On parametrized problems with non-linear boundary conditions

Abstract

We consider a parametrized boundary-value problem containing an unknown parameter both in the non-linear ordinary differential equations and in the non-linear boundary conditions. By using a suitable change of variables, we reduce the original problem to a family of those with linear boundary conditions plus some non-linear algebraic determining equations. We construct a numerical-analytic scheme suitable for studying the solutions of the transformed boundary-value problem. Acknowledgement 1. The first author was partially supported by the scholarship Charles Simonyi.

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

“…The study of the problem then consists of two parts, namely, the analytic part, when the integral Eq. (13) is dealt with by using the method of successive approximations (9), and the numerical one, which consists in finding a values of the 2n unknown parameters from the system of Eqs. (41) and (42).…”

“…This facilitates greatly the construction of iterations compared to formula (9). For the same reason, system (53), which has to be solved numerically, is considerably simpler than (46) …”

“…Indeed, let us construct the first approximation. Using (9) and carrying out computations in Maple, at the first iteration (m ¼ 1), we obtain u 11 ðt; n; gÞ ¼ n 1 þ t 4 400 þ t 3 3 4ðÀn 2 þ g 2 Þ 2 þ 2 5…”

“…The idea that we are going to employ is based on the reduction to a family of simpler auxiliary boundary problems obtained by ''freezing'' certain values of the solution sought for (see, e.g., [8][9][10]). In our case, the auxiliary problems will have two-point linear separated conditions at a and b: uðaÞ ¼ n; uðbÞ ¼ g;…”

“…The vectors n and g in (8) and (9) are treated as unknown parameters. Considering formulae (8) and (9), one arrives immediately at the following Proposition 1. If, for a fixed pair ðn; gÞ 2 D 0  D 1 , the sequence fu m ðÁ; n; gÞ : m P 0g converges to a function u 1 ðÁ; n; gÞ uniformly on ½a; b, then:…”