Manana Mirianashvili

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 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.

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.

We consider a Dirichlet boundary‐value problem for the three‐dimensional convection‐diffusion equations with constant coefficients in the unit cube. A high order compact finite difference scheme is constructed on a 19‐point stencil using the Steklov averaging operators. We prove that the finite difference scheme converges in discrete $W_2^m\left(\omega\right)$‐norm with the convergence rate $O\left(h^{s-m}\right)$, where the real parameter s satisfies the condition $\max\left(1.5,\ m\right)<s\le m+4,\ m\ =0,1,2$ and the exact solution belongs to the Sobolev space $W_2^s\left(\Omega\right)\$.

We consider an initial boundary-value problem for the Regularized Long Wave equation. A three level conservative difference scheme is studied. On the first level a two level scheme is used to find the values of the unknown functions which ensures the expression of the initial energies only by the initial data. The obtained algebraic equations are linear with respect to the values of the unknown function for each new level. The use of the Gronwall lemma does not require any restriction on mesh steps. It is proved that the finite difference scheme converges with the rate $O\left(\tau^2+h^2\right)$ when the exact solution belongs to the Sobolev space $W_2^3$.

We consider the first initial-boundary value problem for linear heat conductivity equation with constant coefficient in $\Omega\times\left(0,\ T\right]$, where $\Omega$ is a unit square. A high order accuracy ADI two level difference scheme is constructed on a 18-point stencil using Steklov averaging operators. We prove that the finite difference scheme converges in the discrete $L_2$-norm with the convergence rate $O\left(h^s+\tau^{\frac{s}{2}}\right)$, when the exact solution belongs to the anisotropic Sobolev space $W_2^{s,\frac{s}{2}}$, $s\in\left(2,4\right]$.

We consider an initial boundary-value problem for the regularized long-wave equation. A three level one-parameter family of conservative difference schemes is studied. Two level schemes are used to find the values of the unknown functions on the first level. A numerical method for selection of artificial boundary conditions is proposed. It is proved that the finite difference scheme converges at rate $O\left(\tau^2+h^2\right)$ when an exact solution belongs to the Sobolev space $W_2^3$.

We consider an initial boundary-value problem for the generalized Benjamin–Bona–Mahony equation. A three-level conservative difference schemes are studied. The obtained algebraic equations are linear with respect to the values of unknown function for each new level. It is proved that the scheme is convergent with the convergence rate of order $k-1$, when the exact solution belongs to the Sobolev space of order $k\ \left(1<k\le3\right)$.

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 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.