A remark on the parametrization of the numerical solution of boundary value problems

Abstract

Here f : R ×R → R and R : R × R → R. Throughout the following, we assume that the conditions imposed on the nonlinear functions f and R of the variable y on [a, b] provide that problem (1.1), (1.2) is solvable. Numerous methods were developed for the solution of boundary value problems of the form (1.1), (1.2). The paper [1] presents approximate methods for operator equations and the investigation of various iterative processes for linear and nonlinear equations. For the choice of an initial approximation for a nonlinear operator equation, one introduces a parameter λ ∈ [0, 1] such that if λ = 0, then the solution of the equation is known or can be readily found, and if λ = 1, then one has the solution of the original equation. Then the desired solution is constructed with the use of continuation with respect to the parameter λ. Three methods for nonlinear boundary value problems were considered in [2] : the shooting method, the method of Green function, and the finite-difference method. The monograph [3] deals with the exposition of numerical-analytic methods for nonlinear systems of differential equations considered under nonseparatable two-point boundary conditions. It was suggested to introduce a parameter in boundary conditions so as to ensure that the solution is known at the first step of the method and the solution of the original boundary value problem is obtained at the last step with respect to the parameter. The nonlinear boundary value problem was preliminarily linearized in the monograph [4]. To improve the composition method, it was suggested to perform Godunov reorthogonalization [5, 6]. To this end, it was suggested to use the Cont algorithm [7]. A universal approach (quasilinearization) to the solution of various nonlinear problems was described in [8]. Bakhvalov [9] suggested to solve a nonlinear boundary value problem (after the linearization) with the use of the Thomas differential orthogonal method [5, 6]. The monograph [10] dealt with various numerical methods for twoand many-point boundary value problems.