SINGULAR PERTURBATION FOR A PROBLEM
From Hydraulics, Narcisa Apreutesei · 2005
Of concern is the comparison of the solution (u,v) to the nonlinear b.v.p. (P) below with the solution (u e ,v e ) of its elliptic regularization (Pe). An asymptotic approximation for (u e ,v e ) is constructed using the boundary layer method of Vishik-Lyusternik. The order of this approximation is found. (BC) u(x,0) = u0 (x) , v (x,0) = v0 (x) , 0 < x < 1 (C) and its elliptic regularization (Pe) � eu e tt − u e = v e + αu e (x, t) − f (x, t) , (x, t) ∈ Q ev e − v e t = u e + βv e (x, t) − g (x, t) , (x, t) ∈ Q, (Se) u e (0, t) = u e (1, t) = 0, 0 < t < T, (BCe) � u e (x,0) = u 0 (x) , v e (x,0) = v0 (x) , 0 < x < 1 u e (x, T) = u1 (x) , v e (x, T) = v1 (x) , 0 < x < 1. (Ce) Problem (P) governs the unsteady fluid flow (water-hammer) with nonlin- ear pipe friction. It is also a model for the fluid flow through a tree-structured system of transmission pipelines (4) . In problem (P) , u denotes the instanta- neous discharge at a point and v is the elevation of hydraulic gradeline. We construct an asymptotic approximation (Y e , Z e ) of (u e , v e ) with the aid of the solution of problem (P) and find the order of this approximation in the spaces L 2 (Q) and C (0, T); L 2 (0,1) � .