On the Models with Partial Distribution of Accuracy

The majority of the models for describing any oscillatory processes have partial distributionof accuracy, i.e. the number of normal mode is higher, the model describes its evolution worse.Therefore the question about convergence of the normal waves series, taking the central place at classical approach, inevitably take out of applicability of model. At such approach this isa lack of models, one of many difficulty in the proof of series convergence and existence ofthe classical solution. In this article we discuss new approach to the description of such models which is simplerclassical: here the proof of convergence of series is replaced with research of uncertainty ofnormal waves amplitudes. The statement was illustrated with a concrete example of theelementary model with partial distribution of accuracy, i.e. problem about string osculations. In such problems there is some uncertainty in initial conditions. So usually the profile of initialvelocity, used for the description of blow by a hammer, we consider as step function or “hat”,but we can consider the whole class of suitable profiles, therefore the whole family of initial-boundary value problems. This uncertainty in initial values gives the chance to estimate an error for each mode separately. As one would expect, the error grows to infinity as numberof a mode tend to infinity. All solutions of considered family of problems are expanded innormal waves series and younger modes have close amplitudes. It allows to keep all classical statements about younger modes and to avoid a investigation of convergence of normal wavesseries, which is technically difficult and take out of applicability of model.

mathematical model, string, oscillations, normal waves, normal modes