Jemal Sanikidze

A computational scheme for approximating a singular operator in the known Lippman-Schwinger equations is suggested. It is based on partial use of Gaussian knots. On the basis of studying the asymptotic properties of the solution and one unclassical estimation of the accuracy of the Gaussian quadrature formulas, the convergence of the scheme, with estimation of the rate of convergence, is proved. The analysis of the computational aspects of the scheme is carried out and the results of calculations of some test examples are given.

The substantiation of a specific computational scheme based on the approximation of singular integrals with the Cauchy kernel in the method of boundary integral equations in the case of arbitrary smooth closed contours is given.

The question of numerical conformal mapping by approximation of the singular integral is considered. On this basis one of the concrete schemes is presented. The convergence of the given scheme is established.

Using the method of integral equations, one concrete numerical scheme for the conformal mapping of domains is constructed. The mentioned scheme is based on a certain approximation of the singular integral.

The question of application of singular integral equations to approximate conformal mappings is considered. On the basis of the so-called modified of discrete rotations concrete calculation algorithm is elaborated. Results of solution of some numerical examples are given.

Certain schemes for approximate calculation of singular integrals with a Cauchy kernel and their application to the numerical solution of the modified Dirichlet problem are offered. Questions of justifying the corresponding computational schemes for domains with Lyapunov boundaries are investigated.

A certain modified scheme of the discrete vortex type for the numerical solution of one class of singular integral equations is studied . Examples of applications to problems of conformal mapping of domains are given.

The question of the numerical solution of the Lippmann-Schwinger integral equations known in nuclear physics is considered. In this paper, we propose one concrete scheme for approximating singular integrals contained in such equations. An estimate of the approximation error is given. For one concrete equation, the numerical results found using this scheme are given.

the concrete computational scheme for the numerical solution of a certain class of singular integral equations that are used in physics, is investigated

Based on the method of approximation of singular integrals, a computational scheme is proposed for the approximate solution of the modified Dirichlet problem in the case of finite doubly connected domains. A substantiation of the scheme is given and an application to numerical conformal mappings is indicated.

A certain calculating scheme for numerical solution of a known in physics Lippman-Shwinger singular integral equation is offered. The corresponding scheme is based on a Gauss type quadrature formula (constructed by the author) with immovable singularity, which is reached by introducing some numerical parameter. The indicated quadrature formula is used together with Chebishev interpolation formula, whose knots are connected in the known way with the singularity point in the corresponding integral equation.

The questions of construction of numerical schemes for some classes of boundary value problems of the plane theory of elasticity are considered.

For some classes of boundary equations of elasticity theory, numerical schemes based on the approximation of singular integrals are considered.

Some approximation scheme for the numerical solution of the first and second major problem in plane elasticity theory for domains with arbitrary Lyapunov contours is presented. Construction of the computational algorithm is mainly based on the method of approximation of singular integrals with Cauchy kernel. Priori estimate error of approximation scheme and rationale of the process is given. The implementation of this algorithm can be equally implemented in both continuous and discrete initial data.

The question of numerical realisation of solution of some boundary problems concerning appendices of a known problem of conjugate of the theory of holomorph functions is considered. In this case the corresponding problem is concretised in the form of a certain class of problems of the theory of elasticity for an infinite plane with the focused rectilinear cuts.

Certain quadrature processes are considered for integrals with kernels $\left(t-z\right)^{-1}$,

$\left(t-z\right)^{-2}$ along piece-wise smooth closed contours, bounding finite or infinite domain $D$ involving $z$. Uniform estimates are given for the corresponding remainder terms namely for the case of arbitrary closeness of $z$ to the boundary of the domain.

Certain quadrature schemes of discrete singularities type with higher algebraic rate of accuracy comparing with similar schemes known earlier are offered. Approximation rate estimates are given for concrete classes of densities of the considered singular integrals.

A quadrature process for Cauchy type singular integrals with Chebyshev weight function is indicated. The error estimate is obtained on some classes of densities.

A question on construction of quadrature formulas for singular integrals with weight functions using the knots of orthogonal polynomials of two consecutive powers is investigated. A case of Chebyshev weight function is considered in detail. The constructed formulas contain the Gauss accuracy quadrature formulas as concrete cases.

The estimation of quadrature formula residual term for Cauchy type singular integrals is given for arbitrary values of Jacobi weight function.

A number of quadrature processes connected with approximation of Cauchy type

singular integrals are considered in relation with approximate solution of boundary problems of certain type. Namely, significant attention is paid to problems of unique solvability of approximating scheme, accuracy and similar questions related with boundary integral problems based on corresponding approximation