Abstract
We continue here the study of asymptotic expansions started in Chapter 2. Our results are inspired by the Vasil’eva theorem providing, for the Tikhonov system
an asymptotic expansion for both variables, uniform on the whole interval [0, T]. This expansion has the form
where \(\tilde x_t^{k,\varepsilon } = \tilde x_{t/\varepsilon }^k,\tilde y_{t/\varepsilon }^k\). The essential property of the “boundary layer functions” \({\tilde x^k}\) and \({\tilde x^k}\) is that they are exponentially decreasing at infinity and this requirement allows us to define them uniquely, using a rather simple algorithm (but calculations are tedious).
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© 2003 Springer-Verlag Berlin Heidelberg
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Kabanov, Y., Pergamenshchikov, S. (2003). Uniform Expansions for Two-Scale Systems. In: Two-Scale Stochastic Systems. Applications of Mathematics, vol 49. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-13242-5_5
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DOI: https://doi.org/10.1007/978-3-662-13242-5_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-08467-6
Online ISBN: 978-3-662-13242-5
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