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EN
A method of solving the integro-differential equations is presented. The discussed equations will be solved by the Taylor differential transformation. By using appropriate properties of this transformation the integro-differential equation will be transformed to a respective recurrence equation. Unfortunately, the high degree of generality and complexity of such defined problem does not allow to obtain the solution in general form. Each equation requires a special method of solution.
EN
This is the continuation of four earlier studies of a scalar fractional differential equation of Riemann-Liouville type [formula] in which we first invert it as a Volterra integral equation [formula] and then transform it into [formula] where R is completely monotone with [formula] and J is an arbitrary positive constant. Notice that when x is restricted to a bounded set, then by choosing J large enough, we can frequently change the sign of the integrand in going from (b) to (c). Moreover, the same kind of transformation will produce a similar effect in a wide variety of integral equations from applied mathematics. Because of that change in sign, we can obtain an a priori upper bound on solutions of (b) with a parameter λ ∈ (0, 1] and then obtain an a priori lower bound on solutions of (c). Using this property and Schaefer’s fixed point theorem, we obtain positive solutions of an array of fractional differential equations of both Caputo and Riemann-Liouville type as well as problems from turbulence, heat transfer, and equations of logistic growth. Very simple results establishing global existence and uniqueness of solutions are also obtained in the same way.
EN
We propose a numerical surface integral method to study complex acoustic systems, for interior and exterior problems. The method is based on a parametric representation in terms of the arc’s lengths in curvilinear orthogonal coordinates. With this method, any geometry that involves quadric or higher order surfaces, irregular objects or even randomly rough surfaces can be considered. In order to validate the method, the modes in cubic, spherical and cylindrical cavities are calculated and compared to analytical results, which produced very good agreement. In addition, as examples, we calculated the scattering in the far field and the near field by an acoustic sphere and a cylindrical structure with a rough cross-section.
EN
We study the existence, uniqueness and approximation of solutions of stochastic differential equations with constraints driven by processes with bounded p-variation. Our main tool are new estimates showing Lipschitz continuity of the deterministic Skorokhod problem in p-variation norm. Applications to fractional SDEs with constraints are given.
5
Application of integral equations for analysis of electric circuit transients
EN
This paper discusses the particularities of integral equations for calculating the electric circuits transients with multiple switching. Each switching in some way affects constraining forces that is taken into account in certain correction of the right side of integral equation. The numerical method is used to solve integral equations.
PL
W pracy zbadano właściwości równań całkowych opisujących stan nieustalony w obwodach z wielokrotnym przełączaniem. Każde przełączenie powoduje modyfikację prawej strony równań. Pokazano metodę numeryczną rozwiązywania równań całkowych opisujących obwody.
6
Mesh-free methods and time integrations for transient heat conduction
EN
The paper deals with transient heat conduction in functionally gradient materials. The spatial variation of the temperature field is approximated by using alternatively two various mesh free approximations,while the time dependence is treated either by the Laplace transform method and/or by the polynomialinterpolation in the time stepping method. The accuracy and convergence of the numerical results as well as the computational e?ciency of various approaches are compared in numerical test example.
7
Approaches to a mathematical modelling of heterogeneous electrical circuits
EN
The possible formulations of models of heterogeneous circuits are compared in this paper. Advantages and disadvantages of the known methods of such tasks solving are analyzed.
PL
Przedmiotem analizy są obwody o niejednorodnych parametrach rozłożonych. Porównano dwa podejścia do analizy takich obwodów: bezpośrednie rozwiązywanie równań cząstkowych i zastąpienie obwodu o parametrach rozłożonych przez łańcuch czwórników skupionych. Rozpatrywano również zastosowanie diakoptyki do analizy takich obwodów.
EN
Integral equations that describe scattering on the structure with step rise parameters, have been numerically solved on example of the trabecular bone model. The model consists of several hundred elements with randomly selected parameters. The spectral distribution of scatter coefficients in subsequent orders of scattering has been presented.
PL
W pracy zajęto się trójfazowymi torami wielkoprądowymi zbudowanymi z jednofazowych przewodów rurowych, z których każdy umieszczony jest w niemagnetycznej osłonie. Przedstawiono model matematyczny obliczeń parametrów elektrodynamicznych w układzie, takich jak: temperatury przewodów i osłon, rozkład indukcji magnetycznej czy wyznaczanie maksymalnej obciążalności prądowej przy zadanych parametrach geometrycznych, materiałowych i termicznych. Wykorzystano w nich metodę równań całkowych oraz metody analityczne. Zamieszczono również wyniki przeprowadzonych przykładowych obliczeń.
EN
The paper deals with three phase heavy-current busways designed in the form of pipe-conductors, each of them located in a non-magnetic shield. A mathematical model is presented serving computation of electrodynamical parameters of the system, like the conductor and shield temperatures, distribution of magnetic flux density, and determining maximal current chargeability under assumed geometric, material, and thermal parameters. The method of integral equations and analytical methods have been used for this purpose. The results of the example calculation are presented.
EN
We consider the model of optimal portfolio of Mertons’ market model. The noises involved in the dynamics of the wealth are fractional white noises. The stochastic optimal control problem is converted into a non-random optimization. An example of problem numerical solution illustrates proposed methodology.
11
A Kneser-type theorem for an integral equation in locally convex spaces
EN
We shall give suffcient conditions for the existence of solutions of the integral equation (1) in locally convex spaces. We also prove that the set of these solutions is a continuum.
EN
In this paper we show that the set of solutions to the following integral equation.
EN
The problem of adhesive contact for a transversely isotropic elastic half-space is considered. The problem is reduced to the solution of two coupled integral equations, and these are solved exactly. Explicit expressions are found for the contact compliance and for coefficients which characterise the singularities of contact stresses at the boundary of the contact region. The numerical results presented for some anisotropic materials show that the influence of anisotropy on the analysed mechanical quantities is significant.
PL
Rozpatrzono adhezyjne zagadnienie kontaktowe dla poprzecznie izotropowej sprężystej półprzestrzeni. Zagadnienie zredukowano do rozwiązania dwóch sprzężonych równań całkowych, które rozwiązano dokładnie. Znaleziono w postaci jawnej wzory na podatność oraz współczynniki określające osobliwości naprężeń kontaktowych na brzegu obszaru kontaktu. Wyniki liczbowe przedstawione dla różnych materiałów pokazują wpływ anizotropii na analizowane wielkości mechaniczne.
EN
By using the complete discrimination system for polynomials, we study the number of positive solutions in C[0,1] to the integral equation phi(x) = integral[...] k(x,y)phi^n(y)dy, where k(x,y) = phi1(x)phi1(y)+phi2(x)phi2(y),[phi]i(x) > 0,[phi]i(y) > 0,0 < x,y < 1,i = 1,2, are continuous functions on [0,1], n is a positive integer. We prove the following results: when n = 1, either there does not exist, or there exist infinitely many positive solutions in C[0,1]; when n [is greater than or equal] 2, there exist at least 1, at most n + 1 positive solutions in C[0,1]. Necessary and sufficient conditions are derived for the cases: 1) n = 1, there exist positive solutions; 2) n [is greater than or equal to] 2, there exist exactly m (m belongs to {1,2,..., n + 1}) positive solutions. Our results generalize the ones existing in the literature, and their usefulness is shown by examples.
15
On nonlinear integral equations in some functions spaces
EN
There are established some conditions for existence of solutions of a nonlinear integral equation Tf =f+g, where T is a convolution-type integral operator.
16
Computational methods for Volterra - Fredholm integral equations
EN
Integral equations in space-time play very important role in mechanics and technology. Particular cases of these equations called mixed integral equations or Volterra-Fredholm integral equations arise in the heat conduction theory [4, 6] and the diffusion theory. Moreover, a current density in electromagnetism is determined by the Volterra-Fredholm integral equations [4]. Nonlinear counterparts of the equations studied in [1] are mathematical models of the spatio-temporal development of an epidemic (the spread of the disease in the given population). Some initial-boundary problems for a number of partial differential equations in physics are reducible to the considered integral equations [2- 3, 6], In this paper the general theory of these equations is used in the projection methods. Presented methods lead to a system of algebraic equations or to a system of Volterra integral equations. The convergence of studied algorithm is proved, the error estimate is established. The presented theory is illustrated by numerical examples.
PL
Rozważa się nieskończoną tarczę z materiału sprężystego, która jest wzmocniona włóknami i ma szczelinę o długości 2a i nieskończenie małej szerokości. Tarcza jest rozciągana równomiernie w kierunku prostopadłym do szczeliny. Po zastosowaniu transformacji całkowej Fouriera problem można sprowadzić do dualnych równań całkowych. Otrzymano zamknięte wzory dla składowych wektorowego pola przemieszczeń oraz składowych tensorowego pola naprężeń. Przeprowadzone analizy numeryczne wykazują, że wzmocnienie tarczy włóknami wpływa w sposób istotny na rozkład naprężeń w otoczeniu szczeliny.
EN
An infinite sheet, which is made of elastic materiał reinforced with fibres and which has a 2a-long crack with zero crack tip separation, has been considered. The sheet is uniformly tensioned in the direction perpendicular to the crack. After applying the Fourier integral transformation the problem was reduced to dual integral equations. Formulas for the components of a vector field of displacements and the components of a tensor field of stresses were obtained. The numerical analysis carried out reveals a significant influence of the reinforcing of the sheet with fibres on stress distribution in the neighbourhood of the crack.
PL
Omówiono podstawy stosowania rezonatorów dielektrycznych jako promienujących elementów antenowych. Podano krótką charakterystykę rodzajów pola elektromagnetycznego potencjalnie możliwych do zastosowania. Zaprezentowano obliczone metodą równań całkowych rozkłady pola elektrycznego i magnetycznego charakterystyczne dla poszczególnych rodzajów oraz obliczone charakterystyki promieniowania. Scharakteryzowano podstawowe sposoby zasilania dielektrycznych anten rezonatorowych.
EN
The bases for application of dielectric resonators as radiating antenna elements are described. Short characteristics of various electromagnetic field modes possible to apply are given. Authors present model electric and magnetic field distributions calculated with the use of the intetral equation method, as well as calculated radiation patterns. Basic ways of excitation of dielectric antennas are characterized.
EN
Two basic integral-equation formulations for scattering by dielectric bodies are discussed: the surface integral equation and the volume integral equation. Special attention is given to the problem of scattering by dielectrics in a multilayer environment, which is important in a variety of contemporary microwave applications. A new formulation for inhomogeneous bodies with rotational symmetry is introduced both for the case of free-space and for layered media. A new type of divergenceless basis functions is developed for this formulation, which together with the mixed-potential integral-equation technique considerably enhances the algorithm efficiency. New formulas for modal decomposition of incident plane wave fields and far-field approximations are also stated for the multilayer environment. The methods presented are validated through a great number of examples concerning scattering, radiation as well as resonance problems. Example applications of the methods are also outlined.
PL