Professor of the Institute of Natural Sciences and Mathematics Valery Karachik has built a solution of the Neumann problem in the unit ball. This and other important results in the field of partial differential equations, obtained jointly with Russian and Chinese colleagues, have been published in top-ranking journals and sparked interest among the complex-analysis specialists.
His article on the “Solvability of the Neumann Boundary Value Problem for the Polyharmonic Equation in a Ball” has been released in the Lobachevskii Journal of Mathematics (Q2), which is being published in English in Kazan.
It covers the topic of partial differential equations. The Neumann problem is a type of a boundary value problem when the boundary conditions are set for normal derivatives of unknown function on the boundary of a defined area. An inhomogeneous polyharmonic equation generalizes the notion of an inhomogeneous harmonic equation, or Poisson's equation, only the Laplacian in this equation is of higher order. The Neumann problem was formulated by a famous Soviet mathematician A.V. Bitsadze in the 80s of the last century.
The main result of this article is finding the conditions of solvability of the Neumann problem, and in case these conditions are met – building of an unknown solution. The method of studying this problem includes converting the Neumann boundary value problem in the unit ball into a relevant classical Dirichlet problem, the solution of which was obtained by the author earlier.
Two other articles by Valery Karachik have been published in the Mathematics journal from MDPI (ТОР 5%). The first article, written jointly with his colleagues from China Hongfen Yuan, Dangting Wang and Tieguo Ji, is titled “On the Growth Orders and Types of Biregular Functions”. The second article is titled “Solutions of Umbral Dirac-Type Equations” and has been published jointly with Hongfen Yuan.
These papers cover the topics of not only differential equations, but also those of the complex analysis, the theory of Clifford algebras generalizing complex numbers, as well as the Umbral calculus (combinatorics and polynimials).
The first article studies the asymptotic growth of biregular functions and gives a method for evaluating it using the Taylor series. In the second article, normalized systems of functions with respect to the Dirac operator in the umbral Clifford analysis are constructed.
In the AIMS Mathematics journal (Q1 as per SCIE), jointly with his colleague from Kazakhstan, Batirkhan Turmetov, an article on the “On solvability of some inverse problems for a nonlocal fourth-order parabolic equation with multiple involution” has been published. It solves two inverse boundary value problems for a nonlocal fourth-order parabolic equation with multiple involution.
In the Boundary Value Problems journal (Q1 as per SCIE), jointly with his colleagues from Kazakhstan, Batirkhan Turmetov and Moldir Muratbekova, an article on the “Bitsadze-Samarsky type problems with double involution” has been published. It examines a new class of nonlocal boundary value problems for the Poisson's equation. Nonlocal conditions are set as the links between the values of unknown function in various points of the area boundary.
Doctor of Sciences (Physics and Mathematics) Valery Karachik is well known as a specialist in the field of mathematical and complex analysis, and his Hirsch index in Scopus is 17. At SUSU, he has been working as the Professor at the Department of Mathematical Analysis and Mathematics Education for more than 20 years now.
References
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MDPI (ТОР-5%) Hongfen Yuan; Valery Karachik, “Solutions of Umbral Dirac-Type Equations”, Mathematics, 12:2 (2024) https://doi.org/10.3390/math12020344
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MDPI (ТОР-5%) H. Yuan, V. Karachik, D. Wang and T. Ji, “On the Growth Orders and Types of Biregular Functions”, Mathematics, 12 (2024), 3804 https://doi.org/10.3390/math12233804
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(Q1 in SCIE edition) Kh. Turmetov, V. V. Karachik, “On solvability of some inverse problems for a nonlocal fourth-order parabolic equation with multiple involution”, AIMS Mathematics, 9:3 (2024), 6832–6849 https://doi.org/10.3934/math.2024333
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(Q1 in SCIE edition) M. Muratbekova, V. Karachik, B. Turmetov, “Bitsadze-Samarsky type problems with double involution”, Boundary Value Problems, 2024:86 (2024), 1-21 http://dx.doi.org/10.1186/s13661-024-01892-w
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(Q2 in SCIE edition) V. Karachik, “Solvability of the Neumann Boundary Value Problem for the Polyharmonic Equation in a Ball”, Lobachevskii Journal of Mathematics, 45:8 (2024), 3559–3571 http://dx.doi.org/10.1134/S1995080224604296