Linear or nonlinear hyperbolic wave problems with input sets (Part I)

Abstract

In this first part, collections of linear hyperbolic initial boundary value problems are treated which are defined via sets of coefficient functions in the differential equations. If the solutions are oscillatory, the nonlinear dependency of the solutions on coefficients becomes more and more ill-conditioned as time progresses, unless there is a sufficiently strong damping term in the differential equation. For the problem of dynamic buckling, the theory of the Neumann series yields a sufficient condition for the uniform boundedness of the oscillatory solutions which are induced by arbitrary continuous transient perturbations whose range is restricted to a suitable interval. In this part I, there is an introductory discussion of the Taylor-representation of sets of solutions in terms of constant coefficients. Via such a Taylor-representation, it is shown that solutions of the ‘distortionless’ telephone line are insensitive to sufficiently small variations of the constant coefficients in this hyperbolic differential equation.