Mifodijus Sapagovas

Difference method of increased order of accuracy for the Poisson equation with nonlocal conditions

In a rectangular domain, we consider the two-dimensional Poisson equation with nonlocal boundary conditions in one of the directions. For this problem, we construct a difference scheme of fourth-order approximation, study its solvability, and justify an iteration method for solving the corresponding system of difference equations. We give a detailed study of the spectrum of the matrix representing this system. In particular, we obtain a criterion for the nondegeneracy of this matrix and conditions for its eigenvalues to be positive.

Modelling of Amperometric Biosensors with Rough Surface of the Enzyme Membrane

A two-dimensional-in-space mathematical model of amperometric biosensors has been developed. The model is based on the diffusion equations containing a nonlinear term related to the Michaelis–Menten kinetic of the enzymatic reaction. The model takes into consideration two types of roughness of the upper surface (bulk solution/membrane interface) of the enzyme membrane, immobilised onto an electrode. Using digital simulation, the influence of the geometry of the roughness on the biosensor response was investigated. Digital simulation was carried out using the finite-difference technique.

Alternating direction method for a two‐dimensional parabolic equation with a nonlocal boundary condition

Abstract The present paper deals with an alternating direction implicit method for a two dimensional parabolic equation in a rectangle domain with a nonlocal boundary condition in one direction. Sufficient conditions of stability for Peaceman‐Rachford method are established. Results of some numerical experiments are presented.

Stability of difference schemes for two-dimensional parabolic equations with non-local boundary conditions

The stability of difference schemes for one-dimensional and two-dimensional parabolic equations, subject to non-local (Bitsadze-Samarskii type) boundary conditions is dealt with. To analyze the stability of difference schemes, the structure of the spectrum of the matrix that defines the linear system of difference equations for a respective stationary problem is studied. Depending on the values of parameters in non-local conditions, this matrix can have one zero, one negative or complex eigenvalues. The stepwise stability is proved and the domain of stability of difference schemes is found.

Alternating direction method for the Poisson equation with variable weight coefficients in an integral condition

The Peaceman-Rachford alternating direction method is used to solve a system of difference equations approximating the Poisson equation in a rectangular domain with integral conditions with fourth-order accuracy. The convergence of the iterative method is studied on the basis of an analysis of the spectrum structure of a one-dimensional difference operator with a nonlocal condition. We study the dependence of the spectrum on the weight functions occurring in the integral conditions. In particular, we discuss the presence of complex eigenvalues with negative real parts in the spectrum of the difference operator with a nonlocal condition. The results of a numerical experiment are presented.

On the stability of an explicit difference scheme for hyperbolic equations with nonlocal boundary conditions

We consider the stability of an explicit finite-difference scheme for a linear hyperbolic equation with nonlocal integral boundary conditions. By studying the spectrum of the transition matrix of the explicit three-layer difference scheme, we obtain a sufficient condition for stability in a special norm.

The Stability of Finite-Difference Schemes for a Pseudoparabolic Equation with Nonlocal Conditions

The stability of an implicit difference scheme for a third-order linear pseudoparabolic equation with nonlocal integral conditions is considered. The analysis of stability is based on the spectral structure for the transition matrix of the system of finite-difference equations.

On The Stability Of Finite-Difference Schemes For Parabolic Equations Subject To Integral Conditions With Applications To Thermoelasticity

Abstract The stability of implicit difference scheme for parabolic equations subject to integral conditions, which correspond to the quasi-static flexure of a thermoelastic rod is considered. The stability analysis is based on the spectral structure of matrix of the difference scheme. The stability conditions obtained here differ from those presented in the articles of other authors..

Numerical spectral analysis of a difference operator with non-local boundary conditions

Abstract The spectrum of a finite difference operator, subject to non-local Robin type boundary conditions, is dealt with. We analyse the spectral properties that relate to the stability of finite difference schemes for parabolic equations. The impact of functions and parameters, defining non-local conditions, on a spectral structure is examined and theoretical study is supported by numerical experiments. Also, for a difference scheme, applied to a parabolic equation with non-local conditions, a sufficient stability criterion, based on spectral properties of the difference operator, is discussed. Numerical evidence suggests that such a criterion is not only sufficient for stability, but necessary, too.

On the spectral properties of three-layer difference schemes for parabolic equations with nonlocal conditions

We consider three-layer difference schemes for a one-dimensional linear parabolic equation with nonlocal integral conditions. A three-layer scheme is written out in an equivalent form of a two-layer scheme. We analyze the dependence of the spectrum of the difference operator on the parameters occurring in the integral conditions. We derive stability conditions for the original three-layer scheme in a specially defined energy norm.