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Note on Logarithmic Switchback Terms in Regular and Singular Perturbation Expansions

Lagerstrom, P. A. and Reinelt, D. A. (1984) Note on Logarithmic Switchback Terms in Regular and Singular Perturbation Expansions. SIAM Journal on Applied Mathematics, 44 (3). pp. 451-462. ISSN 0036-1399. doi:10.1137/0144030.

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The occurrence of logarithmic switchback is studied for ordinary differential equations containing a parameter k which is allowed to take any value in a continuum of real numbers and with boundary conditions imposed at x = ε and x = ∞. Classical theory tells us that if the equation has a regular singular point at the origin there is a family of solutions which varies continuously with k, and the expansion around the origin has log x terms for a discrete set of values of k. It is shown here how nonlinearity enlarges this set so that it may even be dense in some interval of the real numbers. A log x term in the expansion in x leads to expansion coefficients containing log ε (switchback) in the perturbation expansion. If for a given value of k logarithmic terms in x and ε occur they may be obtained by continuity from neighboring values of k. Switchback terms occurred conspicuously in singular-perturbation solutions of problems posed for semi-infinite domain x ≥ ε. This connection is historical rather than logical. In particular we study here switchback terms for a specific example using methods of both singular and regular perturbations.

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Additional Information:©1984 Society for Industrial and Applied Mathematics. Received by the editors August 18, 1983. This author [P.A.L.] gratefully acknowledges the help and hospitality of the Fondation des Treilles.
Issue or Number:3
Record Number:CaltechAUTHORS:LAGsiamjam84
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:11045
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Deposited On:24 Jun 2008
Last Modified:08 Nov 2021 21:13

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