# Ústav matematiky a deskriptivní geometrie

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Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point
(De Gruyter, 2024-02-05) Diblík, Josef; Růžičková, Miroslava
A singular nonlinear differential equation z(sigma) dw/dz = aw + zwf(z , w), where sigma > 1, is considered in a neighbourhood of the point z = 0 z=0 located either in the complex plane C if sigma is a natural number, in a Riemann surface of a rational function if sigma is a rational number, or in the Riemann surface of logarithmic function if sigma is an irrational number. It is assumed that w = w ( z ) w=w\left(z) , a is an element of C { 0 } a, and that the function f f is analytic in a neighbourhood of the origin in C x C . Considering sigma to be an integer, a rational, or an irrational number, for each of the above-mentioned cases, the existence is proved of analytic solutions w = w (z ) w=w(z) in a domain that is part of a neighbourhood of the point z = 0 z=0 in C or in the Riemann surface of either a rational or a logarithmic function. Within this domain, the property lim z -> 0 w (z) = 0 is proved and an asymptotic behaviour of w (z) s established. Several examples and figures illustrate the results derived. The blow-up phenomenon is discussed as well.
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Multiplicity of solutions for nonlinear coercive problems
(Elsevier, 2023-12-01) Diblík, Josef; Galewski, Marek; Radulescu, Vicentiu; Šmarda, Zdeněk
We are concerned in this paper with problems that involve nonlinear potential mappings satisfying condition (S) and whose potentials are coercive. We first provide mild sufficient conditions for the minimizing sequence in the Weierstrass-Tonelli theorem in order to have strongly convergent subsequences. Next, we establish a three critical point theorem which is based on the Pucci-Serrin type mountain pass lemma and which is an infinite dimensional counterpart of the Courant theorem. Ricceri-type three critical point results then follow. Some applications to Dirichlet boundary value problems driven by the perturbed Laplacian are given in the final part of this paper.
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Modelling of crack formation and growth using FEM for selected structural materials at static loading
(WSEAS, 2023-11-01) Kozák, Vladislav; Vala, Jiří
he purpose of this paper is to show the results of a study focused on the occurrence of damage heterogeneous materials, especially on the issue of modelling crack formation and propagation. In the beginning the attention is paid to the direct application of the finite element method to different types of materials in order to find critical parameters determining behaviour of materials at damage process. The applications of damage mechanics and possible approaches to model the origin of a crack propagation through modifications in FEM systems are presented and some practical applications are tested. Main effort is devoted to cement fibre composites and the search for new methods for their more accurate modelling, especially close to the field stress concentrator, respectively ahead of the crack tip. Modified XFEM method has been used as a suitable tool for numerical modelling.
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On a derivative-free optimization approach to some problems of civil engineering
(WSEAS, 2023-09-26) Vala, Jiří; Jarošová, Petra
Development of advanced materials and structures for civil engineering, due to the requirements of green and sustainable building, including the reduction of energy consumption and the balance between occupant comfort and environmental friendliness, needs proper analysis of related physical, chemical, etc. processes, whose mathematical description leads to direct, sensitivity and inverse initial and boundary value problems for nonlinear partial differential equations, analysed numerically using finite element, difference and similar techniques. Design optimization requires to implement a set of additional variable parameters into all related computations, which is very expensive or quite impossible in most cases. Thus realistic computational strategies work with the minimizations of some cost functions with unknown parameters using certain kind of numerical differentiation, like quasi-Newton, inexact Newton or conjugate gradient methods, some derivative-free approach, or, as a much-favoured alternative, some heuristic soft-computing algorithm. A reasonable compromise seems to be the exploitation of an algorithm coming from the non-gradient Nelder-Mead simplex approach. In this paper, referring to the experience with i) the direct problem of thermal design of a residential building and ii) the inverse problem of identification of material characteristics as thermal conductivity and diffusivity from well-Advised laboratory experiments, after several remarks to the history and progress of the Nelder-Mead method and its improvements, we shall demonstrate some convergence properties of such approach, regardless of the highly cited evaluation of the original Nelder-Mead algorithm: "Mathematicians hate it because you cannot prove convergence; engineers seem to love it because it often works.".
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Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type
(De Gruyter, 2023-10-06) Diblík, Josef; Korobko, Evgeniya
The article investigates a second-order nonlinear difference equation of Emden-Fowler type. New conditions with respect to parameters of equation are found such that the equation admits a solution asymptotically represented by a power function that is asymptotically equivalent to the exact solution of the nonlinear second-order differential Emden-Fowler equation. Two-term asymptotic representations are given not only for the solution itself but also for its first- and second-order forward diffrences as well. Previously known results are discussed, and illustrative examples are considered.