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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.
PL
W artykule rozważa się zastosowanie metody rozwiązań podstawowych do rozwiązania problemu Cauchy'ego związanego z ustalonym przepływem ciepła w płaskim dwuspójnym obszarze. Metoda jest testowana w obszarze pierścieniowym, dla którego znane jest ścisłe rozwiązanie. W postaci szeregu wykresów przedstawiono wpływ zaburzenia danych oraz odległości konturu źródłowego od brzegu obszaru. Na podstawie eksperymentów numerycznych wykazano, że dobre wyniki można otrzymać bez regularyzacji, jeśli dane są zakłócone niezbyt mocno.
EN
In this paper, the application of the method of fundamental solution to the Cauchy problem in two-connected plane region for steady heat conduction equation is investigated. The method is testet in annlular region for which exact solution is know. The influence of the disturbances of data and the distances of source contour from boundary contour is presented in thy series of graphs. By numerical experiments it is found that solution considered inverse problem could be obtained without regularization for moderate disturbances of data.
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.
5
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.
6
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.
7
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.
8
Content available remote On solvability of linear differential equations in Rn
EN
We construct xo is an element of RN and a row-finite matrix T = {Ti,j(t)}i,j is an element of N of polynomials of one real variable t such that the Cauchy problem x(t) = Ttx(t), x{0) = xo in the Frechet space RN has no solutions. We also construct a row-finite matrix A = {Aij(t)}ij is an element of N of C°°(R) functions such that the Cauchy problem x{t) = Atx;(t), x(0) = xo in RN has no solutions for any xo infinity RN\ {0}. We provide some sufficient condition of solvability and unique solvability for linear ordinary differential equations x(t) = Ttx(t) with matrix elements Ti,j(t) analytically dependent on t.
EN
Classical solutions of the local Cauchy problem on the Haar pyramid are approximated in the paper by solutions of suitable quasilinear systems of difference equations. The proof of the stability of the difference problem is based on a comparison technique with nonlinear estimates of the Perron type. This new approach to the numerical solving of nonlinear functional differential equations is generated by a quasilinearization method for initial problems. Numerical examples are given.
EN
In this work we give an alternative approach to the study of some singular boundary value problems for a second order differential-operator equation in the space of Holder continuous functions. We prove that the solution can be represented explicitly as the sum u = uR + uS of a regular part and a singular part under some natural assumptions on the data. We then give a complete analysis of uR and uS by using the operational calculus.
EN
This paper deals with the mixed problem for the semilinear parabolic equation of the second order in an unbounded domain with some nonlocal boundary data. We prove that there exists the unique global time solution for any locally integrable initial data and right-hand term: hence, no growth condition at infinity for these functions is required.
EN
The paper is concerned with initial-boundary problems for quasilinear infinite systems of first order partial differential functional equations. The unknown function is the functional variable in the system, the partial derivatives appear in a classical sense. A theorem on the existence and uniqueness of the Caratheodory solution and continuous dependence upon initial-boundary data is proved. The mixed problem is equivalent in a suitable function space to a system of functional integral equations. Infinite differential systems with a deviated argument and differential integral problems can be derived from a general model by specializing given functions.
15
Content available remote An abstract second order Cauchy problem with non-densely defined operator, I
EN
By using the theory of extrapolation space X-1 associated with an operator A which is non densely defined in Banach space X, the existence and uniqueness of solutions of linear second order differential initial value problem (1) is proved.
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