Geodesic
Asymptotics of solution of the Riccati equation
Čelâbinskij fiziko-matematičeskij žurnal, Tome 1 (2016) no. 2, pp. 59-67

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Uniform asymptotics is found for a solution of the initial value problem to the equation $ \varepsilon^2 u '= -u^2 + \varepsilon f (x) $, singularly depending on a small parameter $\varepsilon$. Equations of this type are already well studied, but this equation represents an unexplored case of the right-hand side behavior. By the method of asymptotics matching the three-scale asymptotic expansion for a solution is constructed and is justificated by the method of upper and lower solutions.
Keywords: asymptotic expansion, small parameter, initial value problem, asymptotics matching method, intermediate expansion, Riccati equation. 
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     author = {M. I. Rusanova},
     title = {Asymptotics of  solution of the {Riccati} equation},
     journal = {\v{C}el\^abinskij fiziko-matemati\v{c}eskij \v{z}urnal},
     pages = {59--67},
     publisher = {mathdoc},
     volume = {1},
     number = {2},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHFMJ_2016_1_2_a5/}
}
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M. I. Rusanova. Asymptotics of  solution of the Riccati equation. Čelâbinskij fiziko-matematičeskij žurnal, Tome 1 (2016) no. 2, pp. 59-67. http://geodesic.mathdoc.fr/item/CHFMJ_2016_1_2_a5/