Solutions of nonstandard nth order initial value problems

摘要

Differential equations of the form y(n)=f (t, y, y',…, y(n), where f is not necessarily linear in its arguments, represent certain physical phenomena (e.g., free oscillations of positively damped systems, nonlinear systems subject to harmonic excitations, motion of a particle on a rotating parabola) and have been known for quite some time. Earlier we established the existence of a unique solution of the nonstandard first order initial value problem y' =f (t, y, y'), y(t0) = y0 under certain natural hypotheses on f and developed some linear and quadratic convergent numerical schemes for the construction of approximate solution of the above problem. In this paper we establish existence results and develop some linearly convergent numerical schemes for the construction of solution of nonstandard nth order initial value problems, and we solve two physical examples.