Geodesic
Asymptotics of the solution of a nonlinear Cauchy problem
Čelâbinskij fiziko-matematičeskij žurnal, Tome 1 (2016) no. 1, pp. 43-51

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The solution uniform asymptotics of the initial value problem for equation $\varepsilon u '= x^2-u^2 + \varepsilon f (x)$ singularly depending on small parameter $ \varepsilon $ is considered. The equation contains the unexplored case of the right-hand side, though equations of this type are well studied. The three-scale solution asymptotic expansion is constructed by the matching method, justificated by the upper and lower solutions method.
Keywords: asymptotic expansion, small parameter, initial value problem, matching method, intermediate expansion. 
@article{CHFMJ_2016_1_1_a4,
     author = {J. A. Krutova},
     title = {Asymptotics of the solution of a nonlinear {Cauchy} problem},
     journal = {\v{C}el\^abinskij fiziko-matemati\v{c}eskij \v{z}urnal},
     pages = {43--51},
     publisher = {mathdoc},
     volume = {1},
     number = {1},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHFMJ_2016_1_1_a4/}
}
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J. A. Krutova. Asymptotics of the solution of a nonlinear Cauchy problem. Čelâbinskij fiziko-matematičeskij žurnal, Tome 1 (2016) no. 1, pp. 43-51. http://geodesic.mathdoc.fr/item/CHFMJ_2016_1_1_a4/