Email this article 
ON THE STUDY OF VARIOUS REPRESENTATIONS OF SOLUTIONS OF QUASI-DIFFERENTIAL EQUATIONS IN THE FORM OF SUMS OF SERIES AND THEIR SOME APPLICATIONS
- Authors: Vatolkin M.Y.1
-
Affiliations:
- M.T. Kalashnikov Izhevsk State Technical University
- Issue: Vol 64, No 11 (2024)
- Pages: 2058-2076
- Section: Ordinary differential equations
- URL: https://journals.rcsi.science/0044-4669/article/view/277200
- DOI: https://doi.org/10.31857/S0044466924110041
- EDN: https://elibrary.ru/KGSNLM
- ID: 277200
Cite item
Full Text
Abstract
In the theory of the matrix equation _𝑋 = 𝐴(𝑡)𝑋, the well-known representation of the solution in the form of a series plays an important role. In the case of an 𝑛-th order scalar equation, it is also interesting to get a representation of the solution in the form of a scalar series, the terms of which are constructed from the coefficients of the original equation. Various representations of the fundamental system of solutions of a homogeneous quasi-differential equation of the 𝑛-th order in the form of scalar series, the terms of which are constructed from the coefficients of the original equation, are studied. As an example, in the form of such series, representations of the elements of the fundamental system of solutions of the Bessel equation are constructed, considered in the interval [#,+∞), where # > 0.
About the authors
M. Yu. Vatolkin
M.T. Kalashnikov Izhevsk State Technical University
Email: vmyu6886@gmail.com
Izhevsk, Russia
References
- Шин Д.Ю. О решениях линейного квазидифференциального уравнения -го порядка // Матем. сборник. 1940. Т. 7(49). №3. С. 479–532.
- Шин Д.Ю. О квазидифференциальных операторах в гильбертовом пространстве // Матем. сборник. 1943. Т. 13(55). №1. С. 39–70.
- Everitt W.N., Marcus L. Boundary value problems and symplectic algebra for ordinary differential and quasidifferential operators // Amer. Math. Soc. 1999. V. 61.
- Eckhardt J., Gestezy F., Nichols R., Teschl G. Weyl–Titchmarsh theory for Sturm–Liuville operators with distributional potentials // Opuscula Mathematica. 2013. V. 33(3). P. 467–563.
- Everitt W.N., Race D. The regular representation of singular second order differential expressions using quasiderivatives // Proc. London Math. Soc. (3). 1992. V. 65(2). P. 383–404.
- Xiao xia Lv, Ji-jun Ao, Zettl A. Dependence of eigenvalues of fourth-order differential equations with discontinuous boundary conditions on the problem // J. Math. Anal. Appl. 2017. V. 456(1). P. 671–685.
- Qinglan Bao, Jiong Sun, Xiaoling Hao, Zettl A. Characterization of self-adjoint domains for regular even order Csymmetric differential operators // Electronic J. of Qualitative Theory of Diff. Equat. 2019. V. 62. P. 1–17.
- Zettl A. Sturm-Liouville Theory. Amer. Math. Soc., 2005.
- Zett A. Recent Developments in Sturm-Liouville Theory. Berlin, Boston: De Gruyter, 2021.
- Jianfang Qin, Kun Li, Zhaowen Zheng, Jinming Cai Dependence of eigenvalues of discontinuous fourthorder differential operators with eigenparameter dependent boundary conditions // J. of Nonlinear Math. Phys. 2022. V. 29(4). P. 776–793.
- Таций Р.М., Стасюк М.Ф., Мазуренко В.В., Власий О.О. Обобщенные квазидифференциальные уравнения. Львов: ЛГУ БЖД, 2017.
- Владыкина В.Е., Шкаликов А.А. Асимптотика решений уравнения Штурма–Лиувилля с сингулярными коэффициентами // Матем. заметки. 2015. Т. 98. №6. С. 832–841.
- Мирзоев К.А. Операторы Штурма–Лиувилля // Тр. ММО. 2014. Т. 75. №2. С. 335–359.
- Конечная Н.Н., Мирзоев К.А. Главный член асимптотики решений линейных дифференциальных уравнений с коэффициентами-распределениями первого порядка // Матем. заметки. 2019. Т. 106. №1. С. 74–83.
- Конечная Н.Н., Мирзоев К.А. Об асимптотике решений линейных дифференциальных уравнений нечетного порядка // Вестн. Моск. ун-та. 2020. Сер. 1. Матем., мех. №1. С. 23–28.
- Дерр В.Я. Неосцилляция решений линейного квазидифференциального уравнения // Известия Института математики и информатики УдГУ. 1999. №1(16). С. 3–105.
- Дерр В.Я. Об адекватном описании сопряженного оператора // Вестник Удмуртского университета. Математика. Механика. Компьютерные науки. 2011. №3. С. 43–63.
- Наймарк М.А. Линейные дифференциальные операторы. М.: Наука, 1969.
- Nehary Z. Disconjugate linear differential operators // Trans. Amer. Math. Soc. 1969. V. 129. P. 500–516.
- Elias U. The extremal solutions of the equation 𝐿𝑦 + (𝑥)𝑦 = 0, II // J. of Math. Anal. and Appl. 1976. V. 55. P. 253–265.
- Everitt W.N. Fourth order singular differential equation // J. Math. Anal. 1963. V. 142. P. 320–340.
- Kusano T., Naito M. Oscillation criteria of general linear ordinary differential equations // Pacific J. Math. 1981. V. 92(2). P. 345–355.
- Trench W.F. Canonical forms and principal systems for general disconjugate equations // Trans. Amer. Math. Soc. 1974. 189. P. 319–327.
- Trench W.F. Asymptotic theory perturbed general disconjugate equations // Hiroshima Math. J. 1982. V. 12. P. 43–58.
- Everitt W.N. A note of linear ordinary quasi-differential equations // Proc. of the Royal Soc. of Edinburgh. 1985. V. 101A. P. 1–14.
- Everitt W.N., Key J.D. On some properties of matrices associated with linear ordinary quasi-differential equations // Proc. of the Royal Soc. of Edinburgh. 1984. V. 96A. P. 211–219.
- Nicolson L.S. Disconjugate systems of linear differential equation // J. of Differential Equat. 1970. 7. P. 570–583.
Supplementary files
