Regular Perturbations of Parameters

Abstract

A difference between real and idealized systems is very often reduced to perturbation of the input parameters. For instance, a thickness of a plate (or shell) is described via formula h = h ₀ + εh(x, y) (h ₀ = const, ε ≪ 1); contour of the circle plate slightly differs from a circle via relation r(θ) = r ₀ + ε cos nθ, etc. Although often the considered system does not follow Hook’s principle, but a difference is small. Non-linearity of many systems only slightly differs from linearity, and this system is said to be a quasi-linear one. The material of an object is weakly anisotropic, and so on. In all cited examples an influence of deviations (or perturbations) is small, and it can be estimated applying the method of regular perturbations. A being sought solution can be presented in the form of the following series

$$ f\left( {x,\varepsilon } \right) \sim \sum\limits_{n = 0}^\infty {{\delta _n}} \left( \varepsilon \right){f_n}\left( x \right) $$

, where δ _n (ε) is the asymptotic sequence depends upon the small parameter ε.