On Spurious Solutions of Singular Perturbation Problems

Abstract

A study is made of several nonlinear boundary-value problems of singular perturbation type for which a straightforward application of boundary-layer theory leads to spurious solutions. It is shown that these problems can be treated successfully by a slight modification of the method of matched asymptotic expansions. The analysis leads to several novel features which are not present in routine singular perturbation problems.

Shooting Method for Nonlinear Singularly Perturbed Boundary-Value Problems

“…For instance, in the case when Q(u) = 2u − 2u 3 which was considered by Lange [3], we will show that there exist exactly two solutions u 1,1 (x, ε) and u 1,2 (x, ε) to (1.1) & (1.2) such that…”

“…But, as was shown by Carrier and Pearson [1, p.202], a routine application of this method will not lead to the determination of the locations of the internal layers, thus creating spurious solutions. To overcome this difficulty, Lange [3] extended the method of matched asymptotics by including exponentially small terms in the expansion of the solution; see also MacGillivray [4]. There are two difficulties with Lange's approach; namely, (i) explicit expressions for the internal layer solutions must be known a priorily, (ii) second-order terms in the asymptotic expansions are needed to determine the layer positions of the leading order approximate solutions.…”

Singularly Perturbed Systems of Volterra Equations

“…This result is important because the method of additive decomposition can lead to spurious solutions (see for example Lange [8]). The method is illustrated in Section 4 by an example from Angell and Olmstead [1] and another example whose boundary layer stability condition fails.…”

Higher-order change of the Hamiltonian (energy) for nearly homoclinic orbits

“…First, we consider the failure as t -> +00. The correct scale is derived by assuming e -kt _ Q^e e kt^ as ^ _^ 00 This type of failure of an asymptotic expansion corresponds to shifting the time variable [Lange (1983)] by a logarithmically large amount, -^ Ine [Haberman (1983)]. The correct time variable in the saddle region after the nearly homoclinic orbit is T + : r+=t+ i ln£ - (…”