Abstract
Multiple-scale and averaging methods have a broad range of applicability for systems of ordinary differential equations, as discussed in Chapters 4 and 5. In contrast, asymptotic solution techniques for partial differential equations are more recent and may be implemented, in general, only with multiple-scale expansions. Some of the early use of multiple scales concerned problems where the unperturbed state has a simple, usually periodic, structure, and the leading effect of weak nonlinearities is to introduce a slow modulation of the parameters. A number of representative examples are discussed in Sec. 6.1. In Sec. 6.2, we study systems of conservation laws that are perturbed about a uniform state. The ideas are illustrated using examples from shallow water flow, gas dynamics, and other applications. The final section, 6.3, gives a brief introduction to multiple-scale homogenization.
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Kevorkian, J., Cole, J.D. (1996). Multiple-Scale Expansions for Partial Differential Equations. In: Multiple Scale and Singular Perturbation Methods. Applied Mathematical Sciences, vol 114. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-3968-0_6
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DOI: https://doi.org/10.1007/978-1-4612-3968-0_6
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