Geodesic
Asymptotic estimates for the solution of the Cauchy problem for a differential equation with linear degeneration
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh International Winter Mathematical School "Modern Methods of Function Theory and Related Problems", Voronezh, January 28 - February 2, 2021, Part 2, Tome 207 (2022), pp. 37-47

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Application of the method of separation of variables to problems for the linearly degenerate equation $u''_{xx}+yu''_{yy}+c(y)u'_y-a(x)u=f(x,y)$ in a rectangle leads to problems for the singularly perturbed ordinary differential equation with degeneration $yY''+c(y)Y'-(\pi^2k^2+a(y))Y=f_k(y)$, $k\in\mathbb{N}$. In this paper, we examine the asymptotic behavior of solutions of this equation with given initial data at $0$ and zero right-hand side as $k\to+\infty$ and obtain the leading term of the asymptotics in the explicit form.
Keywords: degenerate differential equation, singularly perturbed differential equation. 
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     author = {D. P. Emel'yanov and I. S. Lomov},
     title = {Asymptotic estimates for the solution of the {Cauchy} problem for a differential equation with linear degeneration},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {37--47},
     publisher = {mathdoc},
     volume = {207},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2022_207_a4/}
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D. P. Emel'yanov; I. S. Lomov. Asymptotic estimates for the solution of the Cauchy problem for a differential equation with linear degeneration. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh International Winter Mathematical School "Modern Methods of Function Theory and Related Problems", Voronezh, January 28 - February 2, 2021, Part 2, Tome 207 (2022), pp. 37-47. http://geodesic.mathdoc.fr/item/INTO_2022_207_a4/