Geodesic
On one method for the analysis of the Cauchy problem for a singularly perturbed inhomogeneous second-order linear differential equation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 10, pp. 1661-1675

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A sequence converging to the solution of the Cauchy problem for a singularly perturbed inhomogeneous second-order linear differential equation is constructed. This sequence is also asymptotic in the sense that the deviation (in the norm of the space of continuous functions) of its nth element from the solution of the problem is proportional to the $(n+1)$th power of the perturbation parameter. A similar sequence is constructed for the case of an inhomogeneous first-order linear equation, on the example of which the application of such a sequence to the justification of the asymptotics obtained by the method of boundary functions is demonstrated. 
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     author = {E. E. Bukzhalev},
     title = {On one method for the analysis of the {Cauchy} problem for a singularly perturbed inhomogeneous second-order linear differential equation},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1661--1675},
     publisher = {mathdoc},
     volume = {57},
     number = {10},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_10_a5/}
}
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E. E. Bukzhalev. On one method for the analysis of the Cauchy problem for a singularly perturbed inhomogeneous second-order linear differential equation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 10, pp. 1661-1675. http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_10_a5/