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The paper describes an iterative method of solution of one-point- and two-point-boundary " trajectory problems" specified in the form of integral equations. The rate of convergence of the process is enhanced by the use of associated Taylor series and/or solutions of simpler problems as first approximants. The accuracy of numerical quadratures need be high only in the final few iterations. The method has strong intuitive appeal, great flexibility, and appears to have an extensive field of potential applications.