The numerical solution of the nonlinear boundary value problem with singularity for the system of delay integrodifferential-algebraic equations

The numerical method for solving the nonlinear boundary value problem for a delay system of integrodifferential-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 difficult 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 finding 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 finite-difference 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 defined by conditions of the problem, were calculated with the help of the Lagrange polynomial with three points. To find the value of the integral components of the problem, the trapezoid method was used.
The results of the numerical study confirm 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 find all possible values of the parameter of the shooting method and to solve the problem.