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

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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/}
}
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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/