Olga Štikonienė

This article investigates the numerical solution of the two-dimensional Poisson equation defined over a rectangular domain subject to a double integral nonlocal boundary condition. We propose a finite difference scheme by discretizing the integral term using the two-dimensional trapezoidal rule. The main difficulty of this problem is that, in the non-classical case, we cannot use the method of separation of variables and decompose the problem into one-dimensional problems. Our approach involves reducing the integral boundary condition from the complete domain to the interior points and strategically partitioning the computational domain into the boundary and interior points. We propose a method that allows us to find a solution by solving the Poisson equation with classical boundary conditions, and using the solutions found to construct a solution to a problem with a nonlocal integral condition. This method requires solving a linear system whose dimension is much smaller than the original. Under certain conditions on the kernel, the proposed method is correct.

This article is the second part of a survey dedicated to M-matrices and the application of the finite difference method to elliptic problems with nonlocal boundary conditions. Here, we examine cases in which the matrix of the resulting system of linear equations is an M-matrix. Here, we address the discrete Sturm–Liouville problem with nonlocal boundary conditions, describing its spectrum in one-dimensional case. This enables us to determine the values of the nonlocality parameters for which the finite difference scheme is represented by an M-matrix.

This article offers a comprehensive overview of the results obtained through numerical methods in solving the minimal surface equation, along with exploring the applications of minimal surfaces in science, technology, and architecture. The content is enriched with practical examples highlighting the diverse applications of minimal surfaces.

In this paper, we investigate the convergence of the Peaceman-Rachford Alternating Direction Implicit method for the system of difference equations, approximating the two-dimensional elliptic equations in rectangular domain with nonlocal integral conditions. The main goal of the paper is the analysis of spectrum structure of difference eigenvalue problem with nonlocal conditions. The convergence of iterative method is proved in the case when the system of eigenvectors is complete. The main results are generalized for the system of difference equations, approximating the differential problem with truncation error O(h⁴). Keyword : elliptic equation, integral boundary conditions, finite-difference method, iterative method, eigenvalue problem

This paper deals with the numerical solution of nonlocal boundary-value problem for two-dimensional pseudoparabolic equation which arise in many physical phenomena. A three-layer alternating direction implicit method is investigated for the solution of this problem. This method generalizes Peaceman–Rachford’s ADI method for the 2D parabolic equation. The stability of the proposed method is proved in the special norm. We investigate algebraic eigenvalue problem with nonsymmetric matrices to prove this stability. Numerical results are presented.

The mathematical model of the electrochemical glucose biosensor based on the enzymatic conversionof the substrate and the diffusion of the substrate was used. The influence of the fluctuations of themembrane thickness, the diffusion coefficients and pH were modelled and their impact was evaluated atdifferent modes of an action of the biosensor.

The present paper deals with a generalization of the alternating-direction implicit(ADI) method for a two dimensional Poisson equation in a rectangle domain with aweighted integral boundary condition in one coordinate direction. We consider the alternatingdirection method for a system of difference equations that approximates Poisson equationwith weighed integral boundary conditions with the fourth-order accuracy. Sufficient conditionsof stability for ADI method are investigated numerically. An analysis of results ofcomputational experiments is presented.

The iterative methods to solve the system of the difference equations derived from the nonlinear elliptic equation with integral condition are considered. The convergence of these methods is proved using the properties of M-matrices, in particular, the regular splitting of an M-matrix. To our knowledge, the theory of M-matrices has not ever been applied to convergence of iterative methods for system of nonlinear difference equations. The main results for the convergence of the iterative methods are obtained by considering the structure of the spectrum of the two-dimensional difference operators with integral condition. *The research was partially supported by the Research Council of Lithuania (grant No. MIP-047/2014).