№ 1(101)/2021-Математика
https://rep.ksu.kz//handle/data/11137
Fri, 21 Jun 2024 03:39:34 GMT2024-06-21T03:39:34ZOn a New Class of Singular Integro-differential Equations
https://rep.ksu.kz//handle/data/11152
On a New Class of Singular Integro-differential Equations
Yuldashev, T.K.; Zarifzoda, S.K.
In this paper for a new class of model and non-model partial integro-differential equations with singularity
in the kernel, we obtained integral representation of family of solutions by aid of arbitrary functions.
Such type of integro-differential equations are different from Cauchy-type singular integro-differential
equations. Cauchy-type singular integro-differential equations are studied by the methods of the theory of
analytic functions. In the process of our research the new types of singular integro-differential operators
are introduced and main property of entered operators are learned. It is shown that the solution of studied
equation is equivalent to the solution of system of two equations with respect to x and y, one of which
is integral equation and the other is integro-differential equation. Further, non-model integro-differential
equations are studied by regularization method. This regularization method for non-model equation is
based on selecting and analysis of a model part of the equation and reduced to the solution of two second
kind Volterra type integral equations with weak singularity in the kernel. It is shown that the presence
of a non-model part in the equation does not affect to the general structure of the solutions. From here
investigation of the model equations for given class of the integro-differential equations becomes important.
In the cases, when the solution of given integro-differential equation depends on any arbitrary functions, a
Cauchy type problems are investigated.
Tue, 30 Mar 2021 00:00:00 GMThttps://rep.ksu.kz//handle/data/111522021-03-30T00:00:00ZOn Boundary Value Problems for a Mixed Type Fractional Differential Equation with Caputo Operator
https://rep.ksu.kz//handle/data/11151
On Boundary Value Problems for a Mixed Type Fractional Differential Equation with Caputo Operator
Yuldashev, T.K.; Islomov, B.I.; Ubaydullaev, U.Sh.
This article is devoted to study the boundary value problems of the first and second kind with respect
to the spatial variable for a mixed inhomogeneous differential equation of parabolic-hyperbolic type with
a fractional Caputo operator in a rectangular domain. In the study of such boundary value problems, we
abandoned the boundary value condition with respect to the first argument and instead it is used additional
gluing condition. In this case, in the justification of the unique solvability of the problems, the conditions
on the boundary domain are removed. This allowed us to weaken the criterion for the unique solvability of
boundary value problems under consideration. The solution is constructed in the form of Fourier series with
eigenfunctions corresponding to homogeneous spectral problems. Estimates for the convergence of Fourier
series are obtained as a regular solution of this mixed equation.
Tue, 30 Mar 2021 00:00:00 GMThttps://rep.ksu.kz//handle/data/111512021-03-30T00:00:00ZAn essential base of the central types of the convex theory
https://rep.ksu.kz//handle/data/11150
An essential base of the central types of the convex theory
Yeshkeyev, A.R.; Omarova, M.T.
In this paper, we consider the model-theoretical properties of the essential base of the central types of
convex theory. Also shows the connection between the center and Jonsson theory in permissible enrichment
signatures. Moreover, the theories under consideration are hereditary. This article is divided into 2 sections:
1) an essential types and an essential base of central types (in this case, the concepts of an essential type and
an essential base are defined using the Rudin-Keisler order on the set of central types of some hereditary
Jonsson theory in the permissible enrichment); 2) the atomicity and the primeness of '(x)-sets. In this
paper, new concepts are introduced: the '(x)-Jonsson set, the APA-set, the APA-existentially closed
model, the '(x)-convex theory, the '(x)-transcendental theory, the APA-transcendental theory. One of the
ideas of this article refers to the fact that in the work of Mustafin T.G. it was noticed that any universal
model of a quasi-transcendental theory with a strong base is saturated, but we generalized this result taking
into account that: the concept of quasi-transcendence will be replaced by the '(x)-transcendence, where
'(x) defines some Jonsson set; and the notion of a strong base is replaced by the notion of an essential
base, but in a permissible enrichment of the hereditary Jonsson theory. The main result of our work shows
that the number of fragments obtained under a closure of an algebraic or definable type does not exceed
the number of homogeneous models of a some Jonsson theory, which is obtained as a result of a permissible
enrichment of the hereditary Jonsson theory.
Tue, 30 Mar 2021 00:00:00 GMThttps://rep.ksu.kz//handle/data/111502021-03-30T00:00:00ZAn algebra of the central types of the mutually model-consistent fragments
https://rep.ksu.kz//handle/data/11149
An algebra of the central types of the mutually model-consistent fragments
Yeshkeyev, A.R.; Mussina, N.M.
In this paper, the model-theoretical properties of the algebra of central types of mutually model-consistent
fragments are considered. Also, the connections between the center and the Jonsson theory in the permissible
signature enrichment are shown, and within the framework of such enrichment, instead of some complete
theory under consideration, we can obtain some complete 1-type, and we will call this type the central
type, while the theories under consideration will be hereditary. Our work is divided into 3 sections: 1) the
outer and inner worlds of the existentially closed model of the Jonsson theory (and the feature between
these worlds is considered for two existentially closed models of this theory); 2) the -comparison of two
existentially closed models (the Schroeder-Bernstein problem is adapted to the study of Jonsson theories in
the form of a JSB-problem); 3) an algebra of central types (we carry over the results of Section 2 for the
algebra (Fr(C); ), where C is the semantic model of the theory T). Also in this article, the following new
concepts have been introduced: the outer and inner worlds of one existentially closed model of the same
theory (as well as the world of this model), a totally model-consistent Jonsson theory. The main result of
our work shows that the properties of the algebra of Jonsson theories for the product of theories are used
as an application to the central types of fixed enrichment. And it is easy to see from the definitions of the
product of theories and hybrids that these concepts coincide if the product of two Jonsson theories gives a
Jonsson theory.
Tue, 30 Mar 2021 00:00:00 GMThttps://rep.ksu.kz//handle/data/111492021-03-30T00:00:00Z