An approach to the choice of the initial approximation of the solution of nonlinear boundary value problem for loaded differential equations

Abstract

On the basis of the parameterization method is investigated nonlinear two - point boundary value problem for systems loaded differential equations. The essence of the parameterization is that the problem of the partition of the interval specified points of loading and introduction of additional parameters is reduced to the equivalent nonlinear two - point boundary value problem with parameter. The introduction of additional parameters allows to get the initial conditions for the unknown functions in the subintervals. For fixed values of the parameters solved the Cauchy problem for systems of ordinary differential equations. Substituting the representation of the solution of the Cauchy problem in the boundary condition and the conditions of continuity of the solution was bult for the entered parameters. Built a system of nonlinear algebraic equations are the basis of the method of parameterization algorithms and allow us to find a good initial approximation to the solution of a nonlinear two - point boundary value problem for loaded differential equations. A one way to choosing a good initial approximation was offered for finding solutions of nonlinear boundary problem. The conditions of existence of isolated solution of the nonlinear two - point boundary value problem for loaded differential equations for suffciently small steps partition.