Ռաֆայել Բարխուդարյանի հրապարակումներ (Full profile)

Trigonometric approximation or interpolation of a non-smooth function on a finite interval has poor convergence properties. This is especially true for discontinuous functions. The case of infinitely differentiable but non-periodic functions with discontinuous periodic extensions onto the real axis has attracted interest from many researchers. In a series of works, we discussed an approach based on quasi-periodic trigonometric basis functions whose periods are slightly bigger than the length of the approximation interval. We proved validness of the approach for trigonometric interpolations. In this paper, we apply those ideas to classical Fourier expansions.

In the present work we deal with the numerical approximation of equations of stationary states for a general class of the spatial segregation of Reaction-diffusion system with two population densities having disjoint supports. We show that the problem gives rise the generalized version of the so-called two-phase obstacle problem and introduce the notion of viscosity solutions for this new model. Then we use quantitative properties of both, solutions and free boundaries, to develop convergent finite difference scheme.

In the current work we consider the numerical solutions of equations of stationary states for a general class of the spatial segregation of reaction–diffusion systems with $m\geq 2$ population densities. We introduce a discrete multi-phase minimization problem related to the segregation problem, which allows to prove the existence and uniqueness of the corresponding finite difference scheme. Based on that scheme, we suggest an iterative algorithm and show its consistency and stability. For the special case $m=2$, we show that the problem gives rise to the generalized version of the so-called two-phase obstacle problem. In this particular case we introduce the notion of viscosity solutions and prove convergence of the difference scheme to the unique viscosity solution. At the end of the paper we present computational tests, for different internal dynamics, and discuss numerical results.

We propose an algorithm to solve the two-phase obstacle problem by finite difference method. We prove the existence and uniqueness of the solution of the discrete nonlinear system and obtain an error estimate for finite difference approximation.

We propose an algorithm to solve the two-phase obstacle problem by finite difference method. We obtain an error estimate for finite difference approximation and prove the convergence of proposed algorithm.