Bulletin of the Institute of Mathematics

Author: Abdullaev O. (Alfraganus University), Duysenbaev R. (Alfraganus University)

Abstract:
The present research is devoted to studying a unique solvability of an inverse problem for a degenerate parabolic-hyperbolic equation involving the Riemann-Liouville fractional derivative in the time variable. In the first step give a solution to the Dirichlet problem for an ordinary differential equation with involution perturbation. Further, using general solution of a wave equation in hyperbolic domain we find right-hand side of a considered equation. The solution of the main problem in a parabolic domain is represented in series form and using the explicit form of the corresponding Cauchy problem, certain properties of the Kilbas-Saigo function, and imposing certain conditions on given functions, we have proved the existence of a unique solution.

Keywords: Inverse problem; degenerate equation; mixed-type equation; involution perturbation; uniqueness and existence of solution; Fourier series.

Author: Bekiev A. (Karakalpak State University), Erejepova Sh. (Karakalpak State University)

Abstract:
In this paper, an initial-boundary value problem for a fourth-order equation with variable coefficients and a loaded term in a rectangular domain is considered. The solution of the problem is constructed in the form of a series with respect to the eigenfunctions of the corresponding spectral problem. The eigenfunctions of the corresponding spectral problem form a complete system and constitute a Riesz basis in L_2(0,1). The uniqueness of the solution follows from the completeness of the system of eigenfunctions. Sufficient conditions on the initial functions are established to guarantee the existence and stability of the solution. The absolute and uniform convergence of the solution series, as well as of the series obtained by termwise differentiation with respect to t and x two and four times, respectively, is proved. The stability of the solution is also established.

Keywords: Fourth-order equation; loaded equation; classical solution; method of separation of variables; uniform convergence of the solution; uniqueness; existence; stability of the solution.

Author: Berdalova Kh. (V.I.Romanovskiy Institute of Mathematics Uzbekistan Academy of Sciences), Jumaniyozov D. (National University of Uzbekistan),(V.I.Romanovskiy Institute of Mathematics Uzbekistan Academy of Sciences)

Abstract:
This paper presents a complete classification of all complex Lie affgebra structures on the three-dimensional Heisenberg algebra \mathfrak{n}_3. Using the characterization of Lie affgebras as Lie algebras equipped with generalized derivations and distinguished elements, we determine the pairs of linear transformations (f,g) satisfying the fundamental compatibility condition that makes the Lie bracket
{x,y} = [x,y] + g(x) + f(y-x) + s into a valid Lie affgebra structure. Through systematic analysis of isomorphism classes under the action of the automorphism group of \mathfrak{n}_3, we identify 62 distinct parametric families of pairwise non-isomorphic Lie affgebras associated with the Heisenberg algebra. This result completes the classification of Lie affgebras with three-dimensional nilpotent Lie algebra fibers.

Keywords: Lie algebras; Lie affgebras; Heisenberg algebra; generalized derivations.

Author: Beshimova Sh. (National University of Uzbekistan)

Abstract:
We study the derivation algebra of the nilpotent Leibniz algebra NF_n\otimes\mathbb F[t]/(t^{m+1}) and subsequently describe all solvable Leibniz extensions. In particular, we prove that no two-dimensional solvable Leibniz extensions exist, and we provide a description of the one-dimensional solvable Leibniz extensions.

Keywords: Leibniz algebra; associative algebra; nilpotent algebra; solvable algebra; tensor products of algebras.

Author: Durdiev U. (Bukhara State University), Odinaev R. (Bukhara State University)

Abstract:
We study the solvability of the initial-boundary value problem for a fourth-order differential equation. The solution is constructed by the Fourier method using the eigenfunctions corresponding to the boundary conditions. This reduces the original fourth-order problem to a second-order equation with conditions involving higher-order derivatives. For the resulting formulation, we derive an equivalent second-order Volterra integral equation. Applying standard results from the theory of integral equations, we prove that this Volterra equation admits a solution. Finally, under appropriate smoothness assumptions on the given data, we conclude that the initial–boundary value problem possesses a classical solution.

Keywords: Integral equation; existence; uniqueness; Fourier method; spectral problem.

Author: Irgashev B. (University of Business and Science)

Abstract:
For a third-order equation involving the second order derivative with a discontinuous coefficient, a Dirichlet-type problem is investigated. Existence and uniqueness theorems are proved. In the course of the study, the problem of “small denominators” arises; the rate at which they tend to zero is determined for certain values of parameters of the studied problem.

Keywords: Equation; third order; conjugation problem; small denominators; existence; uniqueness.

Author: Khakimov O. (V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences), Toshpulatova M. (V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences)

Abstract:
In this paper, we study quantum Markov chains associated with a quantum d-state Potts model on the semi-infinite Cayley tree of order two. The interaction Hamiltonian consists of the classical Potts interaction and a quantum exchange term depending on the parameter \lambda. For \lambda=0, the model reduces to the classical Potts model. Therefore, the main focus of the paper is the genuinely quantum case \lambda\neq0.
For symmetric translation-invariant boundary conditions, we derive the corresponding compatibility equation and prove that this class of boundary conditions is invariant under the compatibility map.
The resulting fixed point equation is analysed explicitly for the two-state and three-state cases. In the two-state case, we obtain a complete description of the parameter regions in which either one or three symmetric translation-invariant quantum Markov chains exist.
In the three-state case, we prove that for \lambda<0 there are no nontrivial symmetric translation-invariant quantum Markov chains. For \lambda>0, the classification within the considered class is expressed in terms of the parameters
z=\frac{e^{\beta J}}{\sinh(\beta\lambda)}, \qquad q=\coth(\beta\lambda),
and a critical value q_*. Depending on the relative positions of q and z, we show that the model admits exactly one, two, or three symmetric translation-invariant quantum Markov chains.

Keywords: Quantum Potts model; quantum Markov chain; Cayley tree; translation-invariant boundary condition; compatibility equation; fixed point equation.

Author: Khamrayev A. (Karshi State University), Daminova M. (Karshi State University)

Abstract:
In this paper, we propose a systematic method for determining the inheritance coefficients that govern transitions between population states and guarantee that the resulting operator maps the simplex of probability distributions into itself. Based on the SOS model, we construct a Volterra quadratic stochastic operator and derive its inheritance coefficients. We then describe the invariant sets and fixed points of the operator and characterize the set of limit points of every orbit.

Keywords: Construction; quadratic stochastic operator; Volterra operator; regular operator.

Author: Kodiraliev A. (Fergana State University)

Abstract: In this paper, we study a degenerate time-fractional partial differential equation involving the Caputo derivative of order \gamma\in(0,1), subject to nonlocal boundary conditions and a nonlocal initial condition of Bitsadze-Samarskii type. We then consider the inverse problem of determining the source term and the initial perturbation jointly with the solution, using two over-determination conditions. Explicit representations of the unknown functions and of the solution are constructed by the Fourier method based on the spectral problem associated with the degenerate operator, and the existence and uniqueness of the solution to the inverse problem are proved. The convergence of the constructed Fourier series is justified by means of Bessel-type inequalities, the Cauchy-Schwarz inequality, and sharp two-sided estimates for the Mittag-Leffler function.

Keywords: Degenerate partial differential equation; Caputo fractional derivative; inverse initial problem; inverse source problem; Fourier method; nonlocal conditions; spectral problem.

Author: Matyakubov A. (National University of Uzbekistan), Mamatov A. (National University of Uzbekistan)

Abstract:
This work is devoted to the analysis of asymptotic regimes and finite-time blow-up effects in nonlinear degenerate diffusion systems involving absorption mechanisms. The considered model consists of a coupled degenerate parabolic system that represents interacting diffusion dynamics between the components. By employing asymptotic techniques in combination with the supsolution method, upper estimates are obtained to describe the qualitative behavior of the solutions. It is demonstrated that the mutual influence of nonlinear diffusion and absorption terms is a determining factor in the emergence of blow-up within a finite time interval. Numerical experiments based on finite-difference schemes are conducted to illustrate the transition between regular and blow-up regimes. The findings enhance the theoretical and computational understanding of nonlinear diffusion-absorption dynamics and contribute to the broader study of nonlinear parabolic systems.

Keywords: Nonlinear parabolic system; self-diffusion; absorption terms; asymptotic behavior; supsolution; blow-up; mathematical modeling; finite-difference method; numerical simulation.

Author: Mirzaeva M. (Fergana State University)

Abstract:
This study is devoted to the unique solvability of the Bitsadze-Samarskii problem for a sub-diffusion equation involving the regularized Prabhakar fractional derivative. First, assuming that a solution to the Bitsadze-Samarskii problem exists, the solution of the corresponding first boundary value problem is derived. Using this solution, the Bitsadze-Samarskii problem is reduced to a Volterra integral equation of the second kind. It is then shown that this integral equation admits a unique solution due to the continuity of its kernel and right-hand side function. A theorem on the existence and uniqueness of the solution is formulated and proved. Finally, the solution to the Bitsadze-Samarskii problem can be represented in terms of the solution of the Volterra integral equation and the solution of the corresponding first boundary value problem.

Keywords: Bitsadze-Samarskii problem; regularized Prabhakar fractional derivative; Prabhakar function; Volterra integral equation of the second kind.

Author: Nortoshev D. (Tashkent State Transport University), Khudoykulov Sh. (Tashkent State Transport University)

Abstract:
In this article, we study the solvability of the generalized Boussinesq–Love equation, a pseudohyperbolic (Sobolev-type) equation, subject to a generalized Ionkin-type nonlocal time condition and a Dirichlet boundary condition. Applying the Fourier method, we reduce the problem to ordinary differential equations for the Fourier coefficients with nonlocal conditions, considering both negative and positive values of the equation’s coefficient. Moreover, we prove the uniqueness and existence of a regular solution by establishing a priori estimates based on Parseval’s identity and Hölder’s inequality, together with the Weierstrass test.

Keywords: Pseudohyperbolic equation; Boussinesq equation; existence and uniqueness of solution; nonlocal problem; method of separation of variables.

Author: Ruzimboev J. (New Uzbekistan University), Boranbaev O. (Karakalpak State University)

Abstract:
In this paper, we characterize the even and odd superderivations of the modified Witt-type Lie superalgebra W over the complex field \mathbb{C}. We also study local even and local odd superderivations on W and prove that every local superderivation on W is a superderivation.

Keywords: Infinite-dimensional Lie algebras; modified Witt-type Lie superalgebra; superderivation; local superderivation.

Author: Sattarov I. (V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences)(Namangan State University)

Abstract:
We consider nonconstant (2,2)-rational functions over the complex p-adic field \mathbb C_p that have three distinct finite fixed points. We derive an affine normal form and exact distance identities at all three fixed points, and use them to classify the fixed points locally as attracting, indifferent, or repelling. To study the orbits, we introduce a real-valued radius map. There are eight possible relative positions of the radius of the second root of the numerator and the two pole radii. We analyze in detail the case in which the poles have the same radius and the second root lies inside their common sphere. This case splits into five parameter subcases. On noncritical spheres we obtain a uniform radial description; on critical spheres, where the next radius may depend on the point, we isolate the corresponding return, invariant, or recurrent sets and state explicitly the remaining point-dependent part of the dynamics. We localize the
pole-preimage set and determine basins of attraction, maximal Siegel disks, and repelling regions whenever these objects are determined by the parameter
data.

Keywords: p-adic rational dynamical system; fixed point; radius map; critical sphere; Siegel disk; basin of attraction; pole preimage.

Author: Sobirov Z. (V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences)(National University of Uzbekistan), Turemuratova A. (Tashkent State University of Economics), Aminov H. (National University of Uzbekistan)

Abstract:
This paper is concerned with an inverse source problem for the fractional heat equation involving a spatial derivative of edge-dependent order, formulated on metric graphs in Sobolev spaces. First, we prove the existence and uniqueness of a strong solution using the unified functional approach based on a priori estimates. Then, we investigate an inverse source problem with an integral overdetermination condition. By transforming the inverse problem into an equivalent operator equation, we show that the corresponding resolvent operator is well-defined.

Keywords: Fractional integral; Caputo derivative; Riemann-Liouville derivative; fractional heat equation; strong solution; inverse source problem; metric graph.

Author: Yusupov B. (V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences)(Urgench State University), Azizov M. (V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences), Mosbahi B. (University of Sfax)

Abstract:
In this paper, we study anti-derivations and biderivations of quasi-filiform non-Lie Leibniz algebras of maximum length. First, we recall the classification of such algebras and describe the general forms of derivations and anti-derivations for the families (M_{1,\delta}), (M_{2,\lambda}), (M_{3,\alpha}), and (M_4). Using these descriptions, we obtain explicit matrix forms of all biderivations of these algebras in the sense of Loday. Moreover, we investigate the relationship between biderivations and Nijenhuis operators. In particular, we show that symmetric bilinear biderivations with abelian images naturally induce Nijenhuis operators on Leibniz algebras. These results provide a connection between the theory of biderivations and deformation theory of Leibniz algebras.

Keywords: Derivations; anti-derivations; biderivations; Nijenhuis operators; Leibniz algebras; nilpotent Leibniz algebras; solvable Leibniz algebras.