Solution of a non-linear discrete boundary value problem by a continuous analog of Newton's method

Abstract

IN this paper we consider the non-linear discrete boundary value problem obtained from a finite difference approximation of the boundary value problem for the ordinary differential equation ϑ = y″ + ƒ(x, y) = 0 (0.1) y(0) = y(1) = 0 (0.2) In [1] a continuous analog of Newton's method was used to find an approximate solution of problem (0.1)–(0.2). The existence of these solutions is assumed. The proposed computational scheme is a realization of Euler's method of solving a Cauchy problem for a certain differential equation in B-space. It reduces to the solution for a fixed value of the parameter t of a boundary value problem for a linear ordinary differential equation, and to a subsequent advancement along the parameter by a simple difference relation. The solution of the boundary value problem for fixed t is assumed to be found numerically, for example, by the use of difference approximations. It is easy to see that this computational scheme is a numerical realization of the continuous analog of Newton's method of solving the difference operator equation approximating to the original boundary value problem (0.1)–(0.2). In the present paper a justification of this method is given: a proof is given of the existence of a solution of the discrete boundary value problem which is the difference analog of problem (0.1)–(0.2), and the convergence of the approximate solution to the solution of the original problem is proved. The result obtained in [2] is used in proving the existence of the solution of the discrete boundary value problem. Similar problems were considered, for example, in [3], pp. 347–388, [4], pp. 59–67. Thus, in [4] a proof is given of the existence and method of finding an approximate solution of problem (0.1)–(0.2) by Newton's method. However, by connecting the questions of the existence and uniqueness of the solution of the original problem with the method of the approximate solution, the author obtains extremely restricted conditions for the applicability of the method.