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  1. 5 Demonstratio Mathematica
  2. 4 International Journal of Applied Mathematics and Computer Science
  3. 3 Fasciculi Mathematici
  4. 3 Opuscula Mathematica
  5. 1 Archives of Control Sciences
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  1. 3 Bartecki K.
  2. 2 Cheng S. S.
  3. 2 Li W.
  4. 1 Abdulazeez Sadeq Taha
  5. 1 Bartłomiejczyk Agnieszka
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  1. 3 2023
  2. 1 2022
  3. 1 2021
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  5. 1 2019
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1
Content available Stability analysis and numerical implementation of the third-order fractional partial differential equation based on the Caputo fractional derivative
Rasheed Sarbast Kamal, Modanli Mahmut, Abdulazeez Sadeq Taha
Journal of Applied Mathematics and Computational Mechanics
|
2023
|
Vol. 22, nr 3
33--42
EN
This paper examines a third-order fractional partial differential equation (FPDE) in the Caputo sense. The Theta difference method (TDM) is utilized to investigate the problem, and a first-order difference scheme is developed. Stability estimates are obtained by applying the Von Neumann analysis method. A test problem is presented as an application, and numerical results are obtained using Matlab software. Error estimates, as well as exact and approximate solutions are presented in a data analysis table. The simulation results are shown through error analysis tables and figures.
2
Metamorphism - an integral transform reducing the order of a differential equation
Kisil Vladimir V.
Journal of Applied Analysis
|
2023
|
Vol. 29, nr 2
219--227
EN
We propose an integral transform, called metamorphism, which allows us to reduce the order of a differential equation. For example, the second-order Helmholtz equation is transformed into a first-order equation, which can be solved by the method of characteristics.
3
Content available Travelling waves for low-grade glioma growth and response to a chemotherapy model
Bartłomiejczyk Agnieszka, Bodnar Marek, Bogdańska Magdalena U., Piotrowska Monika J.
International Journal of Applied Mathematics and Computer Science
|
2023
|
Vol. 33, no. 4
569--581
EN
Low-grade gliomas (LGGs) are primary brain tumours which evolve very slowly in time, but inevitably cause patient death. In this paper, we consider a PDE version of the previously proposed ODE model that describes the changes in the densities of functionally alive LGGs cells and cells that are irreversibly damaged by chemotherapy treatment. Besides the basic mathematical properties of the model, we study the possibility of the existence of travelling wave solutions in the framework of Fenichel’s invariant manifold theory. The estimates of the minimum speeds of the travelling wave solutions are provided. The obtained analytical results are illustrated by numerical simulations.
4
Content available remote Applications of the second-order edge element method versus zero order in optical diffusion tomography
Rymarczyk Tomasz, Sikora Jan
Przegląd Elektrotechniczny
|
2022
|
R. 98, nr 1
107--110
EN
This paper presents an approximation of two boundary elements by analysing numerical aspects of such an approximation. Diffusion optical tomography with defined region geometry was used for this purpose. The Helmholtz equation in the frequency domain was transformed to integral form. The inverse problem was defined as an optimal shape design problem. The conversion of the imaging problem to an inverse task required the solution of the PDE by the BEM. Remesh in the optimisation process is not required in contrast to the solution of the MES. Two different approximations are compared, and the results of the research work are presented. The proposed method depends on the configuration of the object or objects within the region and the starting position and dimensions of the circular inclusion.
PL
W artykule przedstawiono aproksymację dwóch elementów brzegowych analizując numerycznych aspekty takiego przybliżenia. Wykorzystano do tego dyfuzyjną tomografię optyczną z określoną geometrią regionu. Równanie Helmholtza w dziedzinie częstotliwości przekształcono do postaci całkowej. Problem odwrotny został zdefiniowany jako problem projektowania optymalnego kształtu. Konwersja problemu obrazowania na zadanie odwrotne wymagała rozwiązania PDE przez BEM. Remesh w procesie optymalizacji nie jest wymagany w przeciwieństwie do rozwiązania MES. Porównano dwa różne przybliżenia. Proponowana metoda silnie zależy od konfiguracji obiektu lub obiektów wewnątrz regionu oraz pozycji wyjściowej i wymiarów wtrącenia kołowego.
5
Content available On classical symmetries of ordinary differential equations related to stationary integrable partial differential equations
Tsyfra Ivan
Opuscula Mathematica
|
2021
|
Vol. 41, no. 5
685--699
EN
We study the relationship between the solutions of stationary integrable partial and ordinary differential equations and coefficients of the second-order ordinary differen¬tial equations invariant with respect to one-parameter Lie group. The classical symmetry method is applied. We prove that if the coefficients of ordinary differential equation satisfy the stationary integrable partial differential equation with two independent variables then the ordinary differential equation is integrable by quadratures. If special solutions of integrable partial differential equations are chosen, then the coefficients satisfy the stationary KdV equations. It was shown that the Ermakov equation belong to a class of these equations. In the framework of the approach we obtained the similar results for generalized Riccati equations. By using operator of invariant differentiation we describe a class of higher order ordinary differential equations for which the group-theoretical method enables us to reduce the order of ordinary differential equation.
6
Content available Stabilized model reduction for nonlinear dynamical systems through a contractivity-preserving framework
Chaturantabut Saifon
International Journal of Applied Mathematics and Computer Science
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2020
|
Vol. 30, no. 4
615--628
EN
This work develops a technique for constructing a reduced-order system that not only has low computational complexity, but also maintains the stability of the original nonlinear dynamical system. The proposed framework is designed to preserve the contractivity of the vector field in the original system, which can further guarantee stability preservation, as well as provide an error bound for the approximated equilibrium solution of the resulting reduced system. This technique employs a low-dimensional basis from proper orthogonal decomposition to optimally capture the dominant dynamics of the original system, and modifies the discrete empirical interpolation method by enforcing certain structure for the nonlinear approximation. The efficiency and accuracy of the proposed method are illustrated through numerical tests on a nonlinear reaction diffusion problem.
7
Content available Partial linear homogeneous differential equations of the first order and Matheamtica
Czajkowski Andrzej Antoni, Oleszak Wojciech Kazimierz
Problemy Nauk Stosowanych
|
2019
|
T. 10
5--14
EN
Introduction and aims: The paper describes the method of solving first order linear differential homogeneous differential equations using Mathematica program. The purpose of the work is to provide algorithms for analytical and symbolic solutions in Mathematica for three selected examples. Material and methods: The work uses selected literature from first order linear partial differential equations. The method of characteristics was used in analytical solutions, and the Mathematica 5 program in numerical solutions. Results: The characteristics method was used in analytical solutions of selected examples of first order linear partial differential equations. In addition to numerical solutions, graphic interpretation was given using spatial and contour charts. Conclusion: Mathematica program solves the first order linear partial differential equations with given boundary conditions using the pde and DSolve procedures. Mathematica program also allows for first order linear partial differential equations with boundary conditions to show some geometric interpretation of their solutions using the Plot3D and ContourPlot commands.
PL
Wstęp i cele: W pracy opisano metodę rozwiązywania równań różniczkowych cząstkowych liniowych jednorodnych pierwszego rzędu z wykorzystaniem programu Mathematica. Celem pracy jest podanie algorytmów rozwiązań analitycznych i symbolicznych w programie Mathematica dla wybranych trzech różnych przykładów. Materiał i metody: W pracy wykorzystano wybraną literaturę z równań różniczkowych cząstkowych liniowych rzędu pierwszego. W rozwiązaniach analitycznych zastosowano metodę charakterystyk, a w rozwiązaniach numerycznych program Mathematica 5. Wyniki: Metodę charakterystyk zastosowano w rozwiązaniach analitycznych wybranych przykładów równań różniczkowych cząstkowych liniowych rzędu pierwszego. Oprócz rozwiązań numerycznych podano interpretację graficzną stosując wykresy przestrzenne i konturowe. Wnioski: Program Mathematica rozwiązuje liniowe jednorodne równania różniczkowe cząstkowe pierwszego rzędu z zadanymi warunkami brzegowymi stosując procedury pde i DSolve. Program Mathematica umożliwia również dla równań różniczkowych cząstkowych liniowych rzędu pierwszego z warunkami brzegowymi pokazanie geometrycznej interpretacji ich rozwiązań za pomocą poleceń Plot3D i ContourPlot.
8
Content available Feedback design of differential equations of reconstruction for second-order distributed parameter systems
Maksimov V. I., Mordukhovich B. S.
International Journal of Applied Mathematics and Computer Science
|
2017
|
Vol. 27, no. 3
467--475
EN
The paper aims at studying a class of second-order partial differential equations subject to uncertainty involving unknown inputs for which no probabilistic information is available. Developing an approach of feedback control with a model, we derive an efficient reconstruction procedure and thereby design differential equations of reconstruction. A characteristic feature of the obtained equations is that their inputs formed by the feedback control principle constructively approximate unknown inputs of the given second-order distributed parameter system.
9
Content available Mass transfer effects on an unsteady MHD free convective flow of an incompressible viscous dissipative fluid past an infinite vertical porous plate
Reddy B. P.
International Journal of Applied Mechanics and Engineering
|
2016
|
Vol. 21, no. 1
143--155
EN
In this paper, a numerical solution of mass transfer effects on an unsteady free convection flow of an incompressible electrically conducting viscous dissipative fluid past an infinite vertical porous plate under the influence of a uniform magnetic field considered normal to the plate has been obtained. The non-dimensional governing equations for this investigation are solved numerically by using the Ritz finite element method. The effects of flow parameters on the velocity, temperature and concentration fields are presented through the graphs and numerical data for the skin-friction, Nusselt and Sherwood numbers are presented in tables and then discussed.
10
Content available On the summability of divergent power series solutions for certain first-order linear PDES
Hibino M.
Opuscula Mathematica
|
2015
|
Vol. 35, no. 5
595--624
EN
This article is concerned with the study of the Borel summability of divergent power series solutions for certain singular first-order linear partial differential equations of nilpotent type. Our main purpose is to obtain conditions which coefficients of equations should satisfy in order to ensure the Borel summability of divergent solutions. We will see that there is a close affinity between the Borel summability of divergent solutions and global analytic continuation properties for coefficients of equations.
11
Content available remote O jednoznaczności rozwiązań równań pola elektromagnetycznego w obszarze anizotropowym i niestacjonarnym
Spałek D.
Prace Naukowe Politechniki Śląskiej. Elektryka
|
2014
|
z. 4
7--16
PL
Artykuł prezentuje rozważania teoretyczne, dotyczące jednoznaczności rozwiązań równań różniczkowych cząstkowych drugiego rzędu oraz równań Maxwella w środowiskach anizotropowych i niestacjonarnych.
EN
The paper presents theoretical analysis of uniqueness for solution of second order partial differential equations and of Maxwell equations in anisotropic and nonstationary region.
12
Content available Transmitancje operatorowe pewnej klasy układów typu hiperbolicznego z wejściami brzegowymi
Bartecki K.
Pomiary Automatyka Kontrola
|
2014
|
R. 60, nr 1
49--52
PL
W artykule przedstawiono ogólną postać transmitancji operatorowych pewnej klasy układów o parametrach rozłożonych, opisanych dwoma równaniami różniczkowymi cząstkowymi typu hiperbolicznego. Zakładając istnienie w układzie dwóch wejść o charakterze wymuszeń brzegowych typu Dirichleta oraz dwóch wyjść rozłożonych wzdłuż osi zmiennej przestrzennej, przedstawiono wyrażenia opisujące transmitancje operatorowe układu dla dwóch różnych konfiguracji sygnałów wejściowych. Rozważania zilustrowano praktycznym przykładem wymiennika ciepła pracującego w układach: współ- oraz przeciwprądowym.
EN
Transfer function models for a class of distributed parameter systems described by the two hyperbolic partial differential equations defined on a one-dimensional finite spatial domain are considered. Assuming two boundary inputs of Dirichlet type, the closed-form expressions for the individual elements of the 22 transfer function matrix are proposed based on the decoupled canonical representation of the system. The influence of the location of the boundary inputs on the transfer function representation is demonstrated for two different input configurations. The first one is the so-called congruent arrangement, for which both inputs act on the system at the same spatial position, l=0 (Fig. 1). The second one is the incongruent arrangement, where both inputs act on the system at its opposite ends, l=0 and l=L, respectively (Fig. 2). The considerations are illustrated with a practical example of a shell and tube heat exchanger operating in parallel- and countercurrent-flow modes (Fig. 3), which correspond to the two abovementioned boundary input configurations. Based on the transfer function model, both frequency and time responses of the system can be determined, which can be useful e.g. in the case of the model-based fault detection scheme.
13
Układy niecałkowitego rzędu i ich wybrane zastosowania
Mitkowski W.
Hutnik, Wiadomości Hutnicze
|
2014
|
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.
14
Content available A general transfer function representation for a class of hyperbolic distributed parameter systems
Bartecki K.
International Journal of Applied Mathematics and Computer Science
|
2013
|
Vol. 23, no. 2
291--307
EN
Results of transfer function analysis for a class of distributed parameter systems described by dissipative hyperbolic partial differential equations defined on a one-dimensional spatial domain are presented. For the case of two boundary inputs, the closed-form expressions for the individual elements of the 2×2 transfer function matrix are derived both in the exponential and in the hyperbolic form, based on the decoupled canonical representation of the system. Some important properties of the transfer functions considered are pointed out based on the existing results of semigroup theory. The influence of the location of the boundary inputs on the transfer function representation is demonstrated. The pole-zero as well as frequency response analyses are also performed. The discussion is illustrated with a practical example of a shell and tube heat exchanger operating in parallel- and countercurrent-flow modes.
15
Content available Steady-state analysis for a class of hyperbolic systems with boundary inputs
Bartecki K.
Archives of Control Sciences
|
2013
|
Vol. 23, no. 3
295--310
EN
Results of a steady-state analysis performed for a class of distributed parameter systems described by hyperbolic partial differential equations defined on a one-dimensional spatial domain are presented. For the case of the system with two state variables and two boundary inputs, the analytical expressions for the steady-state distribution of the state variables are derived, both in the exponential and in the hyperbolic form. The influence of the location of the boundary inputs on the steady-state response is demonstrated. The considerations are illustrated with a practical example of a shell and tube heat exchanger operating in parallel- and countercurrent-flow modes.
16
Content available remote Stochastic semigroups and their applications to biological models
Pichór K., Rudnicki R., Tyran-Kamińska M.
Demonstratio Mathematica
|
2012
|
Vol. 45, nr 2
463-494
EN
Some recent results concerning generation and asymptotic properties of stochastic semigroups are presented. The general results are applied to biological models described by piecewise deterministic Markov processes: birth-death processes, the evolution of the genome, genes expression and physiologically structured models.
17
Content available remote Estymacja stanu obiektu opisanego równaniami różniczkowymi cząstkowymi typu hiperbolicznego z zastosowaniem sztucznych sieci neuronowych
Kwater T.
Zeszyty Naukowe Politechniki Rzeszowskiej. Elektrotechnika
|
2011
|
z. 31 [281]
77-90
PL
W artykule przedstawiono zagadnienia związane z matematycznym modelowaniem zanieczyszczeń organicznych w rzece oraz zagadnienie estymacji stanu rzeki. Zaprezentowane modele bazują na zwyczajnych równaniach różniczkowych, które rozszerza się do równań różniczkowych cząstkowych, opisujących zjawisko transportu i dyfuzji. Nie zmniejszając dokładności rozważań i wykorzystując naturalną specyfikę rzeki, uzyskano opis zjawiska transportu w postaci zbioru równań różniczkowych zwyczajnych z dyskretnymi pomiarami. Zagadnienie estymacji dla takiego opisu, przy podejściu filtru Kalmana, sprowadza się do etapu filtracji i predykcji. Rozwiązanie uzyskano, stosując narzędzie wspomagające w postaci sztucznych sieci neuronowych. Rezultaty badań symulacyjnych potwierdzają możliwość stosowania zaproponowanego systemu monitorującego stan rzeki długiej, funkcjonującego w oparciu o sztuczne sieci neuronowe. System taki realizowałby monitoring, a także sterowanie napowietrzaniem rzeki dla zapewnienia jej warunków ekologicznych.
EN
In the article the mathematical modeling of organic pollutants in the river and the problem of state estimating of rivers quality was considered. The presented models are based on ordinary differential equations, which extends to partial differential equations with the phenomenon of transport and diffusion. The considered cases concerned on of transport phenomena using the natural characteristics of the river without loss of the accuracy description. The mathematical model was obtained as a set of ordinary differential equations with discrete measurements. The problem of estimation for such a description, using the approach Kalman filter lead to the two stages ie. filtering and prediction. The solution was obtained using a support tool in the form of artificial neural networks. The results of simulation confirm the possibility of applying the proposed system to monitoring the quality states of a long river, realized due to using the artificial neural networks. Such a system would further the monitoring and controlling of aeration of the river to ensure its ecological conditions.
18
Content available A finite difference method for nonlinear parabolic-elliptic systems of second-order partial differential equations
Malec M., Sapa L.
Opuscula Mathematica
|
2007
|
Vol. 27, no. 2
259-289
EN
This paper deals with a finite difference method for a wide class of weakly coupled nonlinear second-order partial differential systems with initial condition and weakly coupled nonlinear implicit boundary conditions. One part of each system is of the parabolic type (degenerated parabolic equations) and the other of the elliptic type (equations with a parameter) in a cube in R1+n. A suitable finite difference scheme is constructed. It is proved that the scheme has a unique solution, and the numerical method is consistent, convergent and stable. The error estimate is given. Moreover, by the method, the differential problem has at most one classical solution. The proof is based on the Banach fixed-point theorem, the maximum principle for difference functional systems of the parabolic type and some new difference inequalities. It is a new technique of studying the mixed-type systems. Examples of physical applications and numerical experiments are presented.
19
Content available remote Using fractional calculus for generation of alpha-stable Lévy probability density functions
Ciesielski M., Leszczyński J.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
|
2005
|
Vol. 4, nr 1
24-33
EN
This paper deals with numerical solutions of a partial differential equation of fractional order. This type of equation describes a process of anomalous diffusion. In this equation a space fractional derivative known as the Riesz operator occurs. We propose a new approach of discretization of the Riesz operator. Using this approach we present a method which utilises this discretization to the solution of equation based on the finite difference method. We also present fundamental solutions of this equation in one-dimensional space which are so-called the class of Lévy stable densities of index α. In the final part of this paper some numerical examples are shown.
20
Content available remote Inequalities useful in the theory of certain partial differential equations
Pachpatte B. G.
Fasciculi Mathematici
|
2002
|
Nr 33
49-56
EN
In this paper we establish some new integral inequalities in two independent variables which can be used as tools in the theory of certain classes of hyperbolic partial differential equations. The analysis used in the proofs is elementry and the results established provide new estimates on these types of inequalities.
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