Goncharuk A. Algebraic constructions in linear differential equations and in the theory of implicit linear difference equations

Українська версія

Thesis for the degree of Doctor of Philosophy (PhD)

State registration number

0823U101429

Thesis Registration Form

0823U101429.pdf

Applicant for

Specialization

  • 111 - Математика

Specialized Academic Board

ID 2915

V.N. Karazin Kharkiv National University

Essay

In the thesis linear differential equations of the n-th order over the a ring of formal power series with coefficients from some commutative rings and implicit difference equations of the n-th order over the the commutative ring. We investigate the existence and uniqueness of solutions of such equations and finding this solution. The question is completely solved for a differential equation with polynomial nonhomogeneity and an implicit difference equation with with finite nonhomogeneity. Sufficient conditions for the existence and uniqueness of a solution of a differential equation in the ring of formal power series with coefficients belonging to the complete valuation ring of a field with non-Archimedean valuation, where the inhomogeneity is not a polynomial, are formulated. We also found this solution in the form of the sum of a series converging under non-Archimedean normalization. The result is specified for an equation over a ring of integers. A special notion of convolution of a formal Laurent series with negative powers and a formal power series is introduced. Using this notion, we find some analogue of the fundamental solution of the operator for the considered equation, and show that, provided that the solution is unique and exists, it has the form of a convolution of the fundamental solution of the corresponding operator with inhomogeneity. Sufficient conditions for the existence and uniqueness of a solution of an implicit linear difference equation over some classes of complete rings, including the ring of p-adic numbers and the ring of formal power series, are formulated in the case when the inhomogeneity of the equation is not finite. We also find this solution in the form of the sum of a series converging under non-Archimedean valuation. For the case of an incomplete ring, under some conditions on the coefficients, it is proved that, provided a solution exists, this solution is unique and equal to the sum of the series under consideration. Sufficient conditions for the existence and uniqueness of a solution in the form of a formal power series for an implicit difference equation whose coefficients are polynomials are proved. For an implicit difference equation of the first order over a ring of polynomials, additional results are proved that allow us to find solutions to specific equations in the ring of polynomials or to prove that such solutions do not exist. We consider a linear implicit difference equation with inhomogeneity in the form of a quasi-polynomial with coefficients belonging to the ring, and find the conditions on the ring for which there exists a unique solution of such an equation. A first-order operator equation with a generalized left shift operator over a ring of integers is considered, for which the first-order differential and difference equations are special cases. For this equation, the criterion of existence and uniqueness of the solution is proved, and the solution is found in the form of the sum of a series converging under a-adic topology. All the equations considered in this thesis can be written in the form of an infinite linear system. It is shown that, under the conditions of uniqueness and existence of a solution to an equation, the solution of a system obtained by Cramer's rule coincides with the unique solution of this equation.

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Research papers

Гефтер, С. Л., Гончарук, А. Б., Півень, О. Л.: Цілочисельні розв'язки векторного неявного лінійного різницевого рівняння. Доповіді НАН України 11, 11–18 (2018) DOI: 10.15407/dopovidi2018.11.011

Goncharuk, A.: Implicit linear difference equation over a non-Archimedean ring. Visnyk of V. N. Karazin Kharkiv National University. Ser. Mathematics, Appl. Math. and Mech. 93, 18–33 (2021) DOI: 10.26565/2221-5646-2021-93-03

Goncharuk, A.: Cramer's rule for implicit linear differential equations over a non-Archimedean ring, Visnyk of V. N. Karazin Kharkiv National University. Ser. Mathematics, Appl. Math. and Mech. 95, 39–48 (2022)

Hefter, S. L., Goncharuk, A. B.: Linear Differential Equation with Inhomogeneity in the Form of a Formal Power Series Over a Ring with Non-Archimedean Valuation. Ukr Math J 74, 1463–1477 (2022) DOI: 10.1007/s11253-023-02163-0 (Scopus Q3)

Gefter, S., Goncharuk A.: The generalized backward shift operator on Z[[x]], Cramer's formula for solving infinite linear systems, and p-adic integers. In: Book of Abstracts of V International Conference ``Analisis and mathematical physics'' dedicated to Vladimir A. Marchenko’s 95th birthday, Kharkiv, Ukraine (2017) DOI: 10.13140/RG.2.2.24135.80805

Goncharuk A.: The generalized backward shift operator on Z[[x]], Cramer's formulas for solving infinite linear systems, and p-adic integers. In: Book of abstracts of The 28th International Workshop on Operator Theory and its Applications (IWOTA), Chemnitz, Germany, pp. 57-58 (2017)

Goncharuk A.: Implicit linear differential equation over the ring of polynomials. Збірник тез доповідей XV Міжнародної наукової конференції студентів та молодих вчених «Сучасні проблеми математики та її застосування в природничих науках та інформаційних технологіях», Харків, с. 5 (2020)

Goncharuk A., Gefter S.: Non-homogeneous implicit linear differential equation over the ring of formal power series. Збірник тез доповідей Міжнародної конференції молодих математиків, Київ, с. 50 (2021)

Goncharuk A.: Implicit difference equation over the ring of polynomials. In: Book of abstracts of Conference on Rings and Polynomials, Graz, Austria, p. 29 (2021)

Goncharuk, A., Gefter, S., Piven', A.: Implicit linear difference equations over commutative rings. In: Book of Abstracts of The 26th International Conference on Difference Equations and Applications (ICDEA 2021), Sarajevo, Bosnia and Herzegovina, p. 154 (2021)

Gefter, S., Goncharuk, A.: Linear differential equations in the ring of formal power series over a topologicl ring. Збірник тез Міжнародної конференції з комплексного і функціонального аналізу, присвяченої пам'яті Богдана Винницького, Дрогобич, с. 19 (2021)

Gefter, S., Goncharuk, A.: Linear differential equations in the ring of formal power series. In: Book of Abstracts of The 5-th International Conference ``Differential Equations and Control Theory'' (DECT 2021), Kharkiv, p. 19 (2021)

Gefter, S. L., Goncharuk, A. B., Piven', A. L.: Quasi-polynomial solutions of implicit linear difference equations over a local commutative ring. In: Book of Abstracts of The International online conference ``Current trends in abstract and applied analysis'', Ivano-Frankivsk, p. 32 (2022)

Gefter, S., Goncharuk, A., Piven', A.: Periodic and quasi-polynomial solutions of implicit linear difference equations over commutative rings. In: Book of Abstracts of The 27th International Conference on Difference Equations and Applications, Paris, p. 137 (2022)

Gefter, S., Goncharuk, A.: Generalized backward shift operators on the ring Z[[x]], Cramer's rule for infinite linear systems, and p-adic integers. In: Bottcher, A., Potts, D., Stollmann, P., Wenzel, D. (eds) The Diversity and Beauty of Applied Operator Theory. Birkhauser, Cham. pp. 247–259 (2018) DOI: 10.1007/978-3-319-75996-8_13

Gefter, S., Goncharuk, A., Piven', A.: Implicit Linear First Order Difference Equations Over Commutative Rings. In: Elaydi, S., Kulenovic, M.R.S., Kalabusic, S. (eds) Advances in Discrete Dynamical Systems, Difference Equations and Applications. ICDEA 2021. Springer Proceedings in Mathematics & Statistics, Springer, Cham. pp. 199–216 (2023) DOI: 10.1007/978-3-031-25225-9_10

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Files

Goncharuk_diss.pdf

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