On parametrization for a non-linear boundary value problem with separated conditions

Abstract

We obtain some results concerning the investigation of two-dimensional non-linear boundary value problems with two-point separated linear boundary conditions. We show that it is useful to reduce the given boundary-value problem, using an appropriate substitution, to a parametrized boundary value problem containing some unknown scalar parameter in the boundary conditions. To study the transformed parametrized problem, we use a method which is based upon special types of successive approximations constructed in an analytic form. The parametrized problem should be investigated together with certain so-called determining equations, which have algebraic or transcendental form.

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

“…The limit function x* satisfies the initial condition x*(0, z, l) = z as well as the boundary conditions in (6), since these conditions are satisfied by all functions x m = x m (t, z, l) of the sequence {x m }. Passing to the limit in the recurrence relation (14) for x m , we show that the limit function x* satisfies the identity (15). If we differentiate this identity, we obtain that x* is a unique solution of the Cauchy problem (16).…”

“…At the beginning, we follow the ideas presented by Rontó and Rontó [14] and by Rontó and Shchobak [15], which contains existence results for a system of two nonlinear differential equations with separated boundary conditions. In order to avoid some technical difficulties, we deal in this article, for simplicity, with nonlinear differential equations with homogeneous Dirichlet boundary conditions.…”

“…In order to avoid some technical difficulties, we deal in this article, for simplicity, with nonlinear differential equations with homogeneous Dirichlet boundary conditions. On the other hand, our basic recurrence relation has the same general form as it is presented in [15]. In Section 2, we state the studied problem and the corresponding setting.…”

Successive approximations and interval halving for integral boundary value problems

“…To replace the boundary conditions (1.5) by certain linear two-point linear separated ones, similarly to [6], [10], [11], [9] we apply a certain "freezing" technique. Namely, we introduce the vectors of parameterś D col.´1;´2; :::;´n/; D col. 1 ; 2 ; :::; n /; Á D col.Á 1 ; Á 2 ; :::; Á n / (3.6)…”

“…R n are a continuous functions in a certain bounded set D and d 2 R n is a given vector. We use an appropriate numerical-analytic approach and a natural interval halving technique which was suggested in [6], [4], [7], [8], [11], [3]. At first, we reduce the given problem (1.4), (1.5) To study the solutions of BVPs (1.6), (1.7) and (1.8), (1.9) we use the special modified form of parameterized successive approximations x m .t;´; / and y m .t; ; Á/ of type (1.3) constructed in analytic form and well defined on the intervals t 2 h a; a C i , respectively.…”