Iterative Schemes for Periodic Boundary-Value Problems with Switchings

We investigate a weakly nonlinear boundary-value problem for a system of nonlinear ordinary differential equations with discontinuous right-hand side and periodic boundary conditions. A modified iterative scheme is proposed for the construction of approximate solutions of this problem. Unlike our previous results, this scheme is based on the refined expansion of the right-hand side of the analyzed differential equation and a new choice of the initial approximation. The proposed iterative procedure is applied to a mathematical model of nonisothermal chemical reaction with periodic switchings of the control functions. In the considered example, the convergence of our iterative scheme to the required solution is illustrated by the analysis of discrepancies in the boundary-value problem. We establish estimates for the range of a small parameter within which the operator used for the construction of the required solutions is contractive. The results of numerical simulations confirm the fact that the proposed computational scheme guarantees better accuracy than the method of simple iterations.