Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2005
Volume
Vol. 25
Number
No. 1
Description
Journal Volume
Opuscula Mathematica
Vol. 25 (2005)
Projects
Pages
Articles
Solution of the Stieltjes truncated matrix moment problem
(2005) Adamân, Vadim Movsevovič; Tkačenko, Igor M.
The truncated Stieltjes matrix moment problem consisting in the description of all matrix distributions $\boldsymbol{\sigma}(t)$ on $[0,\infty)$ with given first $2n+1$ power moments $(\mathbf{C}_j)_{n=0}^j$ is solved using known results on the corresponding Hamburger problem for which $\boldsymbol{\sigma}(t)$ are defined on $(-\infty,\infty)$. The criterion of solvability of the Stieltjes problem is given and all its solutions in the non-degenerate case are described by selection of the appropriate solutions among those of the Hamburger problem for the same set of moments. The results on extensions of non-negative operators are used and a purely algebraic algorithm for the solution of both Hamburger and Stieltjes problems is proposed.
A sufficient condition for Schur stability of the convex combination of the polynomials
(2005) Białas, Stanisław
In this paper is given a simple suffcient condition for Schur stability of the convex combination of the real polynomials.
Monotone iterative methods for infinite systems of reaction-diffusion-convection equations with functional dependence
(2005) Brzychczy, Stanisław
We consider the Fourier first initial-boundary value problem for an infinite system of semilinear parabolic differential-functional equations of reaction-diffusion-convection type of the form $\mathcal{F}^i[z^i](t,x)=f^i(t,x,z),\quad i \in S,$ where $\mathcal{F}^i:=\mathcal{D}_t-\mathcal{L}^i,\quad \mathcal{L}^i:=\sum_{j,k=1}^m a_{jk}^i(t,x)\mathcal{D}^2_{x_jx_k}+\sum_{j=1}^m b_j^i(t,x)\mathcal{D}_{x_j}$ in a bounded cylindrical domain $(0,T] \times G:=D \subset \mathbb{R}^{m+1}$. The right-hand sides of the system are Volterra type functionals of the unknown function z. In the paper, we give methods of the construction of the monotone iterative sequences converging to the unique classical solution of the problem considered in partially ordered Banach spaces with various convergence rates of iterations. We also give remarks on monotone iterative methods in connection with numerical methods, remarks on methods for the construction of lower and upper solutions and remarks concerning the possibility of extending these methods to more general parabolic equations. All monotone iterative methods are based on differential inequalities and, in this paper, we use the theorem on weak partial differential-functional inequalities for infinite systems of parabolic equations, the comparison theorem and the maximum principle. A part of the paper is based on the results of our previous papers. These results generalize the results obtained by several authors in numerous papers for finite systems of semilinear parabolic differential equations to encompass the case of infinite systems of semilinear parabolic differential-functional equations. The monotone iterative schemes can be used for the computation of numerical solutions.
The Abel summation of the Kontorovich-Lebedev integral representation
(2005) Cojuhari, Petru A.; Gomilko, Aleksandr M.
A new result on the summation of the Kontorovich–Lebedev integral representation in the sense of Abel mean is given.
Numerical approximations of difference functional equations and applications
(2005) Kamont, Zdzisław
We give a theorem on the error estimate of approximate solutions for difference functional equations of the Volterra type. We apply this general result in the investigation of the stability of difference schemes generated by nonlinear first order partial differential functional equations and by parabolic problems. We show that all known results on difference methods for initial or initial boundary value problems can be obtained as particular cases of this general and simple result. We assume that the right hand sides of equations satisfy nonlinear estimates of the Perron type with respect to functional variables.