Singular Perturbation Problems

Citation

Latta, Gordon Eric (1951) Singular Perturbation Problems. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/mjtp-g563. https://resolver.caltech.edu/CaltechTHESIS:07012025-171900167

Abstract

The equations considered in this rarer are linear differential equations in one and two independent variables. The problem at hand is to study solutions of boundary value problems for these equations in their dependence on a small parameter ϵ. Specifically, the equations are of the form (A) ϵ Nɸ + Mɸ = 0 where M, N are linear differential expressions, and ϵ > 0 is a small parameter; the order n of N is greater than the order m of M.

It is found, in certain cases, that the solution of a boundary value problem for (A), say ɸ(P, ϵ) tends non uniformly to a function u(P) satisfying the "reduced equation" M ɸ = 0, and even assumes the original boundary values on certain portions of the boundary of the region in question.

When the regions of non uniform convergence are located, an asymptotic expansion in terms of specific functions of ϵ, for ϵ small, is obtained.

Section two deals with a class of ordinary differential equations, while sections three and four deal with partial differential equations. In particular, it arrears from the results of section four, that methods used in this rarer should carry over to the non-linear Navier-Stokes equations of which the Oseen equations of the last section are a linearized approximation. This is being investigated at present.

Thesis Files