In this work, we investigate the solvability and the approximate construction of solutions of certain types of regular non-linear boundary value problems for systems of ordinary differential equations on a compact interval. For this purpose, we construct analytically a uniformly convergent parametrised sequence of functions depending on the properties of the concrete boundary conditions and non-linearities in the given systems. The value of the parameter introduced artificially into the scheme is to be determined by solving a certain system of algebraic or transcendental equations. The text is divided into 10 sections. Sections 1 and 2 contain the notation used in what follows and provide a short introduction. In Section 3, the successive approximation techniques are treated for the investigation of periodic solutions of non-autonomous periodic systems. In Section 4, we apply the method for the study the periodic solutions of autonomous systems by using the appropriate reduction to a non-autonomous system. In Section 5, we establish conditions under which a system of non-linear non-autonomous ordinary differential equations has a family of solutions that are periodic with a common period and possess a certain symmetry property. Sections 6 and 7 deal with the investigation of non-linear two-point boundary value conditions by using a parametrisation that leads one to a family of problem with linear two-point conditions considered together with certain additional algebraic or transcendental equations with respect to certain parameters. In Section 8, we use the parametrisation approach to study some three-point non-linear boundary value problem which, as a result, can be investigated through auxiliary two-point problems. Most of theoretical results are illustrated by examples. Section 9 contains some historical remarks concerning the development and application of the method. Finally, in Section 10, we give several exercises concerning the successive approximation technique under consideration.

Successive Approximation Techniques in Non-Linear Boundary Value Problems for Ordinary Differential Equations

A new approach to non-local boundary value problems for ordinary differential systems

For a constructive analysis of the periodic boundary value problem for systems of non-linear non-autonomous ordinary differential equations, a numerical-analytic approach is developed, which allows one to both study the solvability and construct approximations to the solution. An interval halving technique, by using which one can weaken significantly the conditions required to guarantee the convergence, is introduced. The main assumption on the equation is that the non-linearity is locally Lipschitzian.

/pdf/constructive-analysis-of-periodic-solutions-with-interval-p32er8n4xk.pdf

Constructive analysis of periodic solutions with interval halving

A three-point boundary-value problem for a system of nonlinear differential equations is reduced to a family of two-point problems, whose solutions are investigated by using the numerical-analytic method.

On the parametrization of three-point nonlinear boundary-value problems

We analyze the application of the numerical-analytic method proposed by A.M. Samoilenko in 1965 to multipoint boundary-value problems.

The theory of the numerical-analytic method: Achievements and new trends of development. VI

We establish new improved estimates necessary for the justification of the numerical-analytic method for the investigation of the existence and construction of approximate solutions of nonlinear boundary-value problems for ordinary differential equations

Some remarks concerning the convergence of the numerical-analytic method of successive approximations

Atmospheric precipitation during cyclonic situations was analyzed using weather types classification. Based on data from the period 1991 to 2015, the observed cyclonic situations were analyzed in terms of their frequency of days with a given weather type. Cyclonic situations with airflow direction from the west and northwest, north and northeast, east and southeast, and south and southwest were analyzed. We identified a declining number of days that can be classified as cyclonic situations. The distributions of the mean annual precipitation totals for these cyclonic situations have been investigated. The highest mean annual precipitation totals occurred during the west cyclonic, northeast cyclonic, and east cyclonic weather types. The lowest mean annual precipitation totals were identified during the southwest cyclonic (with fronts moving from north to northeast) and north cyclonic weather types. The percentage of the individual cyclonic weather types and supertypes in the mean annual precipitation total was calculated. The directional supertype west + northwest with the west cyclonic type occurred with the highest percentage, although variations may arise due to windward and leeward effects.

Mean annual totals of precipitation during the period 1991–2015 with respect to cyclonic situations in Slovakia

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Some remarks concerning the convergence of the numerical-analytic method of successive approximations

Mean annual totals of precipitation during the period 1991–2015 with respect to cyclonic situations in Slovakia