical Analysis, 45 (2013), 2421.  doi: 10.1137/120898164.  Google Scholar

[26]

P. A. Markowich, The Stationary Semiconductor Device Equations,, Springer Science & Business Media, (1986).  doi: 10.1007/978-3-7091-3678-2.  Google Scholar

[27]

P. A. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations,, Springer-Verlag: Berlin, (1990).  doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[28]

M. Mojarradi, D. Binkley, B. Blalock, R. Andersen, N. Ulshoefer, T. Johnson and L. Del Castillo, A miniaturized neuroprosthesis suitable for implantation into the brain,, IEEE Transactions on Neural Systems and Rehabilitation Engineering, 11 (2003), 38.  doi: 10.1109/TNSRE.2003.810431.  Google Scholar

[29]

X. Navarro, T.B Krueger, N. Lago, S. Micera, T. Stieglitz and P. Dario, A critical review of interfaces with the peripheral nervous system for the control of neuroprostheses and hybrid bionic systems,, Journal of the Peripheral Nervous System, 10 (2005), 229.  doi: 10.1111/j.1085-9489.2005.10303.x.  Google Scholar

[30]

Y. Ohno, K. Maehashi, Y. Yamashiro and K. Matsumoto, Electrolyte-gated graphene field-effect transistors for detecting pH and protein adsorption,, Nano Letters, 9 (2009), 3318.  doi: 10.1021/nl901596m.  Google Scholar

[31]

W. R. Patterson, Y. Song, C. W. Bull, I. Ozden, A. P. Deangellis, C. Lay, J. L. McKay, A. V. Nurmikko, J. D. Donoghue and B. W. Connors, A microelectrode/microelectronic hybrid device for brain implantable neuroprosthesis applications,, IEEE Transactions on Biomedical Engineering, 51 (2004), 1845.  doi: 10.1109/TBME.2004.831521.  Google Scholar

[32]

I. Peitz and P. Fromherz, Electrical interfacing of neurotransmitter receptor and field effect transistor,, The European Physical Journal E: Soft Matter and Biological Physics, 30 (2009), 223.  doi: 10.1140/epje/i2009-10461-3.  Google Scholar

[33]

R. Popovtzer, A. Natan and Y. Shacham-Diamand, Mathematical model of whole cell based bio-chip: An electrochemical biosensor for water toxicity detection,, Journal of Electroanalytical Chemistry, 602 (2007), 17.  doi: 10.1016/j.jelechem.2006.11.022.  Google Scholar

[34]

M.J. Schöning and A. Poghossian, Bio FEDs (Field-Effect Devices): State-of-the-Art and New Directions,, Electroanalysis, 18 (2006), 1893.   Google Scholar

[35]

W. M. Siu and R. S. C. Cobbold, Basic properties of the electrolyte-SiO2-Si system: Physical and theoretical aspects,, IEEE Transactions on Electron Devices, 26 (1979), 1805.   Google Scholar

[36]

A. Stett, B. Muller and P. Fromherz, Two-way silicon-neuron interface by electrical induction,, Physical Review E, 55 (1997), 1779.  doi: 10.1103/PhysRevE.55.1779.  Google Scholar

[37]

T. Tokuda, Y. L. Pan, A. Uehara, K. Kagawa, M. Nunoshita and J. Ohta, Flexible and extendible neural interface device based on cooperative multi-chip CMOS LSI architecture,, Sensors and Actuators A: Physical, 122 (2005), 88.  doi: 10.1016/j.sna.2005.03.065.  Google Scholar

[38]

R. E. G. van Hal, J. C. T. Eijkel and P. Bergveld, A general model to describe the electrostatic potential at electrolyte oxide interfaces,, Advances in Colloid and Interface Science, 69 (1996), 31.   Google Scholar

[39]

M. W. Shinwari, M. J. Deen and D. Landheer, Study of the electrolyte-insulator-semiconductor field-effect transistor (EISFET) with applications in biosensor design,, Microelectronics Reliability, 47 (2007), 2025.   Google Scholar

[1]

H. L. Smith, X. Q. Zhao. Competitive exclusion in a discrete-time, size-structured chemostat model. Discrete & Continuous Dynamical Systems - B, 2001, 1 (2) : 183-191. doi: 10.3934/dcdsb.2001.1.183
[2]

Julien Arino, Fred Brauer, P. van den Driessche, James Watmough, Jianhong Wu. A final size relation for epidemic models. Mathematical Biosciences & Engineering, 2007, 4 (2) : 159-175. doi: 10.3934/mbe.2007.4.159
[3]

Philipp Fuchs, Ansgar Jüngel, Max von Renesse. On the Lagrangian structure of quantum fluid models. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1375-1396. doi: 10.3934/dcds.2014.34.1375
[4]

Boling Guo, Guangwu Wang. Existence of the solution for the viscous bipolar quantum hydrodynamic model. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3183-3210. doi: 10.3934/dcds.2017136
[5]

Azmy S. Ackleh, Linda J. S. Allen. Competitive exclusion in SIS and SIR epidemic models with total cross immunity and density-dependent host mortality. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 175-188. doi: 10.3934/dcdsb.2005.5.175
[6]

Youcef Amirat, Kamel Hamdache. Steady state solutions of ferrofluid flow models. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2329-2355. doi: 10.3934/cpaa.2016039
[7]

Jianquan Li, Zhien Ma. Stability analysis for SIS epidemic models with vaccination and constant population size. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 635-642. doi: 10.3934/dcdsb.2004.4.635
[8]

L. M. Abia, O. Angulo, J.C. López-Marcos. Size-structured population dynamics models and their numerical solutions. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1203-1222. doi: 10.3934/dcdsb.2004.4.1203
[9]

Weinan E, Jianfeng Lu. Mathematical theory of solids: From quantum mechanics to continuum models. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5085-5097. doi: 10.3934/dcds.2014.34.5085
[10]

Dagny Butler, Eunkyung Ko, R. Shivaji. Alternate steady states for classes of reaction diffusion models on exterior domains. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1181-1191. doi: 10.3934/dcdss.2014.7.1181
[11]

Inom Mirzaev, David M. Bortz. A numerical framework for computing steady states of structured population models and their stability. Mathematical Biosciences & Engineering, 2017, 14 (4) : 933-952. doi: 10.3934/mbe.2017049
[12]

M. Chipot, A. Rougirel. On the asymptotic behaviour of the solution of parabolic problems in cylindrical domains of large size in some directions. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 319-338. doi: 10.3934/dcdsb.2001.1.319
[13]

Ansgar Jüngel, Josipa-Pina Milišić. Full compressible Navier-Stokes equations for quantum fluids: Derivation and numerical solution. Kinetic & Related Models, 2011, 4 (3) : 785-807. doi: 10.3934/krm.2011.4.785
[14]

Guangwu Wang, Boling Guo. Global weak solution to the quantum Navier-Stokes-Landau-Lifshitz equations with density-dependent viscosity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6141-6166. doi: 10.3934/dcdsb.2019133
[15]

E. Almaraz, A. Gómez-Corral. On SIR-models with Markov-modulated events: Length of an outbreak, total size of the epidemic and number of secondary infections. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2153-2176. doi: 10.3934/dcdsb.2018229
[16]

Thomas Lorenz. Nonlocal hyperbolic population models structured by size and spatial position: Well-posedness. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4547-4628. doi: 10.3934/dcdsb.2019156
[17]

Mustapha Mokhtar-Kharroubi, Quentin Richard. Spectral theory and time asymptotics of size-structured two-phase population models. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020048
[18]

Niclas Bernhoff. Boundary layers for discrete kinetic models: Multicomponent mixtures, polyatomic molecules, bimolecular reactions, and quantum kinetic equations. Kinetic & Related Models, 2017, 10 (4) : 925-955. doi: 10.3934/krm.2017037
[19]

José Luis López, Jesús Montejo-Gámez. On viscous quantum hydrodynamics associated with nonlinear Schrödinger-Doebner-Goldin models. Kinetic & Related Models, 2012, 5 (3) : 517-536. doi: 10.3934/krm.2012.5.517
[20]

Nicola Zamponi. Some fluid-dynamic models for quantum electron transport in graphene via entropy minimization. Kinetic & Related Models, 2012, 5 (1) : 203-221. doi: 10.3934/krm.2012.5.203

2018 Impact Factor: 0.871

Tools

Metrics

  • PDF downloads (45)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences