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What is Singular Perturbation mean?
In mathematics, a singular perturbation problem is a problem containing a small parameter that cannot be approximated by setting the parameter value to zero. More precisely, the solution cannot be uniformly approximated by an asymptotic expansion
φ ( x ) ≈ ∑ n = 0 N δ n ( ε ) ψ n ( x ) {\displaystyle \varphi (x)\approx \sum _{n=0}^{N}\delta _{n}(\varepsilon )\psi _{n}(x)\,}as ε → 0 {\displaystyle \varepsilon \to 0} . Here ε {\displaystyle \varepsilon } is the small parameter of the problem and δ n ( ε ) {\displaystyle \delta _{n}(\varepsilon )} are a sequence of functions of ε {\displaystyle \varepsilon } of increasing order, such as δ n ( ε ) = ε n {\displaystyle \delta _{n}(\varepsilon )=\varepsilon ^{n}} . This is in contrast to regular perturbation problems, for which a uniform approximation of this form can be obtained. Singularly perturbed problems are generally characterized by dynamics operating on multiple scales. Several classes of singular perturbations are outlined below.
The term "singular perturbation" was coined in the 1940s by Kurt Otto Friedrichs and Wolfgang R. Wasow.
referencePosted on 21 Nov 2024, this text provides information on Miscellaneous in Physics Related related to Physics Related. Please note that while accuracy is prioritized, the data presented might not be entirely correct or up-to-date. This information is offered for general knowledge and informational purposes only, and should not be considered as a substitute for professional advice.
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