Construction of approximate solutions of systems of differential equations with parameters and restrictions

We propose a new approach to the investigation of boundary-value problems for systems of differential equations with parameters and restrictions. This approach uses ideas of both iterative and projection methods. We develop and substantiate a new modification of the projection-iterative method for the construction of approximate solutions of boundary-value problems that has significant advantages over the existing versions of the method. The solution of numerous problems of contemporary natural sciences is impossible without the construction of various mathematical models, the investigation and analysis of which stimulate the development of many fields of mathematics. Among these mathematical models, an important place is occupied by problems with parameters. There are many papers devoted to the development of the theory of these problems. In particular, in [1‐5], problems of analytic and qualitative theory and development of approximation methods for the construction of solutions of differential, integral, and integro-differential equations with restrictions were considered. In the present paper, we propose a new approach to the investigation of boundary-value problems for systems of differential equations with parameters and restrictions and develop an efficient method for the construction of their solutions.