arXiv subjects

Jemal Rogava

Publications and source records attributed to Jemal Rogava.

5 recordsLinked to original sources

arXiv2026

On the Approximation of the Unitary Operator Group Associated with a Rotation Matrix and Its Applications to Abstract Hyperbolic Equations

The solution of the Cauchy problem for homogeneous abstract hyperbolic equations, together with its derivative, admits a vector representation in terms of a unitary operator group associated with a rotation matrix. A rational approximation of this unitary group is constructed and shown to possess optimal fourth-order convergence. The order of convergence is determined in accordance with the smoothness scale. Based on this rational approximation, a two-layer semi-discrete scheme is constructed for the approximate solution of Cauchy problems for nonhomogeneous abstract hyperbolic equations in both the linear and semi-linear settings. The convergence properties of the scheme are examined in relation to the regularity of the solution.

math.NA↗

arXiv2026

On the Numerical Treatment of an Abstract Nonlinear System of Coupled Hyperbolic Equations Associated with the Timoshenko Model

The present work addresses the Cauchy problem for an abstract nonlinear system of coupled hyperbolic equations associated with the Timoshenko model in a real Hilbert space. Our purpose is to develop and delve into a temporal discretization scheme for approximating a solution to this problem. To this end, we propose a symmetric three-layer semi-discrete time-stepping scheme in which the nonlinear term is evaluated at the temporal midpoint. As a result, at each time step, this approach reduces the original nonlinear problem to a linear one and enables parallel computation of its solution. Convergence is proved, and second-order accuracy with respect to the time-step size is established on a local temporal interval. The proposed scheme is applied to a spatially one-dimensional nonlinear dynamic Timoshenko beam system, and the results obtained for the abstract nonlinear system are extended to this setting. A Legendre-Galerkin spectral approximation is employed for the spatial discretization. By taking differences of Legendre polynomials within the Galerkin framework, the resulting linear system is sparse and can be efficiently decoupled. The convergence of the method is also investigated. Finally, several numerical experiments on carefully chosen benchmark problems are conducted to validate the proposed approach and to confirm the theoretical findings.

math.NA↗

arXiv2023

On Convergence of a Three-Layer Semi-Discrete Scheme for the Nonlinear Dynamic String Equation of Kirchhoff-Type with Time-Dependent Coefficients

This paper considers the Cauchy problem for the nonlinear dynamic string equation of Kirchhoff-type with time-varying coefficients. The objective of this work is to develop a time domain discretization algorithm capable of approximating a solution to this initial-boundary value problem. To this end, a symmetric three-layer semi-discrete scheme is employed with respect to the temporal variable, wherein the value of a nonlinear term is evaluated at the middle node point. This approach enables the numerical solutions per temporal step to be obtained by inverting the linear operators, yielding a system of second-order linear ordinary differential equations. Local convergence of the proposed scheme is established, and it achieves quadratic convergence regarding the step size of the discretization of time on the local temporal interval. We have conducted several numerical experiments using the proposed algorithm for various test problems to validate its performance. It can be said that the obtained numerical results are in accordance with the theoretical findings.

math.AP↗

arXiv2022

On Stability and Convergence of a Three-layer Semi-discrete Scheme for an Abstract Analogue of the Ball Integro-differential Equation

We consider the Cauchy problem for a second-order nonlinear evolution equation in a Hilbert space. This equation represents the abstract generalization of the Ball integro-differential equation. The general nonlinear case with respect to terms of the equation which include a square of a norm of a gradient is considered. A three-layer semi-discrete scheme is proposed in order to find an approximate solution. In this scheme, the approximation of nonlinear terms that are dependent on the gradient is carried out by using an integral mean. We show that the solution of the nonlinear discrete problem and its corresponding difference analogue of a first-order derivative is uniformly bounded. For the solution of the corresponding linear discrete problem, it is obtained high-order a priori estimates by using two-variable Chebyshev polynomials. Based on these estimates we prove the stability of the nonlinear discrete problem. For smooth solutions, we provide error estimates for the approximate solution. An iteration method is applied in order to find an approximate solution for each temporal step. The convergence of the iteration process is proved.

math.NA↗

arXiv2022

Parallel Type Decomposition Scheme for Quasi-Linear Abstract Hyperbolic Equation

Cauchy problem for an abstract hyperbolic equation with the Lipschitz continuous operator is considered in the Hilbert space. The operator corresponding to the elliptic part of the equation is a sum of operators $A_{1},\,A_{2},\,\ldots,\,A_{m}$. Each addend is a self-adjoint and positive definite operator. A parallel type decomposition scheme for an approximate solution of the stated problem is constructed. The main idea of the scheme is that on each local interval classic difference problems are solved in parallel (independently from each other) respectively with the operators $A_{1},\,A_{2},\,\ldots,\,A_{m}$. The weighted average of the received solutions is announced as an approximate solution at the right end of the local interval. Convergence of the proposed scheme is proved and the approximate solution error is estimated, as well as the error of the difference analogue for the first-order derivative for the case when the initial problem data satisfy the natural sufficient conditions for solution existence.

math.NA↗