A FUNCTIONAL EQUATION METHOD FOR ESTIMATING CONVERGENCE INTERVALS AND TRUNCATION ERRORS IN SMALL PARAMETER NONLINEAR OSCILLATIONS.
Interim technical rept.,
CRUFT LAB HARVARD UNIV CAMBRIDGE MASS
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Most current methods of constructing periodic solutions of quasi-linear differential equations with periodic right-hand sides suffer from two important practical defects. Although under appropriate conditions the theory of such equations guarantees the existence of a periodic solution for sufficiently small values of the perturbing parameter, there is usually no indication of how small the parameter must be, and hence no indication whether any particular calculated solution really applies. Secondly, although in principle an exact solution may usually be constructed by infinite iteration of some computation algorithm assuming the parameter to be sufficiently small, there is usually no indication of the error remaining after only a finite number of iterations, and hence no indication of the accuracy of the approximate solution with which one must be satisfied in practice. The present report describes a method of constructing, from the majorized right-hand side of the original differential equation, a certain functional algebraic equation whose solution is a majorant for the periodic solution of the differential equation. This majorant has the property that its power series expansion majorizes the formal power series expansion of the periodic solution term by term. Author
- Operations Research