The numerical solution of the nonlinear boundary value problem with singularity for the system of delay integrodifferential-algebraic equations
M.N. Afanaseva*, E.B. Kuznetsov**
Moscow Aviation Institute (National Research University), Moscow, 125993 Russia
E-mail: *mary.mai.8@yandex.ru, **kuznetsov@mai.ru
Received April 24, 2019
DOI: 10.26907/2541-7746.2019.2.181-190
For citation: Afanaseva M.N., Kuznetsov E.B. The numerical solution of the nonlinear boundary value problem with singularity for the system of delay integrodifferential-algebraic equations. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2019, vol. 161, no. 2, pp. 181–190. doi: 10.26907/2541-7746.2019.2.181-190.
(In Russian)
Abstract
The numerical method for solving the nonlinear boundary value problem for a delay system of integrodifierential-algebraic equations was discussed. The occurrence of a delay argument in the system characterizes the behavior of the studied parameters not only at the current, but also at the previous period of time. Of particular interest are the problems characterized by the existence of singular limit points. It is very di–cult to solve these problems using the classical methods. A numerical solution of the boundary value problem was constructed by the shooting method. The values of the \shooting" parameter were found using a combination of the method of continuation with respect to the best parameter, the method of continuation with respect to the parameter in the Lahaye form, and the Newton method. At each iteration of the shooting method, the initial problem was solved. The computation of the initial problem influences the flnding of the required solution and the continuation of the iterative process of the shooting method. The initial problem was rearranged based on the best parameter - the length of the curve of the solution set, and flnite-difierence representation of derivatives. The resulting problem was solved by the Newton method. The values of the functions at the deviation point, where the values are not deflned by conditions of the problem, were calculated with the help of the Lagrange polynomial with three points. To flnd the value of the integral components of the problem, the trapezoid method was used. The results of the numerical study conflrm the efficiency of the proposed algorithm for solving the studied problem. The obtained numerical solution of the nonlinear boundary value problem with delay has the equation that loses its meaning in singular limit points. Thus, using of the continuation with respect to the best parameter while solving the problem allows to flnd all possible values of the parameter of the shooting method and to solve the problem.
Keywords: boundary value problem, numerical solution, difierential equations with delay, shooting method, method of continuation with respect to best parameter, singular limit points
Acknowledgments. The study was supported by the Russian Science Foundation (project no. 18-19-00474).
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