Zurab Vashakidze
@ug.edu.ge
School of Science and Technology
The University of Georgia (UG)
RESEARCH, TEACHING, or OTHER INTERESTS
Numerical Analysis, Computational Mathematics, Modeling and Simulation, Mathematical Physics
Scopus Publications
Scholar Citations
Scholar h-index
Scopus Publications
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.
Jemal Rogava, Mikheil Tsiklauri, and Zurab Vashakidze
Elsevier BV
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.