Asymptotics for a Class of Perturbed Initial Value Problems

Abstract

Publisher Summary This chapter discusses the initial value problems containing a small nonnegative perturbation parameter. On time intervals initiating the origin one can approximate the exact solution (provided it exists) asymptotically. The asymptotic solutions could be derived from so-called formal asymptotic solutions, i.e., functions that satisfy the differential equation and the initial conditions up to an asymptotic accuracy of certain order. These formal asymptotic solutions should then be compared asymptotically with the exact solution. As an application, the chapter considers a class of perturbed oscillations described by the nonlinear second order ordinary differential equation with slowly varying coefficients. The chapter discusses the concepts of (formal) asymptotic solution in dealing with the vector differential equation, formal asymptotic solutions of oscillation problems, proof of asymptotic correctness, improved formal asymptotic solution of the oscillation problem, oscillation problems with small decay, and various theorems.