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EN
The paper discusses the existence of solutions for Cauchy-type problem of fractional order in the space of Lebesgue integrable functions on bounded interval. Some qualitative properties of solutions are presented such as monotonicity, uniqueness and continuous dependence on the initial data. The main tools used are measure of weak (strong) noncompactness, Darbo fixed point theorem and fractional calculus.
2
Content available remote General remarks on dynamic projection method
EN
A brief history and a mathematical description of the dynamic projection operators technique is presented. An example of the general Cauchy problem for evolution equations in 1+ 1 dimensions is studied in detail. A boundary regime propagation is formulated in terms of operators and illustrated by the simplest one-dimensional diffusion equation. The problem of temperature waves is discussed.
EN
The paper deals with the model of variable-order nonlinear hereditary oscillator based on a numerical finite-difference scheme. Numerical experiments have been carried out to evaluate the stability and convergence of the difference scheme. It is argued that the approximation, stability and convergence are of the first order, while the scheme is stable and converges to the exact solution
EN
We study the k-summability of divergent formal solutions for the Cauchy problem of certain linear partial differential operators with coefficients which are polynomial in t. We employ the method of successive approximation in order to construct the formal solutions and to obtain the properties of analytic continuation of the solutions of convolution equations and their exponential growth estimates.
PL
W pracy rozwiązane zostało zagadnienie odwrotne dla przypadku stacjonarnego pola temperatury w obszarze wielospójnym, które jest ważne z technicznego punktu widzenia i dotyczy problemu chłodzenia łopatek turbin gazowych. Rozwiązane zostało zagadnienie odwrotne testowe dla obszaru pierścienia eliptycznego, w którym znany jest rozkład temperatury oraz współczynnik przejmowania ciepła na brzegu zewnętrznym obszaru. Na tej podstawie wyznaczony został rozkład temperatury oraz gęstości strumienia ciepła na brzegu wewnętrznym pierścienia. W funkcjonale optymalizującym rozwiązanie zagadnienia odwrotnego uwzględniony został człon związany z gradientem temperatury w całym obszarze. Obliczenia przeprowadzono dla znanego rozkładu współczynnika przejmowania ciepła na brzegu zewnętrznym obszaru zaburzonego błędem losowym równym 0, 1, 5 oraz 10%. Zbadano wpływ gradientu temperatury na czas i dokładność obliczeń. Uwzględnienie gradientu temperatury w funkcjonale, który jest minimalizowany w procesie obliczeniowym skróciło czas obliczeń oraz zmniejszyło oscylacje rozkładu temperatury oraz strumienia ciepła na brzegu wewnętrznym obszaru wielospójnego.
EN
In this paper, the inverse problem for the steady-state temperature field in the multiply-connected domain was solved, which is of great importance from technical point of view and concerns the problem of cooling the gas turbine blades. Test inverse problem for domain of the elliptical ring with the known temperature distribution and the heat transfer coefficient on the outer boundary of the domain was solved. On this basis the distributions of temperature and heat flux density on the inner boundary of the ring were determined. The optimization functional of the solution of the inverse problem comprises a term related to the temperature gradient in the whole domain. Calculations were made for the known distribution of the heat transfer coefficient on the outer boundary of the domain disturbed by random error equal of 0, 1, 5 and 10 %. The influence of the temperature gradient on time and the accuracy of calculations was examined. Taking into account the temperature gradient in the functional, which is minimized in the calculation process, reduced the time of calculations and decreased oscillations of the temperature as well as heat flux distributions on the inner boundary of the multiply-connected domain.
EN
The one-dimensional time-fractional advection-diffusion equation with the Caputo time derivative is considered. The fundamental solution to the Cauchy problem is obtained using the integral transform technique. The numerical results are illustrated graphically.
7
Content available remote On nonlinear differential equations in generalized Musielak-Orlicz spaces
EN
We consider ordinary differential equations u′(t)+(I−T)u(t)=0, where an unknown function takes its values in a given modular function space being a generalization of Musielak-Orlicz spaces, and T is nonlinear mapping which is nonexpansive in the modular sense. We demonstrate that under certain natural assumptions the Cauchy problem related to this equation can be solved. We also show a process for the construction of such a solution. This result is then linked to the recent results of the fixed point theory in modular function spaces.
EN
The paper contains an existence theorem for local solutions of an initial value problem for a nonlinear integro-differential equation in Banach spaces. The assumptions and proofs are expressed in terms of measures of noncompactness.
EN
The Cauchy problems for time-fractional diffusion equation with delta pulse initial value of a sought-for function is studied in a circle domain in the axisymmetric case under zero Dirichlet and Neumann boundary conditions, respectively. The Caputo fractional derivative is used. The Laplace and finite Hankel integral transforms are employed. The results are illustrated graphically.
EN
The diffusion-wave equation is a mathematical model of a wide range of important physical phenomena. The first and second Cauchy problems and the source problem for the diffusion-wave equation are considered in cylindrical coordinates. The Caputo fractional derivative is used. The Laplace and Hankel transforms are employed. The results are illustrated graphically.
11
Content available remote Matematyczne aspekty opisu turbulencji
PL
W pracy omówiono wybrane aspekty matematycznego opisu turbulentnych przepływów cieczy. W szczególności, odniesiono się do szóstego Problemu Milenijnego dotyczącego istnienia, jednoznaczności i regularności rozwiązań zagadnienia Cauchy'ego dla równań Naviera-Stokesa. Rozważono rozwiązania klasyczne, słabe w sensie Leray'a oraz - krótko - podejście półgrupowe Kato-Fujity. Zwrócono również uwagę na recepcję tego problemu wśród fizyków teoretyków i przedstawicieli dyscyplin technicznych.
EN
This paper reviews the selected aspects of mathematical description of turbulent fluid flows. In particular, the basic results concerning existence, uniqueness and regularity of the Cauchy problem for the Navier-Stokes equations (NSE) are described (the sixth problem of the Millenium). The classical solutions, the weak formulation of the NSE and semi-group approach of Kato-Fujita are considered. Some remarks about the significance of these problems for theoretical physicists and engineers are also briefly presented.
12
EN
The purpose of this paper is to present some theorems on existence and uniqueness of solution for nonautonomous second order Cauchy problem with a dumping operator and with dependent on t not densely defined operators.
13
Content available remote Implementation and speed up of a parallel algorithm for the cauchy problem
EN
The speed up of a parallel algorithm with respect to sequential one for the case of the Cauchy problem. Four various known numerical methods are applied for solving of the problem. For each method a speed up function is determined. Then a really speed up is given for various number of used processors and points processed by a single processor. The algorithm was implemented on the platform MS .NET in MS Visual C# using a distributed calculation. The obtained results of the really speed up are comparable with theoretical speed up function. The numerical results indicate that efficiency of the parallel computations increases with the number of arithmetical operations needed for one step of used numerical methods.
EN
Using the properties of the Henstock–Kurzweil integral and corresponding theorems, we prove the existence theorem for the equation x(m)(t) = f(t, x) in a Banach space, where f is HL integrable and satis.es certain conditions. Our fundamental tool is the measure of noncompactness developed by Kuratowski and Hausdorff.
15
Content available remote On the unbounded solutions for parabolic differential-functional Cauchy problem
EN
We consider the initial value problem for second order differential–func- tional equation. Functional dependence on an unknown function is of the Hale type. We prove the existence theorem for unbounded classical solution. Our formulation admits a large group of nonlocal problems. We put particular stress on “retarded and deviated” argument as it seems to be the most difficult.
16
Content available remote An existence theorem for set differential inclusions in a semilinear metric space
EN
Using the notion of continuous approximate selections, we establish an existence theorem for set differential inclusions in a semi-linear metric space.
17
Content available remote Semilinear Cauchy problems with almost sectorial operators
EN
Existence of a mild solution to a semilinear Cauchy problem with an almost sectorial operator is studied. Under additional regularity assumptions on the nonlinearity and initial data we also prove the existence of a classical solution to this problem. An example of a parabolic problem in Holder spaces illustrates the abstract result.
EN
The cauchy problem for a quasilinear hyperbolic systems with coefficients functionally dependent on the solutions is studied. We assume that the coefficients are continuous nonlinear operators in the Banach space C1 (R) satisfying some additional assumptions. Under these assumptions we prove the uniqueness and existence of local in time C1 solutions, provided that the initial data are also of class C1.
19
Content available remote Monotonic solutions of multi term fractional differential equations
EN
In this paper, we present an existence of monotonic solutions for a nonlinear multi term non-autonomous fractional differential equation in the Banach space of summable functions. The concept of measure of noncompactness and a fixed point theorem due to G. Emmanuelle is the main tool in carring out our proof.
EN
The analytical properties of dissolving operators related with the Cauchy problem for a class of nonautonomous partial differential equations in Hilbert spaces are studied using theory of bilinear forms in respectively rigged Hilbert spaces triples. Theorems specifying the existence of a dissolving operator for a class of adiabatically perturbed nonautonomous partial differential equations are stated. Some applications of the results obtained are discussed.
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