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1

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

Nemer Ahlem, Kaboul Hanane, Mokhtari Zouhir

Journal of Applied Analysis

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2021

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Vol. 27, nr 2

289--297

EN

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

2

Solving system of linear equations via bicomplex valued metric space

Gnanaprakasam Arul Joseph, Boulaaras Salah Mahmoud, Mani Gunaseelan, Cherif Bahri, Idris Sahar Ahmed

Demonstratio Mathematica

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2021

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Vol. 54, nr 1

474--487

EN

In this paper, we prove some common fixed point theorems on bicomplex metric space. Our results generalize and expand some of the literature’s well-known results. We also explore some of the applications of our key results.

3

Możliwości i ograniczenia informatyki

Mitkowski W.

Automatyka / Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie

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2008

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T. 12, z. 2

307-319

PL

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.

EN

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.

4

Preconditioning GMRES for discontinuous Galerkin approximations

Banaś K., Wheeler M. F.

Computer Assisted Mechanics and Engineering Sciences

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2004

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Vol. 11, No. 1

47-62

EN

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.

5

Sztuczna inteligencja modeluje powierzchnię Ziemi

Olszewski R.

Geodeta : magazyn geoinformacyjny

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2002

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nr 5

28--31

6

Invariant measures for stochastic heat equations

Tessitore G., Zabczyk J.

Probability and Mathematical Statistics

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1998

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Vol. 18, Fasc. 2

271--287

EN

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.