| Title:
|
The stability analysis of a discretized pantograph equation (English) |
| Author:
|
Jánský, Jiří |
| Author:
|
Kundrát, Petr |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
136 |
| Issue:
|
4 |
| Year:
|
2011 |
| Pages:
|
385-394 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The paper deals with a difference equation arising from the scalar pantograph equation via the backward Euler discretization. A case when the solution tends to zero but after reaching a certain index it loses this tendency is discussed. We analyse this problem and estimate the value of such an index. Furthermore, we show that the utilized proof technique enables us to investigate some other numerical formulae, too. (English) |
| Keyword:
|
pantograph equation |
| Keyword:
|
numerical solution |
| Keyword:
|
stability |
| MSC:
|
34K28 |
| MSC:
|
39A06 |
| MSC:
|
39A12 |
| MSC:
|
39A30 |
| MSC:
|
65L03 |
| MSC:
|
65L05 |
| MSC:
|
65L12 |
| MSC:
|
65L20 |
| idZBL:
|
Zbl 1245.65103 |
| idMR:
|
MR2985548 |
| DOI:
|
10.21136/MB.2011.141698 |
| . |
| Date available:
|
2011-11-10T15:51:10Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141698 |
| . |
| Reference:
|
[1] Bakke, V. L., Jackiewicz, Z.: Stability analysis of linear multistep methods for delay differential equations.Int. J. Math. Math. Sci. 9 (1986), 447-458. Zbl 0622.65057, MR 0859113, 10.1155/S0161171286000583 |
| Reference:
|
[2] Bellen, A., Guglielmi, N., Torelli, L.: Asymptotic stability properties of $\Theta$ methods for the pantograph equation.Appl. Numer. Math. 24 (1997), 279-293. Zbl 0878.65064, MR 1464729, 10.1016/S0168-9274(97)00026-3 |
| Reference:
|
[3] Bellen, A., Zennaro, M.: Numerical Methods for Delay Differential Equations.Oxford University Press (2003). Zbl 1038.65058, MR 1997488 |
| Reference:
|
[4] Čermák, J., Jánský, J.: On the asymptotics of the trapezoidal rule for the pantograph equation.Math. Comp. 78 (2009), 2107-2126. Zbl 1198.65112, MR 2521280, 10.1090/S0025-5718-09-02245-5 |
| Reference:
|
[5] Elaydi, S.: An Introduction to Difference Equations.Springer (2005). Zbl 1071.39001, MR 2128146 |
| Reference:
|
[6] Györi, I., Pituk, M.: Comparison theorems and asymptotic equilibrium for delay differential and difference equations.Dynam. Systems Appl. 5 (1996), 277-302. MR 1396192 |
| Reference:
|
[7] Iserles, A.: Exact and discretized stability of the pantograph equation.Appl. Numer. Math. 24 (1997), 295-308. Zbl 0880.65058, MR 1464730, 10.1016/S0168-9274(97)00027-5 |
| Reference:
|
[8] Kundrát, P.: Asymptotic properties of the discretized pantograph equation.Studia Univ. Babeş-Bolyai, Mathematica 50 (2005), 77-84. Zbl 1112.39004, MR 2175107 |
| Reference:
|
[9] Kuruklis, S. A.: The asymptotic stability of $x_{n+1}-ax_n+bx_{n-k}=0$.J. Math. Anal. Appl. 188 (1994), 719-731. MR 1305480 |
| Reference:
|
[10] Liu, Y.: Numerical investigation of the pantograph equation.Appl. Numer. Math. 24 (1997), 309-317. Zbl 0878.65065, MR 1464731, 10.1016/S0168-9274(97)00028-7 |
| . |
