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1

Varying coefficients in parallel shared-memory variational splitting solvers for non-stationary Maxwell equations

Łoś Marcin, Woźniak Maciej, Paszyński Maciej

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2024

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Vol. 72, nr 3

art. no. e149179

EN

Direction-splitting implicit solvers employ the regular structure of the computational domain augmented with the splitting of the partial differential operator to deliver linear computational cost solvers for time-dependent simulations. The finite difference community originally employed this method to deliver fast solvers for PDE-based formulations. Later, this method was generalized into so-called variational splitting. The tensor product structure of basis functions over regular computational meshes allows us to employ the Kronecker product structure of the matrix and obtain linear computational cost factorization for finite element method simulations. These solvers are traditionally used for fast simulations over the structures preserving the tensor product regularity. Their applications are limited to regular problems and regular model parameters. This paper presents a generalization of the method to deal with non-regular material data in the variational splitting method. Namely, we can vary the material data with test functions to obtain a linear computational cost solver over a tensor product grid with non-regular material data. Furthermore, as described by the Maxwell equations, we show how to incorporate this method into finite element method simulations of non-stationary electromagnetic wave propagation over the human head with material data based on the three-dimensional MRI scan.

2

Singular integration in boundary element method for Helmholtz equation formulated in frequency domain

Rymarczyk Tomasz, Sikora Jan

Informatyka, Automatyka, Pomiary w Gospodarce i Ochronie Środowiska

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2021

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T. 11, nr 4

4--8

EN

Two ways of approximation of the BEM kernel singularity are presented in this paper. Based on these approximations extensive error analysis was carried on. As a criterion the preciseness and simplicity of approximation were selected. Simplicity because such approach would be applied for the tomography problems, so time of execution plays particularly significant role. One of the approximations which could be applied for the wide range of the arguments of the kernel were selected.

PL

Dwie metody aproksymacji osobliwości funkcji Greena zaproponowano w tej pracy. Bazując na tych aproksymacjach dokonano wnikliwej analizy błędów. Jako kryterium wybrano dokładność i prostotę zaproponowanych aproksymacji. Prostotę dlatego, że takie podejście będzie proponowane w zagadnieniach tomograficznych. Tak więc czas odgrywa zasadniczą rolę. Wybrano aproksymację, która może być stosowana dla szerokiego zakresu argumentów.

3

Existence and Ulam-Hyers stability of the implicit fractional boundary value problem with ψ-Caputo fractional derivative

Wahash Hanan A., Abdo Mohammed S, Panchal Satish K.

Journal of Applied Mathematics and Computational Mechanics

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2020

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Vol. 19, nr 1

89--101

EN

In this paper, we investigate the existence, uniqueness and Ulam-Hyers stability of solutions for nonlinear implicit fractional differential equations with boundary conditions involving a ψ-Caputo fractional derivative. The obtained results for the proposed problem are proved under a new approach and minimal assumptions on the function ƒ. The analysis is based upon the reduction of the problem considered to the equivalent integral equation, while some fixed point theorems of Banach and Schauder and generalized Gronwall inequality are employed to obtain our results for the problem at hand. Finally, the investigation is illustrated by providing a suitable example.

4

Hyers-Ulam stability of a coupled system of fractional differential equations of Hilfer-Hadamard type

Ahmad Manzoor, Zada Akbar, Alzabut Jehad

Demonstratio Mathematica

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2019

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Vol. 52, nr 1

283--295

EN

In this paper, existence and uniqueness of solution for a coupled impulsive Hilfer-Hadamard type fractional differential system are obtained by using Kransnoselskii’s fixed point theorem. Different types of Hyers-Ulam stability are also discussed. We provide an example demonstrating consistency to the theoretical findings.

5

Positive solution of a fractional differential equation with integral boundary conditions

Abdo M. S., Wahash H. A., Panchal S. K.

Journal of Applied Mathematics and Computational Mechanics

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2018

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Vol. 17, nr 3

5--15

EN

In this paper, we prove the existence and uniqueness of a positive solution for a boundary value problem of nonlinear fractional differential equations involving a Caputo fractional operator with integral boundary conditions. The technique used to prove our results depends on the upper and lower solution, the Schauder fixed point theorem and the Banach contraction principle. The result of existence obtained through constructing the upper and lower control functions of the nonlinear term without any monotone requirement.

6

A new imaging algorithm for electric capacitance tomography

Pańczyk M., Sikora J.

Prace Instytutu Elektrotechniki

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2016

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Z. 274

27--41

EN

A new imaging algorithm of permittivity for Electrical Capacitance Tomography (ECT) was proposed in this paper. Some aspects of sensitivity analysis and the application of adjoint variables were discussed. In the final section of this paper some advantages and disadvantages were shown.

PL

W pracy przedstawiono „nowy” algorytm obrazowania przenikalności elektrycznej w Elektrycznej Tomografii Pojemnościowej. Omówiono aspekty związane z analizą wrażliwościową i zmienna sprzężoną. Przedstawiono jej interpretację fizyczną. Wskazano na zalety i wady zaproponowanej metody obrazowania.

7

Implicit solution of 1d nonlinear porous medium equation using the four-point Newton- EGMSOR iterative method

Chew J. V. L., Sulaiman J.

Journal of Applied Mathematics and Computational Mechanics

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2016

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Vol. 15, nr 2

11--21

EN

The numerical method can be a good choice in solving nonlinear partial differential equations (PDEs) due to the difficulty in finding the analytical solution. Porous medium equation (PME) is one of the nonlinear PDEs which exists in many realistic problems. This paper proposes a four-point Newton-EGMSOR (4-Newton-EGMSOR) iterative method in solving 1D nonlinear PMEs. The reliability of the 4-Newton-EGMSOR iterative method in computing approximate solutions for several selected PME problems is shown with comparison to 4-Newton-EGSOR, 4-Newton-EG and Newton-Gauss-Seidel methods. Numerical results showed that the proposed method is superior in terms of the number of iterations and computational time compared to the other three tested iterative methods.

8

Stanisław Zaremba (1863‒1942) and his results in the field of differential equations

Koroński J.

Czasopismo Techniczne. Nauki Podstawowe

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2015

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Y. 112, iss. 2-NP

207--217

EN

The subject of the paper is presentation of the publications of Stanisław Zaremba in the field of partial differential equations. We present selected results in detail. Zaremba published about 120 works. Among them more than 60 were devoted to partial differential equations.

PL

Artykuł poświęcony jest prezentacji publikacji Stanisława Zaremby w dziedzinie równań różniczkowych cząstkowych. Wybrane rezultaty zaprezentowano bliżej. Zaremba opublikował około 120 prac, z których ponad 60 jest poświęconych równaniom różniczkowym cząstkowym.

9

Stanisław Kępiński (1867‒1908) and his papers in the field of differential equations

Koroński J.

Czasopismo Techniczne. Nauki Podstawowe

|

2015

|

Y. 112, iss. 2-NP

201--206

EN

The subject of this paper is an analysis of the publications of Stanisław Kępiński in the field of ordinary and partial differential equations. In particular we present part I and part II of the monograph (textbook) of Stanisław Kępiński on the ordinary and partial differential equations.

PL

Artykuł poświęcony jest prezentacji publikacji Stanisława Kępińskiego w dziedzinie zwyczajnych i cząstkowych równań różniczkowych. W pracy prezentujemy zwłaszcza dwuczęściową monografię (podręcznik) z równań różniczkowych zwyczajnych i cząstkowych.

10

Układy niecałkowitego rzędu i ich wybrane zastosowania

Mitkowski W.

Hutnik, Wiadomości Hutnicze

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2014

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Vol. 81, nr 1

39--43

PL

W pracy omówiono podstawowe własności układów niecałkowitego rzędu. Podano wybrane przykłady zastosowań z następujących obszarów: modelowanie, identyfikacja, regulacja automatyczna. Przedstawiono również przykłady symulacji komputerowych.

EN

The paper discusses the basic properties of fractional systems. Given some examples of applications of the following areas: modeling, identification, automatic control. It also presents examples of computer simulations.

11

Oscillation for certain impulsive partial difference equations

Özpinar F., Öztürk S., Koçak Z.F.

Demonstratio Mathematica

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2014

|

Vol. 47, nr 1

79--102

EN

In this paper, we obtain some sufficient criteria for the oscillation of the solutions of linear impulsive partial difference equations with continuous variables.

12

Władysław Zajączkowski and differential equations in Poland in the second half on the nineteenth century

Koroński J.

Czasopismo Techniczne. Nauki Podstawowe

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2014

|

R. 111, z. 1-NP

107--117

EN

The paper contains some scientific information on Władysław Zajączkowski (1837–1898) and on his first Polish monograph about ordinary and partial differential equations. Moreover, the aim of this paper is a presentation of selected scientific results of Polish mathematicians publishing in the nineteenth century in the field of ordinary and partial differential equations. Some more details about the publications on differential equations in the 19th century written by Polish mathematicians can be found in [3‒9].

PL

W artykule zawarto pewne informacje naukowe o Władysławie Zajączkowskim i jego pierwszej polskiej monografii z równań różniczkowych zwyczajnych i cząstkowych. Ponadto przedmiotem pracy jest prezentacja wybranych rezultatów naukowych matematyków polskich publikujących w drugiej połowie dziewiętnastego wieku w dziedzinie równań różniczkowych zwyczajnych i cząstkowych. Pewne szczegółowe informacje o publikacjach z równań różniczkowych w dziewiętnastym wieku napisanych przez matematyków polskich można znaleźć w [3‒9].

13

Mirosław Krzyżański (1907-1965) i jego monografia z równań różniczkowych cząstkowych

Koroński J.

Kwartalnik Historii Nauki i Techniki

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2013

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R. 58, nr 4

7--28

EN

The present paper contains a spacious biography of Mirosław Krzyżański (1907-1965) and some information on the monograph about partial differential equ ations of second order, which was edited: in Polish - part I in 1957, and part II in 1962, and in English by PWN in 1971. Moreover in the paper some information on the most important original scientific results and bibliography of M. Krzyżański are given.

14

Modelling of mass transport in watercourses considering mass transfer between phases in unsteady states. Part II. Mass transport during absorption and adsorption processes

Bielski A.

Environment Protection Engineering

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2011

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Vol. 37, nr 4

71-89

EN

Equations describing the rate of adsorption and absorption processes and those based on Whitman's model have been analyzed. In the case of unstable states, the mass flux penetrating to the layer of the river sediment and calculated by means of these equations differs from the mass flux calculated from the mass diffusion equation. In order to minimize the discrepancies between the flux determined by Whitman's model and a real flux, the correction factor has been introduced into the concentration gradient equation originated from Whitman's model. This correction factor can be expressed as a time dependence of the product of a certain parameter and the concentration derivative at the phase boundary (solid phase side). The corrected equation for the concentration gradient has been used to derive another equation, describing a general rate of the absorption and adsorption processes at the linear interfacial equilibrium and the chemical reactions occurring in the liquid and solid phases; the chemical reactions follow the first order monomolecular mechanism in unstable states with reference to the liquid phase. Knowing the general rate of the earlier mentioned processes it is possible to construct an advective-dispersion model of mass transport in a river including these particular processes. Such a model contains a term defined as a correction factor referring to the time dependence of the concentration derivative with respect to time. The described model may be also used for simulation of the transport of pollutants undergoing adsorption and absorption in the layer river sediment; the processes occur with a finite and infinitely large rate through the equilibrium states.

15

Integral equation arising in the theory of partial differential equations

Pachpatte B. G.

Fasciculi Mathematici

|

2010

|

Nr 43

115-126

EN

The aim of the present paper is to study some basic qualitative properties of solutions of a certain integral equation arising in the theory of partial differential equations. The well known Banach fixed point theorem and the new integral inequality with explicit estimate obtained in the present paper are used to establish the results.

16

Solution of non-linear Klein-Gordan equation with a quadratic non-linear term by adomian decomposition method

Basak K. Ch., Ray P. Ch., Bera R. K.

International Journal of Applied Mechanics and Engineering

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2009

|

Vol. 14, no 3

891-897

EN

Nonlinear PDEs are systematically solved by the decomposition method of Adomian for general boundary conditions described by boundary operator equations. In the present case, the solution of the nonlinear Klein-Gordon equation has been considered as an illustration of the decomposition method of Adomian.

17

Growth estimates on mixed Volterra-Fredholm type integral inequalities

Pachpatte B. G.

Fasciculi Mathematici

|

2009

|

Nr 42

63-72

EN

In this paper we establish explicit estimates on some mixed Volterra-Fredholm type integral inequalities which can be used as tools in certain applications. Some applications are also given to illustrate the usefulness of the results.

18

Asymptotic behaviour of the fundamental solution of the equations of transversally isotropic elastic medium

Jaworska K.

Biuletyn Wojskowej Akademii Technicznej

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2003

|

Vol. 52, nr 7

31-40

EN

In this paper we consider the equations arised for harmonic waves in a transversely isotropic elastic medium. Our aim is to investigate the asymptotic behaviour of the solution at large distances.

PL

Praca dotyczy zjawiska rozchodzenia się fal harmonicznych w poprzecznie jednorodnym ośrodku sprężystym opisanego za pomocą pewnego układu równań różniczkowych cząstkowych. Celem było wyznaczenie asymptotyki rozwiązania tego układu równań przy dużych odległościach od źródła.

19

Metoda analizy dynamiki układów napędowych z parametrami rozłożonymi

Sikora-Iliew R., Potrawka S.

Elektrotechnika i Elektronika

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2002

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T. 21, z. 1

82--89

PL

W artykule przedstawiono metodę analizy dynamiki układów elektromechanicznych z parametrami rozłożonymi. Dynamikę układu napędowego z parametrami rozłożonymi zapisano w oparciu o równanie różniczkowe cząstkowe-równanie falowe. Model matematyczny opisujący w sposób ogólny dynamikę napędu stworzono w oparciu o energetyczna funkcję stanu Lagrange'a przy uwzględnieniu energii potencjalnych i koenergii kinetycznych układu elektromechanicznego. Równania opisujące przemianę elektromechaniczną zapisano w oparciu o zasadę najmniejszego działania Hamiltona i równania Eulera—Lagrange'a. Analiza układu poparta została badaniami symulacyjnymi wybranych systemów elektromechanicznych. Przypadek systemu elektromechanicznego powiązanego materiałowo rozważono dla układu napędowego zwijarki. Uzyskane wyniki i wnioski końcowe mogą być przydatne również dla dydaktyki.

EN

The dynamics analysis for distributed-parameters and concentrated- parameters system drive's are presented in this paper. The distributed-parameters system drive's dynamics has described by use partial differential equation - wave equation. This equation for system drive's has solved by change to differential-difference equation. A mathematical model describing drive's system dynamics universally has given in this paper. This model has created by use Lagrange function of state. Equations describing electromechanical conversion have derived by use Hamilton' s principle of least action and Euler - Lagrange equations. The analysis has been backed up by simulation of selected electromechanical systems. The case of material-connected electromechanical system have been also considered by simulation of a coiling machine drive. Obtained results allowed for final conclusions, which can be useful for educational purposes too.

20

Budowa operatorów różnicowych dla równań różniczkowych cząstkowych wyższych rzędów

Włodarczyk M.

Zeszyty Naukowe Politechniki Świętokrzyskiej. Elektryka

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2000

|

Z. 37

129-138

PL

W pracy przedstawiono metody budowy operatorów różnicowych dla pochodnych wyższych rzędów w zastosowaniu do schematu różnicowego dla pewnego równania różniczkowego cząstkowego czwartego rzędu zawierającego pochodne cząstkowe mieszane. W literaturze można znaleźć wyrażenia na pochodne różnicowe trzeciego i czwartego rzędu dla funkcji jednej zmiennej, jednak pochodne cząstkowe mieszane funkcji dwóch zmiennych są bardzo rzadko lub w ogóle nie są rozpatrywane. Podano również błędy obcięcia wyznaczonych pochodnych różnicowych w oparciu o szereg Taylora.

EN

This paper presents methods of construction of difference operators for higher order derivatives to be used in a difference scheme for certain partial differential equation of fourth order containing mixed partial derivatives. In literature one can find derivative expressions on difference derivatives of third and of fourth orders for a function of one variable. However mixed partial derivatives for a function of two variables are examined very seldom. Truncation error of determined difference derivatives on the basis of Taylor series has also been given.