Wyniki wyszukiwania - Biblioteka Nauki

1

Remarks concerning Driver's equation

100%

Herzog G. , Lemmert R.

Annales Polonici Mathematici

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1994

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tom 59

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nr 2

197-202

EN

We consider uniqueness for the initial value problem x' = 1 + f(x) - f(t), x(0) = 0. Several uniqueness criteria are given as well as an example of non-uniqueness.

2

On some dynamical reconstruction problems for a nonlinear system of the second-order

100%

Blizorukova M. , Maksimov V.

Opuscula Mathematica

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2010

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tom Vol. 30, no. 4

399-410

EN

The problem of reconstruction of unknown characteristics of a nonlinear system is considered. Solution algorithms stable with respect to the informational noise and computational errors are specified. These algorithms are based on the method of auxiliary positionally controlled models.

3

On reconstructing an unknown coordinate of a nonlinear system of differential equations

100%

Blizorukowa M. , Kuklin A. , Maksimov V.

Opuscula Mathematica

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2014

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tom Vol. 34, no. 2

257--269

EN

The paper discusses a method of auxiliary controlled models and the application of this method to solving problems of dynamical reconstruction of an unknown coordinate in a nonlinear system of differential equations. The solving algorithm, which is stable with respect to informational noises and computational errors, is presented.

4

The RC Circuit Described by Local Fractional Differential Equations

80%

Zhao X.-H. , Zhang Y. , Zhao D. , Yang X.

Fundamenta Informaticae

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2017

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tom Vol. 151, nr 1/4

419--429

EN

A non-differentiable resistor-capacitor circuit comprised of the capacitor and resistor in the fractal-time domain is first proposed in this article. The solution behavior of the corresponding local fractional ordinary differential equation is presented for the Mittag-Leffler decay defined on Cantor sets. The obtained results reveal the sufficiency of the local fractional calculus in the analysis of the fractal electrical systems.

5

Dependent noise for stochastic algorithms

80%

Doukhan P. , Brandierie O.

Probability and Mathematical Statistics

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2004

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tom Vol. 24, Fasc. 2

381--399

EN

We introduce different ways of being dependent for the input noise of stochastic algorithms. We are aimed to prove that such innovations allow to use the ODE (ordinary differential equation) method. Illustrations to the linear regression frame and to the law of large numbers for triangular arrays of weighted dependent random variables are also given.

6

Do not let the dead bite! Different scenarios of the zombie epidemic reexamined

80%

Tomczyk W.

Zeszyty Naukowe Towarzystwa Doktorantów Uniwersytetu Jagiellońskiego. Nauki Ścisłe

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2015

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nr 11

21-33

EN

The Zombie Epidemic is a fun framework for investigating different scenarios of spreading disease. An extended Kermack – McKendrick model is analyzed. The only thing that can save humanity is to not get bitten or to find a remedy for the ”zombie virus” (both almost impossible).

7

Modelling tumour-immunity interactions with different stimulation functions

80%

Zhivkov P. , Waniewski J.

International Journal of Applied Mathematics and Computer Science

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2003

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tom 13

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nr 3

307-315

EN

Tumour immunotherapy is aimed at the stimulation of the otherwise inactive immune system to remove, or at least to restrict, the growth of the original tumour and its metastases. The tumour-immune system interactions involve the stimulation of the immune response by tumour antigens, but also the tumour induced death of lymphocytes. A system of two non-linear ordinary differential equations was used to describe the dynamic process of interaction between the immune system and the tumour. Three different types of stimulation functions were considered: (a) Lotka-Volterra interactions, (b) switching functions dependent on the tumour size in the Michaelis-Menten form, and (c) Michaelis-Menten switching functions dependent on the ratio of the tumour size to the immune capacity. The linear analysis of equilibrium points yielded several different types of asymptotic behaviour of the system: unrestricted tumour growth, elimination of tumour or stabilization of the tumour size if the initial tumour size is relatively small, otherwise unrestricted tumour growth, global stabilization of the tumour size, and global elimination of the tumour. Models with switching functions dependent on the tumour size and the tumour to the immune capacity ratio exhibited qualitatively similar asymptotic behaviour.

8

Modern Taylor series method in numerical integration

70%

Chaloupka J. , Necasová G. , Veigend P. , Kunovský J. , Šátek V.

Systemy Wspomagania w Inżynierii Produkcji

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2017

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tom Vol. 6, iss. 4

263--273

EN

The paper deals with extremely exact, stable, and fast numerical solutions of systems of differential equations. It also involves solutions of problems that can be reduced to solving a system of differential equations. The approach is based on an original mathematical method, which uses the Taylor series method for solving differential equations in a non-traditional way. Even though this method is not much preferred in the literature, experimental calculations have verified that the accuracy and stability of the Taylor series method exceed the currently used algorithms for numerically solving differential equations. The Modern Taylor Series Method (MTSM) is based on a recurrent calculation of the Taylor series terms for each time interval. Thus, the complicated calculation of higher order derivatives (much criticised in the literature) need not be performed but rather the value of each Taylor series term is numerically calculated. An important part of the method is an automatic integration order setting, i.e. using as many Taylor series terms as the defined accuracy requires. The aim of our research is to propose the extremely exact, stable, and fast numerical solver for modelling technical initial value problems that offers wide applications in many engineering areas including modelling of electrical circuits, mechanics of rigid bodies, control loop feedback (controllers), etc.

CS

Clánek se zabývá presným, stabilním a rychlým rešením soustav diferenciálních rovnic. Soustavou diferenciálních rovnic lze reprezentovat velké množství reálných problému. Numerické rešení je založeno na unikátní numerické metode, která netradicne využívá Taylorovu radu. I presto, že tato metoda není v literature príliš preferována, experimentální výpocty potvrdily, že presnost a stabilita této metody presahuje aktuálne používané numerické algoritmy pro numerické rešení diferenciálních rovnic. Moderní metoda Taylorovy rady je založena na rekurentním výpoctu clenu Taylorovy rady v každém casovém intervalu. Derivace vyšších rádu nejsou pro výpocet prímo využity, derivace jsou zahrnuty do clenu Taylorovy rady, které se pocítají rekurentne numericky. Duležitou vlastností metody je automatická volba rádu metody v závislosti na velikosti integracního kroku, tzn. je využito tolik clenu Taylorovy rady, kolik vyžaduje zadaná presnost výpoctu. Cílem výzkumu je navrhnout velmi presný, stabilní a rychlý nástroj pro modelování technických pocátecních problému využitých v praxi pri modelování elektrických obvodu, mechaniky tuhých teles, problematiky zpetnovazebního rízení a další.

9

Verified solution method for population epidemiology models with uncertainty

70%

Enszer J. , Stadtherr M.

International Journal of Applied Mathematics and Computer Science

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2009

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tom 19

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nr 3

501-512

EN

Epidemiological models can be used to study the impact of an infection within a population. These models often involve parameters that are not known with certainty. Using a method for verified solution of nonlinear dynamic models, we can bound the disease trajectories that are possible for given bounds on the uncertain parameters. The method is based on the use of an interval Taylor series to represent dependence on time and the use of Taylor models to represent dependence on uncertain parameters and/or initial conditions. The use of this method in epidemiology is demonstrated using the SIRS model, and other variations of Kermack-McKendrick models, including the case of time-dependent transmission.

10

A novel interval arithmetic approach for solving differential-algebraic equations with VALENCIA-IVP

70%

Rauh A. , Brill M. , Günther C.

International Journal of Applied Mathematics and Computer Science

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2009

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tom 19

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nr 3

381-397

EN

The theoretical background and the implementation of a new interval arithmetic approach for solving sets of differentialalgebraic equations (DAEs) are presented. The proposed approach computes guaranteed enclosures of all reachable states of dynamical systems described by sets of DAEs with uncertainties in both initial conditions and system parameters. The algorithm is based on VALENCIA-IVP, which has been developed recently for the computation of verified enclosures of the solution sets of initial value problems for ordinary differential equations. For the application to DAEs, VALENCIA-IVP has been extended by an interval Newton technique to solve nonlinear algebraic equations in a guaranteed way. In addition to verified simulation of initial value problems for DAE systems, the developed approach is applicable to the verified solution of the so-called inverse control problems. In this case, guaranteed enclosures for valid input signals of dynamical systems are determined such that their corresponding outputs are consistent with prescribed time-dependent functions. Simulation results demonstrating the potential of VALENCIA-IVP for solving DAEs in technical applications conclude this paper. The selected application scenarios point out relations to other existing verified simulation techniques for dynamical systems as well as directions for future research.

11

A method for determination of flow at the plane, horizontal interface separating streams of two different fluids

70%

Prosnak W. J. , Cześnik P. P.

Archives of Hydro-Engineering and Environmental Mechanics

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2005

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tom Vol. 52, nr 2

89-110

EN

The paper deals with the theoretical investigation of the phenomenon, which consists in generation - by wind - of the movement of water, the bodies of air and of water being separated by a horizontal, plane interface, which is identical with the free surface of water. A set of assumptions defining a physical model of the phenomenon, borrowed from Lock (1951), is introduced, as well as the mathematical description of this model. It reduces to a composite ordinary differential problem, containing two non-linear equations of the third order, which have to satisfy some boundary conditions. A novel method of solution of the differential problem just mentioned is presented in the paper. In the method use is made of exact formulae for coefficients of series representing the solution. The method seems to be competitive with the one given in the paper by Lock (1951).

12

Numerical integration of differential equations in the presence of first integrals: observer method

70%

Busvelle E. , Kharab R. , Maciejewski A. J. , Strelcyn J.-M.

Applicationes Mathematicae

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1993-1995

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tom 22

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nr 3

373-418

EN

We introduce a simple and powerful procedure-the observer method-in order to obtain a reliable method of numerical integration over an arbitrary long interval of time for systems of ordinary differential equations having first integrals. This aim is achieved by a modification of the original system such that the level manifold of the first integrals becomes a local attractor. We provide a theoretical justification of this procedure. We report many tests and examples dealing with a large spectrum of systems with different dynamical behaviour. The comparison with standard and symplectic methods of integration is also provided.

13

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

60%

Koroński J.

Czasopismo Techniczne. Nauki Podstawowe

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2015

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tom 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.

14

Solving boundary value problems in the open source software R: package bvpSolve

60%

Mazzia F. , Cash J. R. , Soetaert K.

Opuscula Mathematica

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2014

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tom Vol. 34, no. 2

387--403

EN

The R package bvpSolve for the numerical solution of Boundary Value Problems (BVPs) is presented. This package is free software which is distributed under the GNU General Public License, as part of the R open source software project. It includes some well known codes to solve boundary value problems of ordinary differential equations (ODEs) and differential algebraic equations (DAEs). In addition to the packages already available for solving initial value problems, the new package now allows non expert users to efficiently solve boundary value problems in the problem solving environment R.

15

Laplace-Carson integral transform for exact solutions of non-integer order initial value problems with Caputo operator

60%

Kumar P. , Qureshi S.

Journal of Applied Mathematics and Computational Mechanics

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2020

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tom Vol. 19, nr 1

57--66

EN

Finding the exact solution to dynamical systems in the field of mathematical modeling is extremely important and to achieve this goal, various integral transforms have been developed. In this research analysis, non-integer order ordinary differential equations are analytically solved via the Laplace-Carson integral transform technique, which is a technique that has not been previously employed to test the non-integer order differential systems. Firstly, it has proved that the Laplace-Carson transform for n-times repeated classical integrals can be computed by dividing the Laplace-Carson transform of the underlying function by n-th power of a real number p which later helped us to present a new result for getting the Laplace-Carson transform for d-derivative of a function under the Caputo operator. Some initial value problems based upon Caputo type fractional operator have been precisely solved using the results obtained thereof.

16

System of second order robot arm problem by an efficient numerical integration algorithm

60%

Ponalagusamy R. , Senthilkumar S.

Archives of Computational Materials Science and Surface Engineering

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2009

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tom Vol. 1, nr 1

38-44

EN

Purpose: The aim of this article is focused on providing numerical solutions for system of second order robot arm problem using the Runge-Kutta Sixth order algorithm. Design/methodology/approach: The parameters governing the arm model of a robot control problem have also been discussed through RK-sixth-order algorithm. The precised solution of the system of equations representing the arm model of a robot has been compared with the corresponding approximate solutions at different time intervals. Findings: Results and comparison show the efficiency of the numerical integration algorithm based on the absolute error between the exact and approximate solutions. The stability polynomial for the test equation γ=λγ (�γ is a complex Number) using RK-butcher algorithm obtained by Murugesan et. al. [1] and Park et. al. [2,3] is not correct and the stability regions for RK-Butcher methods have been absurdly presented. They have made a blunder in determining the range for real parts of �λh (h is a step size) involved in the test equation for RK-Butcher algorithms. Further, they have abruptly drawn the stability region for STWS method assuming that it is based on the Taylor's series technique. Research limitations/implications: It is noticed that STWS algorithm is not based on the Taylor�'s series method and it is an A-stable method. In the present paper, a corrective measure has been taken to obtain the stability polynomial for the case of RK-Butcher algorithm, the ranges for the real part of �λh and to present graphically the stability regions of the RK-Butcher methods. Originality/value: Based on the numerical results and graphs, a thorough comparison is carried out between the numerical algorithms.

17

Application of the Taylor differential transformation in the calculus of variations

60%

Grzymkowski R. , Pleszczyński M. , Hetmaniok E.

Silesian Journal of Pure and Applied Mathematics

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2017

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tom Vol. 7, iss. 1

65--82

EN

The paper presents the method of solving some problems belonging to the area of the calculus of variations, that is the problems of searching for the selected types of functionals which can be transformed to some, nonlinear in general, ordinary differential equations or systems of such equations. The obtained equations are solved on the basis of the Taylor differential transformation.

18

Nonlinear optimal control problem with constraints for general 2-D systems

60%

Biły B.

Matematyka Stosowana : matematyka dla społeczeństwa

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1993

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tom Nr 36

41--53

EN

In this paper a optimal, nonlinear problem for general two-dimensional, stationary, discrete systems is presented. The state and control vectors are subject to restrictions in this optimal problem. Using a method of transformation for the system and the performance index, the problem of finding the optimal sequence of control vectors is solved by a method of mathematical programming. The necessary conditions are formulated. The simple numerical example illustrates the presented method.

PL

W pracy rozpatruje się zadanie sterowania optymalnego dla modelu 2-D przy nieliniowym wskaźniku jakości, przy ograniczeniach na wektory stanu i sterowania. W oparciu o Twierdzenie Podstawowe programowania matematycznego formułuje się warunki konieczne dla tego zadania . Idee zawarte w tej pracy wykorzystywane już były w artykułach [1], [3] przy nieco innych modelach i bez warunków ograniczających.

19

Parallel Algorithm for Solving Systems of Ordinary Differential Equations for NVIDIA CUDA Platform

60%

Pala A.

Zeszyty Naukowe. Elektryka / Politechnika Opolska

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2013

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tom z. 69

73--74

EN

Parallel algorithm for solving systems of ordinary differential equations (ODEs) for Nvidia CUDA technology has been developed. This algorithm is based on concept of dividing systems of equations into individual equations or groups of equations which then are solved by separate threads. This article demonstrates initial results and analysis of working time of the algorithm in few examples of its application.

20

Bezpośrednie rozwiązywanie nieliniowych układów równań różniczkowych zwyczajnych wyższych rzędów za pomocą zespołu procedur BGKODE

60%

Beniak R. , Gardecki A.

Zeszyty Naukowe. Informatyka / Politechnika Opolska

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2005

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tom Vol. 302, z. 2

151-167

PL

Praca zawiera analizę właściwości zespołu procedur BGKODE w odniesieniu do bezpośredniego rozwiązywania nieliniowych równań różniczkowych zwyczajnych wyższych rzędów. W celu sprawdzenia poprawności otrzymanych rozwiązań porównuje się wyniki obliczeń otrzymane za pomocą procedury DIFSUB i BGKODE. Przeprowadzone testy obejmują sześć układów równań różniczkowych, z których ostatni dotyczy modelu silnika indukcyjnego z wyeliminowanymi równaniami wirnika. Przeprowadzone badania potwierdzają możliwość efektywnego wykorzystania procedury BGKODE i jej wysokie możliwości w analizowanym zakresie.

EN

In this paper the authors have presented the BGKODE routine, which is used to direct solutions of nonlinear systems of ordinary differential equations. The research, which was carried out by the authors, is related with analysis of the routine main features considering with direct solutions of high order nonlinear systems. In order to obtain an appropriate estimation of abilities of the routine in this case, the authors have compared the results of calculations with the results of calculations that were done using the Gear DIFSUB subroutine. This subroutine utilises the normal form of first order ordinary differential equations system. Six nonlinear equations of high orders [8] are used to comparison. There are: the Riemann P-differential equation (13), the Chebyshev differential equation (15), the Duffing differential equation (17), the differential equation (19) obtained from the Van der Pol equation by differentiation, the tested equation (21), which was obtained from the ODEcalcTM and differential equation described the second order model of induction motor (23). In comparison of efficiency of the BGKODE routine and the DIFSUB subroutine subscripts m and n were used. The subscripts define adequately the number of calls of subroutine and the number of calls of procedure, which defines the right side of differential equations. The result of comparison of m and n subscripts has been located in Table 1 and 2. The result of research guides to a conclusion that the BGKODE routine may be used to solving high order differential equations in a direct way. This possibility might be important in case when we solve a very complicated system of ordinary differential equations, when it is hard to obtain the normal form of ordinary differential equations system. Needless to say that the calculating efficiency of BGKODE routines is higher then DIFSUB routines. The figures that have been enclosed present the course of relative error, which was obtained owing to the comparison of the numerical calculations of differential equations with the BGKODE routines and the DIFSUB routines, show how similar the results are. Therefore, the BGKODE routines might be used to fast solving not only the first order nonlinear differential equations systems, but also high order ones.