Geodesic
The discrete continuation in the boundary value problem for systems of nonlinear differential equations with deviation argument
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 21 (2019) no. 3, pp. 309-316.

Voir la notice de l'article provenant de la source Math-Net.Ru

he solution of the boundary value problems for system of nonlinear differential equations with argument delay is considered in the article. The solution is based on the shooting method. Within its framework the method of continuation with respect to parameter in the Lahaye form, method of the best parametrization and the Newton method are implemented that allow to find possible solutions. To solve the Cauchy problem at each step of the shooting method the discrete continuation method with respect to the best parameter combined with the Newton method is applied. This approach allows to build the solution in the case when singular limit points exist. That provides continuation of Newton iteration process. The algorithm is completed by calculating the Lagrange polynomial to obtain the values of function in the delay points. The example given in the article represents the advantages of the proposed method.
Keywords: numerical solution, equations with delay, boundary value problem, the best parameter, discrete continuation, shooting method. 
@article{SVMO_2019_21_3_a0,
     author = {M. N. Afanaseva and E. B. Kuznetsov},
     title = {The discrete continuation in the boundary value problem for systems of nonlinear differential equations with deviation argument},
     journal = {\v{Z}urnal Srednevol\v{z}skogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {309--316},
     publisher = {mathdoc},
     volume = {21},
     number = {3},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVMO_2019_21_3_a0/}
}
TY  - JOUR
AU  - M. N. Afanaseva
AU  - E. B. Kuznetsov
TI  - The discrete continuation in the boundary value problem for systems of nonlinear differential equations with deviation argument
JO  - Žurnal Srednevolžskogo matematičeskogo obŝestva
PY  - 2019
SP  - 309
EP  - 316
VL  - 21
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SVMO_2019_21_3_a0/
LA  - ru
ID  - SVMO_2019_21_3_a0
ER  - 
%0 Journal Article
%A M. N. Afanaseva
%A E. B. Kuznetsov
%T The discrete continuation in the boundary value problem for systems of nonlinear differential equations with deviation argument
%J Žurnal Srednevolžskogo matematičeskogo obŝestva
%D 2019
%P 309-316
%V 21
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SVMO_2019_21_3_a0/
%G ru
%F SVMO_2019_21_3_a0
M. N. Afanaseva; E. B. Kuznetsov. The discrete continuation in the boundary value problem for systems of nonlinear differential equations with deviation argument. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 21 (2019) no. 3, pp. 309-316. http://geodesic.mathdoc.fr/item/SVMO_2019_21_3_a0/

[1] Y. F. Dolgii, P. G. Surkov, Mathematical models of delay dynamical systems, Ural Federal University Publ., Ekaterinburg, 2012, 119 pp. (In Russ.)

[2] M. N. Afanaseva, E. B. Kuznetsov, “Numerical method for solving nonlinear boundary value problem for differential equations with retarded argument”, Trudy MAI, 88 (2016) (In Russ.)

[3] N. S. Bakhvalov, N. P. Zhidkov, G. M. Kobel'kov, Numerical Methods, Nauka Publ., Moscow, 1987, 600 pp. (In Russ.) | MR

[4] V. I. Shalashilin, E. B. Kuznetsov, Parametric Continuation and Optimal Parametrization in Applied Mathematics and Mechanics, Editorial URSS, Moscow, 1999, 224 pp. (In Russ.)

[5] E. M. Budkina, E. B. Kuznetsov, “Modeling of technological process for aircraft structural components manufacturing based on the best parametrization and boundary value problem for nonlinear differential-algebraic equations”, Aerospace MAI Journal, 23:1 (2016), 189–196 (In Russ.)

[6] S. D. Krasnikov, E. B. Kuznetsov, “Parametrization of the numerical solution of boundary value problems for nonlinear differential equations”, Comput. Math. Math. Phys., 45:12 (2005), 2148–2158 | MR | Zbl

[7] S. S. Dmitriev, E. B. Kuznetsov, “Numerical solution to systems of delay integrodifferential algebraic equations”, Comput. Math. Math. Phys., 48:3 (2008), 430–444 | MR | Zbl

[8] M. S. Roberts, J. S. Shipman, Two-point boundary value problems: shooting methods, Elsevier, New York, 1972, 269 pp. | MR | Zbl

[9] A. M. Samoilenko, N. I. Ronto, Numerical-analytic methods in theory of boundary-value problems, Naukova Dumka, Kiev, 1986, 222 pp. (In Russ.) | MR

[10] M. E. Lahaye, “Une metode de resolution d’une categorie d’equations transcendentes”, Compter Rendus hebdomataires des seances de L’Academie des sciences, 198:21 (1934), 1840–1842

[11] E. B. Kuznetsov, “Optimal parametrization in numerical construction of curve”, Journal of the Franklin Institute, 344:21 (2007), 658–671 | DOI | MR | Zbl