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1

Computation of solution of integral equations via fixed point results

Alqudah Manar A., Garodia Chanchal, Uddin Izhar, Nieto Juan J.

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

772--785

EN

The motive of this article is to study a modified iteration scheme for monotone nonexpansive mappings in the class of uniformly convex Banach space and establish some convergence results. We obtain weak and strong convergence results. In addition, we present a nontrivial numerical example to show the convergence of our iteration scheme. To demonstrate the utility of our scheme, we discuss the solution of nonlinear integral equations as an application, which is again supported by a nontrivial example.

2

Procedural generation of aesthetic patterns from dynamics and iteration processes

Gdawiec K.

International Journal of Applied Mathematics and Computer Science

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2017

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Vol. 27, no. 4

827--837

EN

Aesthetic patterns are widely used nowadays, e.g., in jewellery design, carpet design, as textures and patterns on wallpapers, etc. Most of the work during the design stage is carried out by a designer manually. Therefore, it is highly useful to develop methods for aesthetic pattern generation. In this paper, we present methods for generating aesthetic patterns using the dynamics of a discrete dynamical system. The presented methods are based on the use of various iteration processes from fixed point theory (Mann, S, Noor, etc.) and the application of an affine combination of these iterations. Moreover, we propose new convergence tests that enrich the obtained patterns. The proposed methods generate patterns in a procedural way and can be easily implemented on the GPU. The presented examples show that using the proposed methods we are able to obtain a variety of interesting patterns. Moreover, the numerical examples show that the use of the GPU implementation with shaders allows the generation of patterns in real time and the speed-up (compared with a CPU implementation) ranges from about 1000 to 2500 times.

3

Weak convergence for nonself nearly asymptotically nonexpansive mappings by iterations

Khan S. H.

Demonstratio Mathematica

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2014

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Vol. 47, nr 2

371--381

EN

In this paper, we obtain a couple of weak convergence results for nonself nearly asymptotically nonexpansive mappings. Our first result is for the Banach spaces satisfying Opial condition and the second for those whose dual satisfies the Kadec-Klee property.

4

The improvement of numerical modelling of airflow in ventilated room

Ciuman P., Lipska B.

Architecture Civil Engineering Environment

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2014

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Vol. 7, no. 3

77--83

EN

In the paper possibility of improvement of airflow modeling in ventilated room was checked. Standard perpendicular room was examined. Into it, through the inlet, isothermal jet was supplied. Numerical simulations were carried out with the use of ANSYS CFX software, based on CFD method. Calculations were executed for steady conditions, with the use of k-ε turbulence model. Results of experiments, conducted in room’s physical model, in a region of supplying jet, with the benefit of laser doppler anemometer, were used for the validation of numerical simulations. Simulations were carried out for various variants of discretization grid. Cells size and their refinement were checked in terms of conformity to experiments results. For chosen best variant, effect of iteration process on jet flow’s pattern was observed.

PL

W artykule sprawdzono możliwość poprawy dokładności modelowania przepływu powietrza w pomieszczeniu wentylowanym. Badano typowe prostopadłościenne pomieszczenie, do którego przez prostokątną kratkę nawiewano izotermiczną strugę powietrza. Obliczenia numeryczne wykonano za pomocą programu ANSYS CFX bazującego na metodzie CFD. Obliczenia przeprowadzono dla warunków ustalonych przy wykorzystaniu modelu turbulencji k-ε. Do walidacji prognoz numerycznych wykorzystano wyniki pomiarów przeprowadzonych w modelu fizykalnym obiektu, w rejonie strugi na wiewanej, za pomocą dopplerowskiego anemometru laserowego. Symulacje przeprowadzono dla różnych wariantów siatki dyskretyzacji, badając wpływ wymiarów i zagęszczenia oczek na zgodność wyników z pomiarami. Dla wybranego najlepszego wariantu obserwowano wpływ przebiegu procesu iteracji na obraz przepływu strugi.

5

A note on the theorem of V. Berinde

Rafiq A.

Demonstratio Mathematica

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2006

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Vol. 39, nr 4

863-867

EN

V. Berinde presented convergence theorem [1] in normed spaces for Ishikawa iterations associated with Zamfirescue operators, which is an extension of the theorem of Rhoades [11]. In this note, we point out some misprints in the calculations and established new inequalities involved in the proof of the theorem. Consequently we give better estimation in proving the convergence theorem.