Geodesic
The method of continuous continuation by a parameter for solving boundary-value problems for nonlinear systems of differential-algebraic equations with delay that have singular points
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 3, Tome 192 (2021), pp. 38-45.

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In this paper, we consider a numerical method for solving a nonlinear boundary-value problem for a system of differential-algebraic equations with a delayed argument that have singular limit points. For a numerical solution of the boundary-value problem, the shooting method is used. The value of the shooting parameter is calculated by the Newton method. We consider the case where the problem is ill-posed and hence the method may diverge. In this case, the solution is constructed by the method of the best parameter, namely, the length of the curve of the set of solutions. The solution of the initial problem for each value of the shooting parameter is calculated using the method of continuous continuation by the best parameter.
Keywords: numerical method, boundary-value problem, differential equation with delay, shooting method, method of continuation by the best parameter, singularly perturbed equation. 
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M. N. Afanaseva; E. B. Kuznetsov. The method of continuous continuation by a parameter for solving boundary-value problems for nonlinear systems of differential-algebraic equations with delay that have singular points. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 3, Tome 192 (2021), pp. 38-45. http://geodesic.mathdoc.fr/item/INTO_2021_192_a3/

[1] Bakhvalov N. S., Zhidkov N. P., Kobelkov G. M., Chislennye metody, Nauka, M., 1987

[2] Budkina E. M., Kuznetsov E. B., “Chislennoe reshenie kraevykh zadach dlya sistem integrodifferentsialno-algebraicheskikh uravnenii”, Mat. XI Mezhdunar. konf. po neravnovesnym protsessam v soplakh i struyakh (NPNJ'2016) (Alushta, 25-31 maya 2016 g.), Izd-vo MAI, M., 2016, 417–419

[3] Budkina E. M., Kuznetsov E. B., “Modelirovanie tekhnologicheskogo protsessa proizvodstva uzlov letatelnykh apparatov na osnove nailuchshei parametrizatsii kraevoi zadachi dlya nelineinykh differentsialno-algebraicheskikh uravnenii”, Vestn. MAI., 23:1 (2016), 189–196

[4] Dmitriev S. S., Kuznetsov E. B., “Chislennoe reshenie sistem integrodifferentsialno-algebraicheskikh uravnenii s zapazdyvayuschim argumentom”, Zh. vychisl. mat. mat. fiz., 48:3 (2008), 430–444

[5] Kamenskii G. A., Skubachevskiii A. L., Lineinye kraevye zadachi dlya differentsialno-raznostnykh uravnenii, Izd-vo MAI, M., 1992

[6] Krasnikov C. D., Kuznetsov E. B., “Parametrizatsiya chislennogo resheniya kraevykh zadach dlya nelineinykh differentsialnykh uravnenii”, Zh. vychisl. mat. mat. fiz., 45:12 (2005), 2148–2158

[7] Kuznetsov E. B., “Nailuchshaya parametrizatsiya pri postroenii krivykh”, Zh. vychisl. mat. mat. fiz., 44:9 (2004), 1540–1551

[8] Samoilenko A. M., Ronto N. I., Chislenno-analiticheskie metody issledovaniya reshenii kraevykh zadach, Naukova Dumka, Kiev, 1986

[9] Shalashilin V. I., Kuznetsov E. B., Metod prodolzheniya resheniya po parametru i nailuchshaya parametrizatsiya v prikladnoi matematike i mekhanike, Editorial URSS, M., 1999

[10] Lahaye M. E., “Une metode de resolution d'une categorie d'equations transcendentes”, C. R. Hebdomat. Seances Acad. Sci., 198:21 (1934), 1840–1842