Pointwise modeling and solving Cauchy problems for nonlinear diff erential equations

Osipov Vladimir Vladimirovich, Osipova Vera Alexandrovna

The method of pointwise representations (pointwise modeling) as a method of mathematical modeling of differential and integral equations using spline step models, and a pointwise representation of functions and operators is considered in the work. The finite-dimensional arising in this case models are homomorphic images of the corresponding objects, which have the highest possible degree of adequacy, which increases as soon as dimension increases until the complete equivalence. Algebraic structures of pointwise representations are considered. It is shown that the algebra AM (0, T) in the space of all piecewise continuous bounded functions defined on a finite time interval [0, T] with a binary operation of ordinary multiplication can naturally be used as the basis for pointwise modeling of linear and nonlinear processes described by differential and integral equations of various types. Pointwise representations for various operations on functions are obtained. A pointwise model of a homogeneous nonlinear differential equation is constructed, a theorem on the existence of its point solution is proved.

Keywords:Approximately operator methods and analytical modeling, the method of point representations, modeling point of differential equations, vector representing the point