The best parametrization for solving the boundary value problem for the system of differential-algebraic equations with delay

Abstract

Abstract In this paper, we considered the numerical approach for solving a nonlinear boundary value problem for the system of differential-algebraic equations with delay argument. The shooting method is used to solve the boundary value problem. The Newton method is used to find the parameter of shooting. To overcome the difficulties associated with the choice of the initial approximation we apply E. Lahaye’s parameter continuation method. If the curve of the solution contains limit points, the method diverges. Then to find the parameter we used the method of continuation with respect to the best parameter - the length of the curve of the solution set. The solution is constructed by advancing the sequence of values of the parameter. With a discrete continuation, the initial-value problem is transformed by a finite-difference representation of the derivatives and entering the best argument and the corresponding equation of hypersphere. The resulting system is solved using the Newton method. To find the values of the functions at the delay point Lagrange polynomial with three points is used. An example of the behavior of an elastoviscoplastic rod is considered.