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On ill-posedness for the generalized BBM equation
Delay-dependent stability criteria for neutral delay differential and difference equations
1. | Institute of Mathematics, Brno University of Technology, Technická 2, CZ-61669 Brno, Czech Republic, Czech Republic |
References:
[1] |
A. Bellen, N. Guglielmi and L. Torelli, Asymptotic stability properties of $\Theta$-methods for the pantograph equation, Appl. Numer. Math., 24 (1997), 279-293.
doi: 10.1016/S0168-9274(97)00026-3. |
[2] |
A. Bellen, Z. Jackiewicz and M. Zennaro, Stability analysis of one-step methods for neutral delay-differential equations, Numer. Math., 52 (1988), 605-619.
doi: 10.1007/BF01395814. |
[3] |
A. Bellen and M. Zennaro, Numerical Methods For Delay Differential Equations, Oxford University Press, Oxford, 2003.
doi: 10.1093/acprof:oso/9780198506546.001.0001. |
[4] |
W. E. Brumley, On the asymptotic behavior of solutions of differential-difference equations of neutral type, J. Differential Equations, 7 (1970), 175-188.
doi: 10.1016/0022-0396(70)90131-2. |
[5] |
M. Calvo and T. Grande, On the asymptotic stability of $\Theta$-methods for delay differential equations, Numer. Math., 54 (1988), 257-269.
doi: 10.1007/BF01396761. |
[6] |
J. Čermák, The stability and asymptotic properties of the $\Theta$-methods for the pantograph equation, IMA J. Numer. Anal., 31 (2011), 1533-1551.
doi: 10.1093/imanum/drq021. |
[7] |
J. Čermák and J. Hrabalová, On stability regions for some delay differential equations and their discretizations,, Period. Math. Hung., ().
|
[8] |
J. Čermák, J. Jánský and P. Kundrát, On necessary and sufficient conditions for the asymptotic stability of higher order linear difference equations, J. Difference Equ. Appl., 18 (2012), 1781-1800.
doi: 10.1080/10236198.2011.595406. |
[9] |
S. Elaydi, An Introduction to Difference Equations, Springer, New York, 2005. |
[10] |
H. I. Freedman and Y. Kuang, Stability switches in linear scalar neutral delay equations, Funkcial. Ekvac., 34 (1991), 187-209. |
[11] |
P. S. Gromova, Stability of solutions of nonlinear equations of the neutral type in the asymptotically critical case, Math. Notes, 1 (1967), 715-726. |
[12] |
N. Guglielmi, Delay dependent stability regions of $\Theta$-methods for delay differential equations, IMA J. Numer. Anal., 18 (1998), 399-418.
doi: 10.1093/imanum/18.3.399. |
[13] |
N. Guglielmi, Asymptotic stability barriers for natural Runge-Kutta processes for delay equations, SIAM J. Numer. Anal., 39 (2001), 763-783.
doi: 10.1137/S0036142900375396. |
[14] |
N. Guglielmi, On the qualitative behaviour of numerical methods for delay differential equations of neutral type. A case study: $\Theta$-methods, Recent Trends in Numerical Analysis (L. Brugnano and D. Trigiante, eds.), 3 (2001), 175-184. |
[15] |
N. D. Hayes, Roots of the transcendental equations associated with certain difference-differential equations, J. London Math. Soc., 25 (1950), 226-232. |
[16] |
C. Huang, Delay-dependent stability of high order Runge-Kutta methods, Numer. Math., 111 (2009), 377-387.
doi: 10.1007/s00211-008-0197-z. |
[17] |
A. Iserles, Exact and discretized stability of the pantograph equation, Appl. Numer. Math., 24 (1997), 295-308.
doi: 10.1016/S0168-9274(97)00027-5. |
[18] |
Z. Jackiewicz, Asymptotic stability analysis of $\Theta$-methods for functional differential equations, Numer. Math., 43 (1984), 389-396.
doi: 10.1007/BF01390181. |
[19] |
S. Junca and B. Lombard, Stability of a critical nonlinear neutral delay differential equation, J. Differential Equations, 256 (2014), 2368-2391.
doi: 10.1016/j.jde.2014.01.004. |
[20] |
V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 1999.
doi: 10.1007/978-94-017-1965-0. |
[21] |
J. Kuang and Y. Cong, Stability of Numerical Methods for Delay Differential Equations, Science Press, Beijing, 2005. |
[22] |
Y. Liu, On the $\Theta$-method for delay differential equations with infinite lag, J. Comput. Appl. Math., 71 (1996), 177-190.
doi: 10.1016/0377-0427(95)00222-7. |
[23] |
H. Matsunaga, Stability switches in a system of linear differential equations with diagonal delay, Appl. Math. Comput., 212 (2009), 145-152.
doi: 10.1016/j.amc.2009.02.010. |
[24] |
H. Matsunaga and H. Hashimoto, Asymptotic stability and stability switches in a linear integro-differential system, Differ. Equ. Appl., 3 (2011), 43-55.
doi: 10.7153/dea-03-04. |
[25] |
W. Snow, Existence, Uniqueness and Stability for Nonlinear Differential-Difference Equations in the Neutral Case, Thesis (Ph.D.)–New York University. 1964. 79 pp. |
[26] |
V. V. Vlasov and D. A. Medvedev, Functional-differential equations and related problems in spectral theory, J. Math. Sci. (N. Y.), 164 (2010), 659-841.
doi: 10.1007/s10958-010-9768-5. |
show all references
References:
[1] |
A. Bellen, N. Guglielmi and L. Torelli, Asymptotic stability properties of $\Theta$-methods for the pantograph equation, Appl. Numer. Math., 24 (1997), 279-293.
doi: 10.1016/S0168-9274(97)00026-3. |
[2] |
A. Bellen, Z. Jackiewicz and M. Zennaro, Stability analysis of one-step methods for neutral delay-differential equations, Numer. Math., 52 (1988), 605-619.
doi: 10.1007/BF01395814. |
[3] |
A. Bellen and M. Zennaro, Numerical Methods For Delay Differential Equations, Oxford University Press, Oxford, 2003.
doi: 10.1093/acprof:oso/9780198506546.001.0001. |
[4] |
W. E. Brumley, On the asymptotic behavior of solutions of differential-difference equations of neutral type, J. Differential Equations, 7 (1970), 175-188.
doi: 10.1016/0022-0396(70)90131-2. |
[5] |
M. Calvo and T. Grande, On the asymptotic stability of $\Theta$-methods for delay differential equations, Numer. Math., 54 (1988), 257-269.
doi: 10.1007/BF01396761. |
[6] |
J. Čermák, The stability and asymptotic properties of the $\Theta$-methods for the pantograph equation, IMA J. Numer. Anal., 31 (2011), 1533-1551.
doi: 10.1093/imanum/drq021. |
[7] |
J. Čermák and J. Hrabalová, On stability regions for some delay differential equations and their discretizations,, Period. Math. Hung., ().
|
[8] |
J. Čermák, J. Jánský and P. Kundrát, On necessary and sufficient conditions for the asymptotic stability of higher order linear difference equations, J. Difference Equ. Appl., 18 (2012), 1781-1800.
doi: 10.1080/10236198.2011.595406. |
[9] |
S. Elaydi, An Introduction to Difference Equations, Springer, New York, 2005. |
[10] |
H. I. Freedman and Y. Kuang, Stability switches in linear scalar neutral delay equations, Funkcial. Ekvac., 34 (1991), 187-209. |
[11] |
P. S. Gromova, Stability of solutions of nonlinear equations of the neutral type in the asymptotically critical case, Math. Notes, 1 (1967), 715-726. |
[12] |
N. Guglielmi, Delay dependent stability regions of $\Theta$-methods for delay differential equations, IMA J. Numer. Anal., 18 (1998), 399-418.
doi: 10.1093/imanum/18.3.399. |
[13] |
N. Guglielmi, Asymptotic stability barriers for natural Runge-Kutta processes for delay equations, SIAM J. Numer. Anal., 39 (2001), 763-783.
doi: 10.1137/S0036142900375396. |
[14] |
N. Guglielmi, On the qualitative behaviour of numerical methods for delay differential equations of neutral type. A case study: $\Theta$-methods, Recent Trends in Numerical Analysis (L. Brugnano and D. Trigiante, eds.), 3 (2001), 175-184. |
[15] |
N. D. Hayes, Roots of the transcendental equations associated with certain difference-differential equations, J. London Math. Soc., 25 (1950), 226-232. |
[16] |
C. Huang, Delay-dependent stability of high order Runge-Kutta methods, Numer. Math., 111 (2009), 377-387.
doi: 10.1007/s00211-008-0197-z. |
[17] |
A. Iserles, Exact and discretized stability of the pantograph equation, Appl. Numer. Math., 24 (1997), 295-308.
doi: 10.1016/S0168-9274(97)00027-5. |
[18] |
Z. Jackiewicz, Asymptotic stability analysis of $\Theta$-methods for functional differential equations, Numer. Math., 43 (1984), 389-396.
doi: 10.1007/BF01390181. |
[19] |
S. Junca and B. Lombard, Stability of a critical nonlinear neutral delay differential equation, J. Differential Equations, 256 (2014), 2368-2391.
doi: 10.1016/j.jde.2014.01.004. |
[20] |
V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 1999.
doi: 10.1007/978-94-017-1965-0. |
[21] |
J. Kuang and Y. Cong, Stability of Numerical Methods for Delay Differential Equations, Science Press, Beijing, 2005. |
[22] |
Y. Liu, On the $\Theta$-method for delay differential equations with infinite lag, J. Comput. Appl. Math., 71 (1996), 177-190.
doi: 10.1016/0377-0427(95)00222-7. |
[23] |
H. Matsunaga, Stability switches in a system of linear differential equations with diagonal delay, Appl. Math. Comput., 212 (2009), 145-152.
doi: 10.1016/j.amc.2009.02.010. |
[24] |
H. Matsunaga and H. Hashimoto, Asymptotic stability and stability switches in a linear integro-differential system, Differ. Equ. Appl., 3 (2011), 43-55.
doi: 10.7153/dea-03-04. |
[25] |
W. Snow, Existence, Uniqueness and Stability for Nonlinear Differential-Difference Equations in the Neutral Case, Thesis (Ph.D.)–New York University. 1964. 79 pp. |
[26] |
V. V. Vlasov and D. A. Medvedev, Functional-differential equations and related problems in spectral theory, J. Math. Sci. (N. Y.), 164 (2010), 659-841.
doi: 10.1007/s10958-010-9768-5. |
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