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

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

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

The article deals with approximate solutions of a nonlinear ordinary differential equation with homogeneous Dirichlet boundary conditions. We provide a scheme of numerical-analytic method based upon successive approximations constructed in analytic form. We give sufficient conditions for the solvability of the problem and prove the uniform convergence of the approximations to the parameterized limit function. We provide a justification of the polynomial version of the method with several illustrating examples. 2000 Mathematics Subject Classification: 34B15; 65L10.

/pdf/numerical-analytic-technique-for-investigation-of-solutions-d3xk2218ce.pdf

Numerical-analytic technique for investigation of solutions of some nonlinear equations with Dirichlet conditions

We obtain some results concerning the investigation of the solutions of three-point Cauchy-Nicoletti type boundary value problems for a certain class of linear functional differ- ential equations. We show that it is useful to reduce the given problem to the parametrized two-point boundary value problem for a suitably perturbed system containing some artificially introduced parameters both in the constructed inhomogeneous two-point boundary conditions and in the modified functional differential equations. To study the transformed parametrized two-point problem, we use a method which is based upon special type of successive approximations constructed in an analytic form. We prove the uniform convergence of these approximations to the parametrized limit function. Our technique leads to a certain system of algebraic equations with respect to the introduced parameters whose solutions provide those numerical values of the parameters that correspond to the solutions of the given three-point boundary value problem.

/pdf/on-a-cauchy-nicoletti-type-three-point-boundary-value-lwd0w12ppa.pdf

On a cauchy-nicoletti type three-point boundary value problem for linear differential equations with argument deviations

In this paper the comparison method and the choosing of the appropriate norm of the comparison operator is used to establish the solvability of the integral-functional equation as a result of the application of the numerical-analytic method of solving of boundary value problems for ordinary differential-delay equations of the neutral type.

Some remarks on an integral equation arising in applications of numerical-analytic method of solving of boundary value problems

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Some remarks on an integral equation arising in applications of numerical-analytic method of solving of boundary value problems