-
Previous Article
Stability of oscillatory gravity wave trains with energy dissipation and Benjamin-Feir instability
- DCDS-B Home
- This Issue
-
Next Article
A Neumann Boundary Value Problem in Two-Ion Electro-Diffusion with Unequal Valencies
From a PDE model to an ODE model of dynamics of synaptic depression
1. | Department of Mathematics, Faculty of Electrical Engineering and Computer Science, Lublin University of Technology, Nadbystrzycka 38A, 20-618 Lublin |
2. | Faculty of Mathematics, Physics and Computer Science, Maria Curie-Skŀodowska University in Lublin, Pl. Marii Curie-Skŀodowskiej 1, 20-031 Lublin, Poland |
References:
[1] |
F. Aristizabal and M. I. Glavinovič, Simulation and parameter estimation of dynamics of synaptic depression, Biol. Cybern., 90 (2004), 3-18.
doi: 10.1007/s00422-003-0432-8. |
[2] |
A. Bielecki and P. Kalita, Model of neurotransmitter fast transport in axon terminal of presynaptic neuron, J. Math. Biol., 56 (2008), 559-576.
doi: 10.1007/s00285-007-0131-5. |
[3] |
A. Bobrowski, Degenerate convergence of semigroups, Semigroup Forum, 49 (1994), 303-327.
doi: 10.1007/BF02573493. |
[4] |
A. Bobrowski, A note on convergence of semigroups, Ann. Polon. Math., 69 (1998), 107-127. |
[5] |
A. Bobrowski, "Functional Analysis for Probability and Stochastic Processes. An Introduction," Cambridge University Press, Cambridge, 2005.
doi: 10.1017/CBO9780511614583. |
[6] |
A. Bobrowski, Degenerate convergence of semigroups related to a model of stochastic gene expression, Semigroup Forum, 73 (2006), 345-366, with correction in 77 (2008), 520-521 (MR2457335).
doi: 10.1007/s00233-006-0633-2. |
[7] |
A. Bobrowski, On limitations and insufficiency of the Trotter-Kato theorem, Semigroup Forum, 75 (2007), 317-336.
doi: 10.1007/s00233-006-0676-4. |
[8] |
R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 3. Spectral Theory and Applications," With the collaboration of Michel Artola and Michel Cessenat, Translated from the French by John C. Amson, Springer-Verlag, Berlin, 1990.
doi: 10.1002/zamm.19920720316. |
[9] |
T. Eisner, "Stability of Operators and Operator Semigroups," Operator Theory: Advances and Applications, 209, Birkahäuser Verlag, Basel, 2010. |
[10] |
K.-J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000. |
[11] |
K.-J. Engel and R. Nagel, "A Short Course on Operator Semigroups," Universitext, Springer, New York, 2006. |
[12] |
S. N. Ethier and T. G. Kurtz, "Markov Processes. Characterization and Convergence," Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. |
[13] |
W. J. Ewens, "Mathematical Population Genetics," Biomathematics, 9, Springer-Verlag, Berlin-New York, 1979, Second edition, 2004. |
[14] |
W. Feller, Diffusion processes in genetics, in "Proc. of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950," University of California Press, Berkeley and Los Angeles, 1951. |
[15] |
W. Feller, Two singular diffusion problems, Ann. of Math. (2), 54 (1951), 173-182.
doi: 10.2307/1969318. |
[16] |
W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math. (2), 55 (1952), 468-519.
doi: 10.2307/1969644. |
[17] |
W. Feller, Diffusion processes in one dimension, Trans. Amer. Math. Soc., 77 (1954), 1-31.
doi: 10.1090/S0002-9947-1954-0063607-6. |
[18] |
G. Greiner, Perturbing the boundary conditions of a generator, Houston J. of Mathematics, 13 (1987), 213-229. |
[19] |
K. Itô and H. P. McKean, Jr., "Diffusion Processes and their Sample Paths," reprint of the 1974 edition, Classics in Mathematics, Springer, 1996.
doi: 10.1214/aoms/1177699390. |
[20] |
I. Karatzas and S. E. Shreve, "Brownian Motion and Stochastic Calculus," Second edition, Graduate Texts in Mathematics, 113, Springer-Verlag, New York, 1991.
doi: 10.1002/zamm.19890691124. |
[21] |
P. Mandl, "Analytical Treatment of One-Dimensional Markov Processes," Die Grundlehren der mathematischen Wissenschaften, Band 151, Academia Publishing House of the Czechoslovak Academy of Sciences, Prague, Springer-Verlag New York, Inc., New York, 1968. |
[22] |
E. Neher and R. S. Zucker, Multiple calcium-dependant process related to secretion in bovine chromaffin cells, Neuron, 10 (1993), 2-30.
doi: 10.1016/0896-6273(93)90238-M. |
[23] |
D. Revuz and M. Yor, "Continuous Martingales and Brownian Motion," 3$^rd$ edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293, Springer-Verlag, Berlin, 1999.
doi: 10.1214/aop/1176989417. |
[24] |
K. Taira, "Semigroups, Boundary Value Problems and Markov Processes," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004. |
[25] |
A. D. Ventcel', On boundary conditions for multi-dimensional diffusion processes, (in Russian), Teoriya Veroyat. i Primen., 4 (1959), 172-185, English translation: Theory Prob. and its Appl., 4 (1959), 164-177. |
show all references
References:
[1] |
F. Aristizabal and M. I. Glavinovič, Simulation and parameter estimation of dynamics of synaptic depression, Biol. Cybern., 90 (2004), 3-18.
doi: 10.1007/s00422-003-0432-8. |
[2] |
A. Bielecki and P. Kalita, Model of neurotransmitter fast transport in axon terminal of presynaptic neuron, J. Math. Biol., 56 (2008), 559-576.
doi: 10.1007/s00285-007-0131-5. |
[3] |
A. Bobrowski, Degenerate convergence of semigroups, Semigroup Forum, 49 (1994), 303-327.
doi: 10.1007/BF02573493. |
[4] |
A. Bobrowski, A note on convergence of semigroups, Ann. Polon. Math., 69 (1998), 107-127. |
[5] |
A. Bobrowski, "Functional Analysis for Probability and Stochastic Processes. An Introduction," Cambridge University Press, Cambridge, 2005.
doi: 10.1017/CBO9780511614583. |
[6] |
A. Bobrowski, Degenerate convergence of semigroups related to a model of stochastic gene expression, Semigroup Forum, 73 (2006), 345-366, with correction in 77 (2008), 520-521 (MR2457335).
doi: 10.1007/s00233-006-0633-2. |
[7] |
A. Bobrowski, On limitations and insufficiency of the Trotter-Kato theorem, Semigroup Forum, 75 (2007), 317-336.
doi: 10.1007/s00233-006-0676-4. |
[8] |
R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 3. Spectral Theory and Applications," With the collaboration of Michel Artola and Michel Cessenat, Translated from the French by John C. Amson, Springer-Verlag, Berlin, 1990.
doi: 10.1002/zamm.19920720316. |
[9] |
T. Eisner, "Stability of Operators and Operator Semigroups," Operator Theory: Advances and Applications, 209, Birkahäuser Verlag, Basel, 2010. |
[10] |
K.-J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000. |
[11] |
K.-J. Engel and R. Nagel, "A Short Course on Operator Semigroups," Universitext, Springer, New York, 2006. |
[12] |
S. N. Ethier and T. G. Kurtz, "Markov Processes. Characterization and Convergence," Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. |
[13] |
W. J. Ewens, "Mathematical Population Genetics," Biomathematics, 9, Springer-Verlag, Berlin-New York, 1979, Second edition, 2004. |
[14] |
W. Feller, Diffusion processes in genetics, in "Proc. of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950," University of California Press, Berkeley and Los Angeles, 1951. |
[15] |
W. Feller, Two singular diffusion problems, Ann. of Math. (2), 54 (1951), 173-182.
doi: 10.2307/1969318. |
[16] |
W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math. (2), 55 (1952), 468-519.
doi: 10.2307/1969644. |
[17] |
W. Feller, Diffusion processes in one dimension, Trans. Amer. Math. Soc., 77 (1954), 1-31.
doi: 10.1090/S0002-9947-1954-0063607-6. |
[18] |
G. Greiner, Perturbing the boundary conditions of a generator, Houston J. of Mathematics, 13 (1987), 213-229. |
[19] |
K. Itô and H. P. McKean, Jr., "Diffusion Processes and their Sample Paths," reprint of the 1974 edition, Classics in Mathematics, Springer, 1996.
doi: 10.1214/aoms/1177699390. |
[20] |
I. Karatzas and S. E. Shreve, "Brownian Motion and Stochastic Calculus," Second edition, Graduate Texts in Mathematics, 113, Springer-Verlag, New York, 1991.
doi: 10.1002/zamm.19890691124. |
[21] |
P. Mandl, "Analytical Treatment of One-Dimensional Markov Processes," Die Grundlehren der mathematischen Wissenschaften, Band 151, Academia Publishing House of the Czechoslovak Academy of Sciences, Prague, Springer-Verlag New York, Inc., New York, 1968. |
[22] |
E. Neher and R. S. Zucker, Multiple calcium-dependant process related to secretion in bovine chromaffin cells, Neuron, 10 (1993), 2-30.
doi: 10.1016/0896-6273(93)90238-M. |
[23] |
D. Revuz and M. Yor, "Continuous Martingales and Brownian Motion," 3$^rd$ edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293, Springer-Verlag, Berlin, 1999.
doi: 10.1214/aop/1176989417. |
[24] |
K. Taira, "Semigroups, Boundary Value Problems and Markov Processes," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004. |
[25] |
A. D. Ventcel', On boundary conditions for multi-dimensional diffusion processes, (in Russian), Teoriya Veroyat. i Primen., 4 (1959), 172-185, English translation: Theory Prob. and its Appl., 4 (1959), 164-177. |
[1] |
Claudio Marchi. On the convergence of singular perturbations of Hamilton-Jacobi equations. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1363-1377. doi: 10.3934/cpaa.2010.9.1363 |
[2] |
Hugo Beirão da Veiga. A challenging open problem: The inviscid limit under slip-type boundary conditions.. Discrete and Continuous Dynamical Systems - S, 2010, 3 (2) : 231-236. doi: 10.3934/dcdss.2010.3.231 |
[3] |
José M. Arrieta, Simone M. Bruschi. Very rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a non uniformly Lipschitz deformation. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 327-351. doi: 10.3934/dcdsb.2010.14.327 |
[4] |
Robert Denk, Yoshihiro Shibata. Generation of semigroups for the thermoelastic plate equation with free boundary conditions. Evolution Equations and Control Theory, 2019, 8 (2) : 301-313. doi: 10.3934/eect.2019016 |
[5] |
Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Existence and applications to the level-set approach. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1047-1069. doi: 10.3934/dcds.2008.21.1047 |
[6] |
Christian Lax, Sebastian Walcher. Singular perturbations and scaling. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 1-29. doi: 10.3934/dcdsb.2019170 |
[7] |
Chérif Amrouche, Yves Raudin. Singular boundary conditions and regularity for the biharmonic problem in the half-space. Communications on Pure and Applied Analysis, 2007, 6 (4) : 957-982. doi: 10.3934/cpaa.2007.6.957 |
[8] |
Alain Miranville, Sergey Zelik. The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 275-310. doi: 10.3934/dcds.2010.28.275 |
[9] |
Laurence Cherfils, Stefania Gatti, Alain Miranville. Long time behavior of the Caginalp system with singular potentials and dynamic boundary conditions. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2261-2290. doi: 10.3934/cpaa.2012.11.2261 |
[10] |
Franck Davhys Reval Langa, Morgan Pierre. A doubly splitting scheme for the Caginalp system with singular potentials and dynamic boundary conditions. Discrete and Continuous Dynamical Systems - S, 2021, 14 (2) : 653-676. doi: 10.3934/dcdss.2020353 |
[11] |
E. C.M. Crooks, E. N. Dancer, Danielle Hilhorst. Fast reaction limit and long time behavior for a competition-diffusion system with Dirichlet boundary conditions. Discrete and Continuous Dynamical Systems - B, 2007, 8 (1) : 39-44. doi: 10.3934/dcdsb.2007.8.39 |
[12] |
Andreas Henrici. Symmetries of the periodic Toda lattice, with an application to normal forms and perturbations of the lattice with Dirichlet boundary conditions. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 2949-2977. doi: 10.3934/dcds.2015.35.2949 |
[13] |
Jihoon Lee, Nguyen Thanh Nguyen. Gromov-Hausdorff stability of reaction diffusion equations with Robin boundary conditions under perturbations of the domain and equation. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1263-1296. doi: 10.3934/cpaa.2021020 |
[14] |
Zvi Artstein. Invariance principle in the singular perturbations limit. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3653-3666. doi: 10.3934/dcdsb.2018309 |
[15] |
Aibin Zang. Kato's type theorems for the convergence of Euler-Voigt equations to Euler equations with Drichlet boundary conditions. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 4945-4953. doi: 10.3934/dcds.2019202 |
[16] |
Ciprian G. Gal, Alain Miranville. Robust exponential attractors and convergence to equilibria for non-isothermal Cahn-Hilliard equations with dynamic boundary conditions. Discrete and Continuous Dynamical Systems - S, 2009, 2 (1) : 113-147. doi: 10.3934/dcdss.2009.2.113 |
[17] |
Jan Březina, Eduard Feireisl, Antonín Novotný. On convergence to equilibria of flows of compressible viscous fluids under in/out–flux boundary conditions. Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3615-3627. doi: 10.3934/dcds.2021009 |
[18] |
Ciprian G. Gal, Maurizio Grasselli. Singular limit of viscous Cahn-Hilliard equations with memory and dynamic boundary conditions. Discrete and Continuous Dynamical Systems - B, 2013, 18 (6) : 1581-1610. doi: 10.3934/dcdsb.2013.18.1581 |
[19] |
Maurizio Grasselli, Alain Miranville, Giulio Schimperna. The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 67-98. doi: 10.3934/dcds.2010.28.67 |
[20] |
Kin Ming Hui, Sunghoon Kim. Existence of Neumann and singular solutions of the fast diffusion equation. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 4859-4887. doi: 10.3934/dcds.2015.35.4859 |
2021 Impact Factor: 1.497
Tools
Metrics
Other articles
by authors
[Back to Top]