Successive approximations and interval halving for integral boundary value problems

Abstract

Abstract. We show how a suitable interval halving and parametrization technique can help to essentially improve the sufficient convergence condition for the successive approximations dealing with solutions of nonlinear non-autonomous systems of ordinary differential equations under integral boundary conditions.

Successive Approximation Technique in the Study of a Nonlinear Fractional Boundary Value Problem

“…This paper disclosed the recent results in the study of a system of nonlinear FDEs of the real order, subjected to essentially nonlinear two-point boundary constraints. For the analytical representation of a solution, we suggested a modified successive approximation technique (see earlier results in [6][7][8][9][10]), based on the so-called dychotomy-type approach ( [13][14][15][16]). The modification aimed to reduce the a priori error of the method for its more efficient application to the nonlinear problems discussed herein.…”

“…To solve this problem, we used the so-called "freezing" (or parametrization) technique (see discussions in [11,12,16]), coupled with the modification of the numerical-analytic method [5]. The aim of such an approach consists of the introduction of an appropriate parametrization with further reduction of the original FBVP ( 6) and ( 7) with nonlinear boundary conditions to two problems, containing already linear separated boundary constraints.…”

“…In the current paper, we provide a new view on the successive approximations approach [5], recently used for the study of FBVPs for periodic, Cauchy-Nicoletti-type, and interpolation boundary conditions (see [6][7][8][9][10]). An original parametrization technique, initially suggested for the reduction of nonlinearities in boundary restrictions (see discussions [11,12]), and a dichotomy-type approach (see results in [13][14][15][16]) led to the investigation of solutions of two "model"-type FBVPs, containing some artificially introduced parameters. The approximate solutions of these problems were constructed analytically, while the numerical values of the parameters were determined as solutions of the so-called "bifurcation" equations.…”

On investigation of some non-linear integral boundary value problem

“…This article uses the approach proposed in [2], [5], [4] in the case of the following non-linear boundary value problem with mixed two-point and integral restrictions dx. and K 1 K 5 are non-negative square matrices of dimension n: The inequalities between vectors are understood componentwise.…”

Study of a nonlinear fractional boundary value problem via the dichotomy-type technique

“…xxx-xxx Cauchy-Nicoletti type boundary conditions (see [2]- [5]). An original 'freezing' technique, initially suggested for the nonlinear systems of ordinary differential equations (see discussions [8], [7]), and a dichotomy-type approach (see [9], [10]) lead to investigation of solutions of two 'model'-type FBVPs, containing some artificially introduced parameters. The approximate solutions of these problems are constructed analytically, while the numerical values of parameters are determined as solutions of the so-called 'bifurcation' equations.…”