https://oldena.lpnu.ua/handle/ntb/40450| Title: | A fitted numerical method for singularly perturbed integro differential equations with delay |
| Authors: | Erdogan, Fevzi Sakar, Mehmet Giyas |
| Affiliation: | Yuzuncu Yil University, Faculty of Sciences, Department of Mathematics, Van, Turkey |
| Bibliographic description (Ukraine): | Erdogan F. A fitted numerical method for singularly perturbed integro differential equations with delay / Fevzi Erdogan, Mehmet Giyas Sakar // Litteris et Artibus : proceedings, 23–25 November, 2017. — Lviv : Lviv Polytechnic Publishing House, 2017. — P. 422–424. — (9th International academic conference «Computer science & engineering 2017» (CSE-2017)). |
| Bibliographic description (International): | Erdogan F. A fitted numerical method for singularly perturbed integro differential equations with delay / Fevzi Erdogan, Mehmet Giyas Sakar // Litteris et Artibus : proceedings, 23–25 November, 2017. — Lviv : Lviv Polytechnic Publishing House, 2017. — P. 422–424. — (9th International academic conference «Computer science & engineering 2017» (CSE-2017)). |
| Is part of: | Litteris et Artibus : матеріали, 2017 Litteris et Artibus : proceedings, 2017 |
| Conference/Event: | 7th International youth science forum «Litteris et Artibus» |
| Journal/Collection: | Litteris et Artibus : матеріали |
| Issue Date: | 23-Dec-2017 |
| Publisher: | Видавництво Львівської політехніки Lviv Polytechnic Publishing House |
| Place of the edition/event: | Львів Lviv |
| Temporal Coverage: | 23–25 листопада 2017 року 23–25 November, 2017 |
| Keywords: | Singularly perturbed problems integrodifferential equation difference schemes uniformly convergent |
| Number of pages: | 3 |
| Page range: | 422-424 |
| Start page: | 422 |
| End page: | 424 |
| Abstract: | This study deals with the singularly perturbed initial value problems for a quasilinear first-order integrodifferential equations with delay. A numerical method is generated on a grid that is constructed adaptively from a knowledge of the exact solution, which involves appropriate piecewise-uniform mesh on each time subinterval. An error analysis shows that the discrete solutions are uniformly convergent with respect to the perturbation parameter. The parameter uniform convergence is confirmed by numerical computations. |
| URI: | https://ena.lpnu.ua/handle/ntb/40450 |
| ISBN: | 978-966-941-108-2 |
| Copyright owner: | © Національний університет “Львівська політехніка”, 2017 |
| References (Ukraine): | [1] R. Bellman, K.L. Cooke, Differential-Difference Equations, Academy Press, New York, 1963. [2] R.D. Driver, Ordinary and Delay Differential Equations, Belin-Heidelberg, New York, Springer, 1977. [3] A. Bellen, M. Zennaro, Numerical methods for delay differential equations, Oxford University Press, Oxford, 2003. [4] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993. [5] S.N.Chow, J.Mallet-Paret, Singularly perturbed delay-differential equations, in: J.Chandra, A.C.Scott (Eds.), Coupled Nonlinear Oscillators, North-Holland, Amsterdam, 1983, pp.7-12. [6] A. Longtin, J. Milton, Complex oscillations in the human pupil light reflex with mixed and delayed feedback, Math. Biosci. 90 (1988)183-199. [7] M.C. Mackey, L. Glass, Oscillation and chaos in physiological control systems, Science, 197(1977)287-289. [8] G.M. Amiraliyev, F. Erdogan, Uniform numerical method for singularly perturbed delay differential equations, J.Comput. Math. Appl. 53(2007)1251-1259. [9] G.M. Amiraliyev, F. Erdogan, Difference schemes for a class of singularly perturbed initial value problems for delay differential equations, Numer. Algorithms, 52, 4(2009) 663-675. [10] I.G. Amiraliyeva, Fevzi Erdogan, G.M.Amiraliyev, A uniform numerical method for dealing with a singularly perturbed delay initial value problem, AppliedMathematics Letters, 23,10(2010)1221-1225. [11] S. Maset, Numerical solution of retarded functional differential equations as abstract Cauchy problems, J. Comput. Appl. Math. 161(2003)259-282. [12] B.J.MacCartin, Exponential fitting of the delayed recruitment/renewal equation. J. Comput. Appl. Math., 136(2001)343-356. [13] M.K. Kadalbajoo, K.K. Sharma, $\varepsilon- $uniform fitted mesh method for singularly perturbed differential difference equations with mixed type of shifts with layer behavior, Int. J. Comput. Math. 81 (2004)49-62. [14] C.G. Lange, R.M. Miura, Singular perturbation analysis of boundary-value problems for differential difference equations, SIAM J. Appl. Math. 42(1982)502-531. [15] H. Tian, The exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag, J. Math. Anal. Appl. 270(2002)143-149. [16] E. R. Doolan, J.J.H. Miller, and W. H. A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole, Press, Dublin, (1980). [17] P.A. Farrell, A.F. Hegarty, J.J.H. Miller, E.O'Riordan and G.I.Shishkin, Robust Computational Techniques for Boundary Layers, Chapman- Hall/CRC, New York, (2000). [18] H.G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Convection-Diffusion and Flow Problems, Springer Verlag, Berlin, (1996). [19] G.M. Amiraliyev and H. Duru, A note on a parameterized singular perturbation problem, J.Comput. Appl. Math. 182 (2005) 233-242. [20] G.M. Amiraliyev, The convergence of a finite difference method on layer-adapted mesh for a singularly perturbed system, Applied Mathematics and Computation, 162,3(2005)1023-1034 |
| References (International): | [1] R. Bellman, K.L. Cooke, Differential-Difference Equations, Academy Press, New York, 1963. [2] R.D. Driver, Ordinary and Delay Differential Equations, Belin-Heidelberg, New York, Springer, 1977. [3] A. Bellen, M. Zennaro, Numerical methods for delay differential equations, Oxford University Press, Oxford, 2003. [4] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993. [5] S.N.Chow, J.Mallet-Paret, Singularly perturbed delay-differential equations, in: J.Chandra, A.C.Scott (Eds.), Coupled Nonlinear Oscillators, North-Holland, Amsterdam, 1983, pp.7-12. [6] A. Longtin, J. Milton, Complex oscillations in the human pupil light reflex with mixed and delayed feedback, Math. Biosci. 90 (1988)183-199. [7] M.C. Mackey, L. Glass, Oscillation and chaos in physiological control systems, Science, 197(1977)287-289. [8] G.M. Amiraliyev, F. Erdogan, Uniform numerical method for singularly perturbed delay differential equations, J.Comput. Math. Appl. 53(2007)1251-1259. [9] G.M. Amiraliyev, F. Erdogan, Difference schemes for a class of singularly perturbed initial value problems for delay differential equations, Numer. Algorithms, 52, 4(2009) 663-675. [10] I.G. Amiraliyeva, Fevzi Erdogan, G.M.Amiraliyev, A uniform numerical method for dealing with a singularly perturbed delay initial value problem, AppliedMathematics Letters, 23,10(2010)1221-1225. [11] S. Maset, Numerical solution of retarded functional differential equations as abstract Cauchy problems, J. Comput. Appl. Math. 161(2003)259-282. [12] B.J.MacCartin, Exponential fitting of the delayed recruitment/renewal equation. J. Comput. Appl. Math., 136(2001)343-356. [13] M.K. Kadalbajoo, K.K. Sharma, $\varepsilon- $uniform fitted mesh method for singularly perturbed differential difference equations with mixed type of shifts with layer behavior, Int. J. Comput. Math. 81 (2004)49-62. [14] C.G. Lange, R.M. Miura, Singular perturbation analysis of boundary-value problems for differential difference equations, SIAM J. Appl. Math. 42(1982)502-531. [15] H. Tian, The exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag, J. Math. Anal. Appl. 270(2002)143-149. [16] E. R. Doolan, J.J.H. Miller, and W. H. A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole, Press, Dublin, (1980). [17] P.A. Farrell, A.F. Hegarty, J.J.H. Miller, E.O'Riordan and G.I.Shishkin, Robust Computational Techniques for Boundary Layers, Chapman- Hall/CRC, New York, (2000). [18] H.G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Convection-Diffusion and Flow Problems, Springer Verlag, Berlin, (1996). [19] G.M. Amiraliyev and H. Duru, A note on a parameterized singular perturbation problem, J.Comput. Appl. Math. 182 (2005) 233-242. [20] G.M. Amiraliyev, The convergence of a finite difference method on layer-adapted mesh for a singularly perturbed system, Applied Mathematics and Computation, 162,3(2005)1023-1034 |
| Content type: | Conference Abstract |
| Appears in Collections: | Litteris et Artibus. – 2017 р. |
| File | Description | Size | Format | |
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| 2017_Erdogan_F-A_fitted_numerical_method_422-424.pdf | 91.98 kB | Adobe PDF | View/Open | |
| 2017_Erdogan_F-A_fitted_numerical_method_422-424__COVER.png | 559.11 kB | image/png | View/Open |
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