arXiv subjects

David Kapanadze

Publications and source records attributed to David Kapanadze.

5 recordsLinked to original sources

arXiv2024

Propagation of Waves from Finite Sources Arranged in Line Segments within an Infinite Triangular Lattice

This paper examines the propagation of time-harmonic waves in a two-dimensional triangular lattice with a lattice constant $a = 1$. The sources are positioned along line segments within the lattice. Specifically, we investigate the discrete Helmholtz equation with a wavenumber $k \in \left( 0,2\sqrt{2} \right)$, 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.

math.AP↗

arXiv2023

Exterior diffraction problems for a triangular lattice

Exterior Dirichlet problems for two-dimensional lattice waves on the semi-infinite triangular lattice are considered. Namely, we study Dirichlet problems for the two-dimensional discrete Helmholtz equation in a plane with a hole. New results are obtained for the existence and uniqueness of the solution in the case of the real wave number $k \in (0, 2\sqrt{2})$ without passing to a complex wave number. Besides, Green's representation formula for the solution is derived with the help of difference potentials. To demonstrate the results, we propose a method for numerical calculation.

math.AP↗

arXiv2022

Half-plane diffraction problems on a triangular lattice

We investigate thin-slit diffraction problems for two-dimensional lattice waves. The peculiar structure allows us to consider the problems on the semi-infinite triangular lattice, consequently, we study Dirichlet problems for the two-dimensional discrete Helmholtz equation in a half-plane. In view of the existence and uniqueness of the solution, we provide new results for the real wave number $k\in (0,3)\backslash\{2\sqrt{2}\}$ without passing to the complex wave number and derive an exact representation formula for the solution. For this purpose, we use the notion of the radiating solution. Finally, we propose a method for numerical calculation. The efficiency of our approach is demonstrated in an example related to the propagation of wave fronts in metamaterials through two small openings.

math-ph↗

arXiv2014

Exact solution to a nonlinear heat conduction problem in doubly periodic 2D composite materials

An analytic solution to a stationary heat conduction problem in 2D unbounded doubly periodic composite materials with temperature dependent conductivities of their components is given. Corresponding nonlinear boundary value problem is reduced a Laplace equation with nonlinear transmission conditions. For special relationships between the conductivity coefficients of the matrix and inclusions, the problem is transformed to fully linear boundary value problem for doubly periodic analytic functions. This allows to reconstruct the solution of the originally nonlinear composite and to find its effective properties. The results are illustrated by numerical examples.

math.AP↗

arXiv2013

Improved algorithm for analytical solution of the heat conduction problem in doubly periodic 2D composite materials

We consider a boundary value problem in unbounded 2D doubly periodic composite with circular inclusions having arbitrary constant conductivities. By introducing complex potentials, the boundary value problem for the Laplace equation is transformed to a special R-linear BVP for doubly periodic analytic functions. This problem is solved with use of the method of functional equations. The R-linear BVP is transformed to a system of functional equations. A new improved algorithm for solution of the system is proposed. It allows one not only to compute the average property but to reconstruct the solution components (temperature and flux) at an arbitrary point of the composite. Several computational examples are discussed in details demonstrating high efficiency of the method. Indirect estimate of the algorithm accuracy has been also provided.

math-ph↗