A convergence theorem on the iterative solution of non-linear two-point boundary value systems
- Paper ID
1963-70
- author
- company
Grumman Aircraft Engineering corporation
- country
U.S.A.
- year
1963
- abstract
The nonlinear two-point boundary value problem occurs quite naturally in studies in flight-mechanic s, e.g., in the numerical application of the theory of the calculus of variations and in intercept and rendezvous studies. Computationally, obtaining solutions to this class of problems is fraught with difficulties such that no really satisfactory systematic procedure is currently available. A widely used procedure for approaching nonlinear problems in diverse branches of mathematics is the substitution of a sequence of linear problems for the nonlinear problem in such a manner that the sequence of solutions to the 1inear problems approach in a limiting sense the solution of the nonlinear prob- . lem. By means of an analog to the simple but powerful geometric notion underlying the Newton-Raphson method this procedure is applied to systems of nonlinear differential equations with two- point boundary conditions. The theorem proved here yields sufficient conditions for convergence and error bounds (which serve as stopping criteria for automatic machine computation) for the resulting strongly convergent sequence. The proof of the theorem which shows the convergence to be uniform and rapid (quadratic), is based on the Contraction Mapping Principle, as given for example by Lyusternik, and is in fact an existence and uniqueness proof for the solution of the nonlinear-system. The motivation for this approach is provided by the fact that the linear two-point boundary value problem is considerably more tractable numerically, so that the theorem proved here provides support for what may be a powerful method for solving nonlinear boundary value problems by means of high speed computers. This may be particularly so with regard to systems of higher dimension (of great importance in flight mechanics). A few examples of the application of the method under study here are available for the case of a single nonlinear equation (Hestenes, Kalaba, et al.). However, very little is known concerning the application to systems of higher order. A numerical example of the application of the method to systems of equations is given in this paper.