Boundary problem with impulse action (second order critical case)

Abstract

This abstract is devoted to the problem of finding the constructive conditions of existence and iterative algorithms for constructing a solution of the boundary value problem with a pulsed action in the critical case of the second order.

It is shown that the existence of an initial boundary value problem depends on the condition, compiled with the help of nonlinearity and the first approximation to the desired solution.

Conditions for the construction of a boundary value problem with a pulsed action to the operator system are found, which is sufficient to prove the existence of the solution of such boundary value problem. The theorem is formulated, which allows us to find a single solution, which is determined by the iterative process.

In the process of finding a sufficient condition for the existence of a solution of the boundary value problem, we obtain an algebraic system with two orthoprojectors, one of which projects r - dimensional Euclidean space to zero space of the matrix, and the other - d - dimensional space on the zero space of the matrix. We formulate the necessary and sufficient condition for the existence of a solution of an algebraic system, which can be written in the form of the corresponding equality. We find that a sufficient condition for the implementation of this equality is the requirement that the intersection of zero-spaces be zero.

After that we can go to the operator system, which does not belong to the class of systems, for solving which the method of simple iterations is used. Selecting an additional variable, such a system is regularized. In this case, the exponent r - the measured constant in the direct sum of two, determined in different quantities, the dimension of the operator system increases. This will enable, under the conditions discussed in this article, to reduce (2n + r) - a measurable system to a regular 2 (n + r) - measurable operator system for which a convergent simple iteration method is used.

To study the boundary-value problem with impulse action in the general case, when the conditions formulated in the article take place, we write in explicit form linear additions which are included in the decomposition of the vector function and arrive at the equivalent of a piecewise-continuously differentiable t-set with discontinuities of the first kind of continuous vector-functions of the operator system.

The resulting system belongs to the class of operator systems for which the method of simple iterations is used. The scheme of research and solution of this system is similar to the scheme of research and solution of the system considered in this article to prove the existence of the solution of the original boundary value problem.

The iterative process of finding the solution for the resulting system is considered under the conditions defined in this article in the expansion of the vector-function. First we find the first approximation of the solution. The definition of a generalized Green operator gives a vector-valued function, which is the only solution of the boundary-value problem with impulse action. Similarly, the second and third approximations are assigned and the only solution to the boundary value problem is determined. The existence of such a solution is ensured by the choice of a vector constant from the condition of solvability of the above boundary value problem.

The convergence of the iterative process is established by the method of the majorant Lyapunov equations. However, as in the non-critical case and in the critical case of the first order, it is enough to prove the existence of a solution of the boundary-value problem with impulse action to determine the conditions for its reduction to the operator system.

The obtained result allows us to assume that in the case where the boundary-value problem with impulse action satisfies the above conditions in such a way that there is a critical case and the corresponding generating boundary-value problem with impulse action under the condition d = m - n1 and only if it has r - parametric a family of generating solutions, then for each value of the vector cr, which satisfies the equation for generating amplitudes, under the conditions under consideration, the boundary value problem with a pulsed action has a unique solution that turns to zero. This solution can be found using a concurrent iterative process. The boundary-value problem with impulse action has thus the only solution that rotates in the generative solution of the boundary value problem. This solution is determined by the iterative process and the formula. On the basis of the obtained theorem is formulated, which is the main result of this work.

In the considered critical case of boundary value problems, the condition formulated in the theorem is a necessary and sufficient condition for the existence of a solution, which is a second approximation to the desired one. If this condition is not fulfilled, then the desired solution of the boundary-value problem with impulse action, which is determined by the method of simple iterations, does not exist. In this case, we can only talk about the pseudorabilization of the boundary-value problem with pulsed action. In the case considered in this article, only a sufficient condition is a restriction that ensures the existence of a solution and the possibility of its construction by means of an iterative analogue of the method of the Lyapunov-Poincare small parameter.

Keywords: heterogeneous boundary value problem, homogeneous boundary value problem with impulse action, orthogonal projection, pseudo-inverse matrix, generalized Green operator, operator system, critical case of second order