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An adapted integration method for Volterra integral equation of the second kind with weakly singular kernel

100%

Nemer A. , Kaboul H. , Mokhtari Z.

2021

tom Vol. 27, nr 2

289--297

In this paper, we consider general cases of linear Volterra integral equations under minimal assumptions on their weakly singular kernels and introduce a new product integration method in which we involve the linear interpolation to get a better approximate solution, figure out its effect and also we provide a convergence proof. Furthermore, we apply our method to a numerical example and conclude this paper by adding a conclusion

Invariant measures for stochastic heat equations

100%

Tessitore G. , Zabczyk J.

Probability and Mathematical Statistics

1998

tom Vol. 18, Fasc. 2

271--287

The paper is concerned with the asymptotic behaviour of solutions to the nonlinear stochastic heat equations, with spatially homogeneous noise, in the whole space. Sufficient conditions for the existence of invariant measures, in weighted spaces of locally square-integrable functions, are given. For linear equations with multiplicative noise an invariant measure, supported by positive functions, is constructed. The existence of a stationary solution to the vector Burgers equations is obtained as an application of the general theory.

Preconditioning GMRES for discontinuous Galerkin approximations

75%

Banaś K. , Wheeler M. F.

Computer Assisted Mechanics and Engineering Sciences

2004

tom Vol. 11, No. 1

47-62

The paper presents an implementation and the performance of several preconditioners for the discontinuous Galerkin approximation of diffusion dominated and pure diffusion problems. The preconditioners are applied for the restarted GMRES method and test problems are taken mainly from subsurface flow modeling. Discontinuous Galerkin approximation is implemented within an hp-adaptive finite element code that uses hierarchical 3D meshes. The hierarchy of meshes is utilized for multi-level (multigrid) preconditioning. The results of numerical computations show the necessity of using multi-level preconditioning and insufficiency of simple stationary preconditioners, like Jacobi or Gauss-Seidel. Successful preconditioners comprise a multi-level block ILU algorithm and a special multi-level block Gauss-Seidel method.

Ring-like operations is pseudocomplemented semilattices

75%

Chajda I. , Länger H.

Discussiones Mathematicae - General Algebra and Applications

2000

tom 20

nr 1

87-95

Ring-like operations are introduced in pseudocomplemented semilattices in such a way that in the case of Boolean pseudocomplemented semilattices one obtains the corresponding Boolean ring operations. Properties of these ring-like operations are derived and a characterization of Boolean pseudocomplemented semilattices in terms of these operations is given. Finally, ideals in the ring-like structures are defined and characterized.

Możliwości i ograniczenia informatyki

63%

Mitkowski W.

2008

tom T. 12, z. 2

307-319

Obecnie informatyka przenika wiele dziedzin. Sprzyja temu stosunkowo łatwy dostęp do coraz lepszego i przyjaznego dla użytkownika, łatwego w obsłudze i dobrze oprogramowanego, sprzętu komputerowego. Wizualizacja wyników obliczeń numerycznych wspomaga wyobraźnię i w konsekwencji stanowi dobrą podpowiedź przy stawianiu hipotez naukowych. Równocześnie zwrócono uwagę na ograniczenia poznawcze przy stosowaniu metod komputerowych.

In the paper possibilities and limitations of applied computer science are considered. An attention is drawn to cognitive limitations. On the other hand analysis of computer simulations develops our imagination and makes it easier to formulate research hypothesis.

Sztuczna inteligencja modeluje powierzchnię Ziemi

63%

Olszewski R.

2002

tom nr 5

28--31