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Item Open Access Grobner-Shirshov bases theory for Zinbiel stperalgebras(Faculty of Engineering and Natural Sciences, 2023) Meiirbek K.This thesis is a collection of 6 chapters .The Grébner-Shirshov basis is an impor- tant mathematical apparatus in algebra and commutative algebra, which is used to study and analyze polynomials and their ideals. The Grdébner-Shirshov basis has a number of important properties that make it a powerful tool for solving various algebraic problems, such as searching for ideals, solving systems of equa- tions and determining the basic invariants of polynomials. In this paper we will construct a Grobner-Shirshov basis for Zinbiel algebras. Algebra with the identity (ab) c = a(bc) + a(cb) is called the Zinbiel algebra. In the process of construct- ing the Grebner-Shirshov basis, two compositions are found and the composition lemma is proved. The method of mathematical induction is used to prove the lemma.Item Open Access Laguerre polynomials in axisymmetric heat problems with a free boundary(Faculty of engineering and natural sciences, 2019) Jabbarkhanov Kh.The aim of the thesis is to consider solving heat equation with free boundaries using by heat polynomials method, in particular, using by Laguerre polynomials. There are two problems were considered. It is spherical inverse and direct problems the mcthod of thermal polynomials is appropriate. As exactly as the approximate solutions. The inverse two-phase spherical Stefan problem for unknown boundary heat. flux is solved by the method of the heat polynomials. Side by side with exact solution two methods for the approximate solution, collocation and variational methods, convenient for engineering applications are presented and compared.