Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2014
Volume
Vol. 34
Number
No. 2
Description
Journal Volume
Opuscula Mathematica
Vol. 34 (2014)
Projects
Pages
Articles
Existence and regularity of solutions for hyperbolic functional differential problems
(2014) Kamont, Zdzisław
A generalized Cauchy problem for quasilinear hyperbolic functional differential systems is considered. A theorem on the local existence of weak solutions is proved. The initial problem is transformed into a system of functional integral equations for an unknown function and for their partial derivatives with respect to spatial variables. The existence of solutions for this system is proved by using a method of successive approximations. We show a theorem on the differentiability of solutions with respect to initial functions which is the main result of the paper.
Local error structures and order conditions in terms of Lie elements for exponential splitting schemes
(2014) Auzinger, Winfried; Herfort, Wolfgang
We discuss the structure of the local error of exponential operator splitting methods. In particular, it is shown that the leading error term is a Lie element, i.e., a linear combination of higher-degree commutators of the given operators. This structural assertion can be used to formulate a simple algorithm for the automatic generation of a minimal set of polynomial equations representing the order conditions, for the general case as well as in symmetric settings.
On reconstructing an unknown coordinate of a nonlinear system of differential equations
(2014) Blizorukova, Marina; Kuklin, Aleksandr Anatol'evič; Maksimov, Vâčeslav
The paper discusses a method of auxiliary controlled models and the application of this method to solving problems of dynamical reconstruction of an unknown coordinate in a nonlinear system of differential equations. The solving algorithm, which is stable with respect to informational noises and computational errors, is presented.
On a singular nonlinear Neumann problem
(2014) Chabrowski, Jan
We investigate the solvability of the Neumann problem involving two critical exponents: Sobolev and Hardy-Sobolev. We establish the existence of a solution in three cases: $\text{(i)}\;\ 2\lt p+1\lt 2^*(s),$, $\text{(ii)}\;\ p+1=2^*(s)$ and $\text{(iii)}\;\ 2^*(s)\lt p+1 \leq 2^*,$ where $2^*(s)=\frac{2(N-s)}{N-2},$ $0\lt s\lt 2,$ and $2^*=\frac{2N}{N-2}$ denote the critical Hardy-Sobolev exponent and the critical Sobolev exponent, respectively.