Zurab Vashakidze

@ug.edu.ge

School of Science and Technology
The University of Georgia (UG)

https://researchid.co/z.vashakidze

RESEARCH, TEACHING, or OTHER INTERESTS

Numerical Analysis, Computational Mathematics, Modeling and Simulation, Mathematical Physics

Scopus Publications

Scholar Citations

Scholar h-index

Scopus Publications

Propagation of waves from finite sources arranged in line segments within an infinite triangular lattice
David Kapanadze and Zurab Vashakidze
Walter de Gruyter GmbH
Abstract This paper examines the propagation of time-harmonic waves in a two-dimensional triangular lattice with a lattice constant a = 1 {a=1} . The sources are positioned along line segments within the lattice. Specifically, we investigate the discrete Helmholtz equation with a wavenumber k ∈ ( 0 , 2 ⁢ 2 ) {k\\in(0,2\\sqrt{2})} , where input data is prescribed on finite rows or columns of lattice sites. We focus on two main questions: the efficacy of the numerical methods employed in evaluating the Green’s function, and the necessity of the cone condition. Consistent with a continuum theory, we employ the notion of radiating solution and establish a unique solvability result and Green’s representation formula using difference potentials. Finally, we propose a numerical computation method and demonstrate its efficiency through examples related to the propagation problems in the left-handed two-dimensional inductor-capacitor metamaterial.

On convergence of a three-layer semi-discrete scheme for the non-linear dynamic string equation of Kirchhoff-type with time-dependent coefficients
Jemal Rogava and Zurab Vashakidze
Wiley
AbstractThis paper considers the Cauchy problem for the non‐linear 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 non‐linear 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.

On stability and convergence of a three-layer semi-discrete scheme for an abstract analogue of the Ball integro-differential equation
Jemal Rogava, Mikheil Tsiklauri, and Zurab Vashakidze
Elsevier BV

On the convergence of a three-layer semi-discrete scheme for the nonlinear dynamic Kirchhoff string equation
Zurab Vashakidze
Walter de Gruyter GmbH
Abstract In this work, the initial-boundary value problem is considered for the dynamic Kirchhoff string equation u t ⁢ t - ( α ⁢ ( t ) + β ⁢ ∫ - 1 1 u x 2 ⁢ d x ) ⁢ u x ⁢ x = f u_{tt}-\\bigl{(}\\alpha(t)+\\beta\\int_{-1}^{1}u_{x}^{2}\\,\\mathrm{d}x\\bigr{)}u_{xx}=f . Here α ⁢ ( t ) \\alpha(t) is a continuously differentiable function, α ⁢ ( t ) ≥ c 0 > 0 \\alpha(t)\\geq\\mathrm{c}_{0}>0 and 𝛽 is a positive constant. For solving this problem approximately, a symmetric three-layer semi-discrete scheme with respect to the temporal variable is applied, in which the value of a nonlinear term is taken at the middle point. This approach allows us to find numerical solutions per temporal steps by inverting the linear operators. In other words, applying this scheme, a system of linear ordinary differential equations is obtained. The local convergence of the scheme is proved. The results of numerical computations using this scheme for different test problems are given for which the Legendre–Galerkin spectral approximation is applied with respect to the spatial variable.

Numerical Solution of Anti-Plane Problems of the Elasticity Theory for Composite Isotropic Plane Slackened by Linear Crack

An application of the legendre polynomials for the numerical solution of the nonlinear dynamical kirchhoff string equation

Approximate solution of anti-plane problem of elasticity theory for composite bodies weakened by cracks by integral equation method

RECENT SCHOLAR PUBLICATIONS

Propagation of waves from finite sources arranged in line segments within an infinite triangular lattice
D Kapanadze, Z Vashakidze
Georgian Mathematical Journal 2025

On convergence of a three‐layer semi‐discrete scheme for the non‐linear dynamic string equation of Kirchhoff‐type with time‐dependent coefficients
J Rogava, Z Vashakidze
ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift fr Angewandte 2024

On stability and convergence of a three-layer semi-discrete scheme for an abstract analogue of the Ball integro-differential equation
J Rogava, M Tsiklauri, Z Vashakidze
Journal of Mathematical Analysis and Applications 518 (1), 126664 2023

Numerical solution for J. Ball’s beam equation with velocity–dependent Effective viscosity
A Papukashvili, G Geladze, Z Vashakidze, M Sharikadze
Rep. Enlarged Sess. Semin. I. Vekua Appl. Math 37, 35-38 2023

On the convergence of a three-layer semi-discrete scheme for the nonlinear dynamic Kirchhoff string equation
Z Vashakidze
Georgian Mathematical Journal 29 (4), 615-627 2022

On the Algorithm of an Approximate Solution and Numerical Computations for J. Ball Nonlinear Integro-Differential Equation
A Papukashvili, G Geladze, Z Vashakidze, M Sharikadze
Rep. Enlarged Sess. Semin. I. Vekua Appl. Math 36, 75-78 2022

An application of the Legendre polynomials for the numerical solution of the nonlinear dynamical Kirchhoff string equation
Z Vashakidze
Mem. Differ. Equ. Math. Phys 79, 107-119 2020

ვარიაციულ-სხვაობიანი სქემა კირხჰოფის ორგანზომილებიანი არაწრფივი დინამიური განტოლებისათვის
ზურაბ ვაშაკიძე
ივანე ჯავახიშვილის სახელობის თბილისის სახელმწიფო უნივერსიტეტი 2017

Variational−Difference Scheme for Kirchhoff Two−Dimensional Nonlinear Dynamical Equation
J Rogava, A Papukashvili, Z Vashakidze
VIII Annual International Conference of the Georgian Mathematical Union, 160 2017

Numerical Computation of the Kirchhoff type Nonlinear Static Beam Equation by Iterative Method
A Papukashvili, J Peradze, Z Vashakidze
VIII Annual International Conference of the Georgian Mathematical Union, 143−144 2017

On one method of approximate solution of Kirchhoff type static beam nonlinear integro−differential equation
A Papukashvili, Z Vashakidze, M Sharikadze
Reports of Enlarged Session of the Seminar of I. Vekua Institute of Applied 2017

Approximate Solution of Boundary Value Problems for the Ordinary Second-Order Differential Equation with Variable Coefficients by Means of Operator Interpolation Method
A Papukashvili, Z Vashakidze, V Muladze
Bull. Georg. Natl. Acad. Sci 10 (3) 2016

The numerical solution of a two−point boundary value problem with a non−constant coefficient by means of operator interpolation method
A Papukashvili, B Tezelishvili, Z Vashakidze
Reports of Enlarged Session of the Seminar of I. Vekua Institute of Applied 2016

Approximate solution of some boundary value problem of antiplane elasticity theory by collocation method for composite bodies weakened by cracks
A Papukashvili, T Davitashvili, Z Vashakidze
VI Annual Meeting of the Georgian Mechanical Union, 35−36, 69−70 2015

On one numerical method of research of the stress−deformed condition of some multystructures with difficult geometry
A Papukashvili, J Rogava, Z Vashakidze
TICSSAM−2015, 140−146 2015

On Crack Behavior in the Composite Piece-wise Homogeneous Body
T DAVITASHVILI, A PAPUKASHVILI, Z VASHAKIDZE
Proceedings of the 4th International Conference on Applied and Computational 2015

Approximate solution of anti-plane problem of elasticity theory for composite bodies weakened by cracks by integral equation method
A Papukashvili, T Davitashvili, Z Vashakidze
Bull. Georg. Natl. Acad. Sci 9 (3) 2015

On one method of approximate solution of Dirchlet boundary value problem of Poisson’s equation for two dimensional body having cross form
A Papukashvili, Z Vashakidze
V Annual International Conference of the Georgian Mathematical Union, 133 2014

TO NUMERICAL REALIZATIONS AND STABILITY OF CALCULATING PROCESS OF SOME PROBLEMS OF THEORY OF ELASTICITY FOR CROSS-SHAPED REGIONS
A Papukashvili, YF Gulver, Z Vashakidze
Reports of Enlarged Session of the Seminar of I. Vekua Institute of Applied 2014

ON THE NUMERICAL SOLUTION OF CONTACT PROBLEM FOR POISSONS AND KIRCHHOFF EQUATION SYSTEM
A Papukashvili, J Rogava, Z Vashakidze
V ANNUAL MEETING OF THE GEORGIAN MECHANICAL UNION, 30−31, 59−60 2014

MOST CITED SCHOLAR PUBLICATIONS

An application of the Legendre polynomials for the numerical solution of the nonlinear dynamical Kirchhoff string equation
Z Vashakidze
Mem. Differ. Equ. Math. Phys 79, 107-119 2020
Citations: 5

On the convergence of a three-layer semi-discrete scheme for the nonlinear dynamic Kirchhoff string equation
Z Vashakidze
Georgian Mathematical Journal 29 (4), 615-627 2022
Citations: 1

ON THE NUMERICAL SOLUTION OF CONTACT PROBLEM FOR POISSONS AND KIRCHHOFF EQUATION SYSTEM
A Papukashvili, J Rogava, Z Vashakidze
V ANNUAL MEETING OF THE GEORGIAN MECHANICAL UNION, 30−31, 59−60 2014
Citations: 1