Abstract
In constructing the asymptotic expansions of slow integral manifolds it is assumed that the degenerate equation (\(\varepsilon = 0\)) allows one to find the slow surface explicitly. In many problems this is not possible due to the fact that the degenerate equation is either a high degree polynomial or transcendental. In this situation many authors suggest the use of numerical methods. However, in many problems the slow surface can be described in parametric form, and then the slow integral manifold can be found in parametric form as asymptotic expansions. If this is not possible, it is necessary to use an implicit slow surface and obtain asymptotic representations for the slow integral manifold in an implicit form. Model examples, as well as examples borrowed from combustion theory, are treated.
Keywords
- Asymptotic Expansion
- Order Approximation
- Parametric Form
- Invariant Manifold
- Slow Variable
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References
Gorelov, G.N., Sobolev, V.A.: Mathematical modeling of critical phenomena in thermal explosion theory. Combust. Flame 87, 203–210 (1991)
Mortell, M.P., O’Malley, R.E., Pokrovskii, A., Sobolev, V.A. (eds.): Singular Perturbation and Hysteresis. SIAM, Philadelphia (2005)
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© 2014 Springer International Publishing Switzerland
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Shchepakina, E., Sobolev, V., Mortell, M.P. (2014). Representations of Slow Integral Manifolds. In: Singular Perturbations. Lecture Notes in Mathematics, vol 2114. Springer, Cham. https://doi.org/10.1007/978-3-319-09570-7_4
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DOI: https://doi.org/10.1007/978-3-319-09570-7_4
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Publisher Name: Springer, Cham
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