| Title:
|
Stability analysis of reducible quadrature methods for Volterra integro-differential equations (English) |
| Author:
|
Bakke, Vernon L. |
| Author:
|
Jackiewicz, Zdzisław |
| Language:
|
English |
| Journal:
|
Aplikace matematiky |
| ISSN:
|
0373-6725 |
| Volume:
|
32 |
| Issue:
|
1 |
| Year:
|
1987 |
| Pages:
|
37-48 |
| Summary lang:
|
English |
| Summary lang:
|
Russian |
| Summary lang:
|
Czech |
| . |
| Category:
|
math |
| . |
| Summary:
|
Stability analysis for numerical solutions of Voltera integro-differential equations based on linear multistep methods combined with reducible quadrature rules is presented. The results given are based on the test equation $y'(t)=\gamma y(t) + \int^t_0(\lambda + \mu t + vs) y(s) ds$ and absolute stability is deffined in terms of the real parameters $\gamma, \lambda, \mu$ and $v$. Sufficient conditions are illustrated for $(0;0)$ - methods and for combinations of Adams-Moulton and backward differentiation methods. (English) |
| Keyword:
|
backward-differentiation-formula method |
| Keyword:
|
Volterra integro-differential equations |
| Keyword:
|
theta method |
| Keyword:
|
test equation |
| Keyword:
|
stability |
| Keyword:
|
linear multistep methods |
| Keyword:
|
reducible quadrature formulas |
| Keyword:
|
linear difference equation |
| Keyword:
|
Adams-Moulton methods |
| Keyword:
|
stability of numerical solution |
| MSC:
|
45J05 |
| MSC:
|
45M10 |
| MSC:
|
65Q05 |
| MSC:
|
65R20 |
| idZBL:
|
Zbl 0624.65140 |
| idMR:
|
MR0879328 |
| DOI:
|
10.21136/AM.1987.104234 |
| . |
| Date available:
|
2008-05-20T18:31:33Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/104234 |
| . |
| Reference:
|
[1] С. T. H. Baker A. Makroglou E. Short: Regions of stability in the numerical treatment of Volterra integro-differential equations.SIAM J. Numer. Anal., Vol. 16, No. 6, December, 1979. MR 0551314 |
| Reference:
|
[2] V. L. Bakke Z. Jackiewicz: Stability of reducible quadrature methods for Volterra integral equations of the second kind.Numer. Math. 47 (1985), 159-173. MR 0799682, 10.1007/BF01389707 |
| Reference:
|
[3] V. L. Bakke Z. Jackiewicz: Boundedness of solutions of difference equations and application to numerical solutions of Volterra integral equations of the second kind.J. Math. Anal. Appl., 115 (1986), 592-605. MR 0836249, 10.1016/0022-247X(86)90018-1 |
| Reference:
|
[4] H. Brunner: A survey of recent advances in the numerical treatment of Volterra integral and integro-differential equations.J. Comput. App. Math., Vol. 8, No. 3, 1982. Zbl 0485.65087, MR 0682889 |
| Reference:
|
[5] H. Brunner J. D. Lambert: Stability of numerical methods for Volterra integro-differential equations.Computing 12, 75-89 (1974). MR 0418490, 10.1007/BF02239501 |
| Reference:
|
[6] C. J. Gladwin R. Jeltsch: Stability of quadrature rule methods for first kind Volterra integral equations.BIT 14, 144-151 (1974). MR 0502108, 10.1007/BF01932943 |
| Reference:
|
[7] P. Linz: Linear multistep methods for Volterra integro-differential equations.J. Assoc. Comput. Mach., 16 (1969), 295-301. Zbl 0183.45002, MR 0239786, 10.1145/321510.321521 |
| Reference:
|
[8] J. Matthys: A-stable linear multistep methods for Volterra integro-differential equations.Numer. Math. 27, 85-94 (1976). Zbl 0319.65072, MR 0436638, 10.1007/BF01399087 |
| Reference:
|
[9] D. Sanchez: A short note on asymptotic estimates of stability regions for a certain class of Volterra integro-differential equations.Manuscript, Department of Mathematics and Statistics, University of New Mexico, May, 1984. |
| Reference:
|
[10] L. M. Milne-Thompson: The calculus of finite differences.MacMillan& Co., London, 1933. |
| Reference:
|
[H] P. H. M. Wolkenfelt: The construction of reducible quadrature rules for Volterra integral and integro-differential equations.IMA Journal of Numerical Analysis, 2, 131-152 (1982). Zbl 0481.65084, MR 0668589, 10.1093/imanum/2.2.131 |
| Reference:
|
[12] P. H. M. Wolkenfelt: On the numerical stability of reducible quadrature methods for second kind Volterra integral equations.Z. Angew. Math. Mech., 61, 399-401 (1981). Zbl 0466.65073, MR 0638029, 10.1002/zamm.19810610808 |
| . |
