Geodesic
The asymptotics of solutions of a singularly perturbed equation with a of fractional turning point
The Bulletin of Irkutsk State University. Series Mathematics, Tome 21 (2017), pp. 108-121 Cet article a éte moissonné depuis la source Math-Net.Ru

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We develop the classical Vishik–Lyusternik–Vasil'eva–Imanaliev boundary-value method for constructing uniform asymptotic expansions of solutions of singularly perturbed equations with singular points. In this paper, by modernizing the classical method of boundary functions, uniform asymptotic expansions of solutions of singularly perturbed equations with a fractional turning point are constructed. As we know, problems with turning points are encountered in the Schrodinger equation for the tunnel junction, problems with a classical oscillator, problems of continuum mechanics, the problem of hydrodynamic stability, the Orr–Sommerfeld equation, and also in the determination of heat to a pipe, etc. Determination of the behavior of solving similar problems with aspiration small (large) parameter to zero (to infinity) is an actual problem. We study the Cauchy and Dirichlet problems for singularly perturbed linear inhomogeneous ordinary differential equations of the first and second order, respectively. Here it is proved that the principal terms of the asymptotic expansions have negative fractional powers with respect to a small parameter. As practice shows, solutions to most singularly perturbed equations with singular points have this property. The constructed decompositions of the solutions are asymptotic in the sense of Erdey, when the small parameter tends to zero. Estimates for the remainder terms of the asymptotic expansions are obtained. The asymptotic expansions are justified. The idea of modifying the method of boundary functions is realized for ordinary differential equations, but it can also be used in constructing the asymptotic of the solution of singularly perturbed partial differential equations with singularities.
Keywords: singularly perturbed, turning point, bisingular problem, Cauchy problem, Dirichlet problem. 
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D. A. Tursunov; K. G. Kozhobekov. The asymptotics of solutions of a singularly perturbed equation with a of fractional turning point. The Bulletin of Irkutsk State University. Series Mathematics, Tome 21 (2017), pp. 108-121. http://geodesic.mathdoc.fr/item/IIGUM_2017_21_a7/

[1] Alymkulov K., Tursunov D. A., “On a method of construction of asymptotic decompositions of bisingular perturbed problems”, Russian Mathematics, 60:12 (2016), 1–8 | DOI | MR | Zbl

[2] Bobochko V. N., “An Unstable Differential Turning Point in the Theory of Singular Perturbations”, Russian Mathematics, 49:4 (2005), 6–14 | MR | Zbl

[3] Bobochko V. N., “Uniform Asymptotics of a Solution of an Inhomogeneous System of Two Differential Equations with a Turning Point”, Russian Mathematics, 50:5 (2006), 6–16 | MR | Zbl

[4] Zimin A. B., “The Cauchy problem for a second order linear equation with small parameter which degenerates in the limit to an equation with singular points”, Differential Equations, 5:9 (1969), 1583–1593 (in Russian) | MR | Zbl

[5] Ilin A. M., Matching of asymptotic expansions of solutions of boundary value problems, AMS, Providence, Rhode Island, 1992, 296 pp. | MR | MR | Zbl

[6] Ilin A. M., Danilin A. R., Asymptotic Methods in Analysis, Fizmatlit, M., 2009, 248 pp. (in Russian)

[7] Lomov S. A., Introduction to the General Theory of Singular Perturbations, AMS, Providence, Rhode Island, 1992 | MR | MR | Zbl

[8] Tursunov D. A., “Asymptotic expansion for a solution of an ordinary second-order differential equation with three turning points”, Tr. IMM UrO RAN, 22, no. 1, 2016, 271–281 (in Russian)

[9] Tursunov D. A., “The asymptotic solution of the bisingular Robin problem”, Sib. Electron. Mat. Reports, 14 (2017), 10–21 (in Russian) | DOI | MR | Zbl

[10] Fedoryuk M. V., Asymptotic analysis: linear ordinary differential equations, Springer-Verlag, Berlin, 1993, 363 pp. | DOI | MR | MR | Zbl

[11] J. D. Cole, Perturbation Methods in Appled Mathematics, Blaisdell, Waltham, MA, 1968 | MR

[12] V. Ekhaus, Matched Asymptotic Expansions and Singular Perturbation, North-Holland, Amsterdam, 1973 | MR

[13] A. Fruchard, R. Schafke, Composite Asymptotic Expansions, Springer-Verlag, Berlin–Heidelberg, 2013 | DOI | MR | Zbl

[14] W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Dover publications, INC, Mineola, N.Y., 1965 | MR

[15] W. Wasow, Linear turning point theory, Springer-Verlag, N. Y., 1985 | DOI | MR | Zbl

[16] A. M. Watts, “A singular perturbation problem with a turning point”, Bull. Austral. Math. Soc., 5 (1971), 61–73 | DOI | MR | Zbl