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

Abstract

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. Statement of the ProblemConsider a system of ordinary differential equations of the form(1) whose solutions satisfy the conditionswhere P .t /; C.t/; and S.t / are m m; m l; and l m matrices, respectively, whose elements are square summable on the segment OE0; T : Furthermore, the columns of the matrix C.t/ are linearly independent, D is a constant m m matrix, f 2 L 2 .OE0; T ; R m /;˛2 R l ; and 2 R m : A solution of problem (1), (2) is understood as a vector 2 R l and vector function x 2 W 1 2 .OE0; T ; R m / that satisfy Eq. (1) and conditions (2). If this is not true, then the problem is inconsistent. In what follows, for convenience, we use the term "function" instead of "vector function."

Parametrization for some boundary value problems of interpolation type

“…In [1,2,3,4,5], various iteration schemes are used. The works [6,7,8,9,10,11] develop the method of averaging of functional corrections and other projection-iterative methods. Topological methods based upon the application of degree theory are used in [12,13,14] in studies of the solvability of boundary value problems.…”