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1

On classical symmetries of ordinary differential equations related to stationary integrable partial differential equations

Tsyfra Ivan

Opuscula Mathematica

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2021

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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.

2

The fourth-order ordinary differential equation with the fractional initial/boundary conditions

Siedlecki Jarosław

Journal of Applied Mathematics and Computational Mechanics

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2020

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Vol. 19, nr 1

79--87

EN

The initial/boundary value problem for the fourth-order homogeneous ordinary differential equation with constant coefficients is considered. In this paper, the particular solutions an ordinary differential equation with respect to the set of boundary conditions are studied. At least one of the boundary conditions is described by a fractional derivative. Finally, a few illustrative examples of particular solutions to the considered problem are shown.

3

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

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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.

4

Impact of Energy Slope Averaging Methods on Numerical Solution of 1D Steady Gradually Varied Flow

Artichowicz W., Prybytak D.

Archives of Hydro-Engineering and Environmental Mechanics

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2015

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Vol. 62, nr 3-4

101--119

EN

In this paper, energy slope averaging in the one-dimensional steady gradually varied flow model is considered. For this purpose, different methods of averaging the energy slope between cross-sections are used. The most popular are arithmetic, geometric, harmonic and hydraulic means. However, from the formal viewpoint, the application of different averaging formulas results in different numerical integration formulas. This study examines the basic properties of numerical methods resulting from different types of averaging.

5

A mathematical model for fluid-glucose-albumin transport in peritoneal dialysis

Cherniha R., Stachowska-Piętka J., Waniewski J.

International Journal of Applied Mathematics and Computer Science

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2014

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Vol. 24, no. 4

837--851

EN

A mathematical model for fluid and solute transport in peritoneal dialysis is constructed. The model is based on a three-component nonlinear system of two-dimensional partial differential equations for fluid, glucose and albumin transport with the relevant boundary and initial conditions. Our aim is to model ultrafiltration of water combined with inflow of glucose to the tissue and removal of albumin from the body during dialysis, by finding the spatial distributions of glucose and albumin concentrations as well as hydrostatic pressure. The model is developed in one spatial dimension approximation, and a governing equation for each of the variables is derived from physical principles. Under some assumptions the model can be simplified to obtain exact formulae for spatially non-uniform steady-state solutions. As a result, the exact formulae for fluid fluxes from blood to the tissue and across the tissue are constructed, together with two linear autonomous ODEs for glucose and albumin concentrations in the tissue. The obtained analytical results are checked for their applicability for the description of fluid-glucose-albumin transport during peritoneal dialysis.

6

Teaching, modeling and visualisation of ordinary differential equations

Mrozek Z.

Czasopismo Techniczne. Nauki Podstawowe

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2014

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R. 111, z. 2-NP

99--111

EN

Advances in computer technology and increased interest in dynamical systems influence the way of teaching ordinary differential equations. The paper presents inquiry oriented teaching, usage of modeling, visualisation and interactive web services. Last chapter describes the ways of using MATLAB or public domain software (e.g. Octave) to solve ordinary differential equations.

PL

Postęp w technologii komputerowej oraz wzrost zainteresowania modelowaniem systemów dynamicznych wpływa na sposób nauczania matematyki, w tym równań różniczkowych zwyczajnych. Przedstawiono podejścia: nauczania przez zadawanie pytań, wykorzystanie modelowania, wizualizacji i interaktywnych usług sieci web. Ostatni rozdział opisuje sposoby wykorzystania środowiska MATLAB lub oprogramowania dostępnego jako public domain (np. Octave) do rozwiązywania równań różniczkowych zwyczajnych.

7

On nonlinear differential equations in generalized Musielak-Orlicz spaces

Kozlowski W. M.

Commentationes Mathematicae

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2013

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Vol. 53, No. 2

13--33

EN

We consider ordinary differential equations u′(t)+(I−T)u(t)=0, where an unknown function takes its values in a given modular function space being a generalization of Musielak-Orlicz spaces, and T is nonlinear mapping which is nonexpansive in the modular sense. We demonstrate that under certain natural assumptions the Cauchy problem related to this equation can be solved. We also show a process for the construction of such a solution. This result is then linked to the recent results of the fixed point theory in modular function spaces.

8

A dynamic prictionless contact problem with adhesion and damage

Selmani M., Selmani L.

Bulletin of the Polish Academy of Sciences. Mathematics

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2007

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Vol. 55, no 1

17-34

EN

We consider a dynamic frictionless contact problem for a viscoelastic material with damage. The contact is modeled with normal compliance condition. The adhesion of the contact surfaces is considered and is modeled with a surface variable, the bonding field, whose evolution is described by a first order differential equation. We establish a variational formulation for the problem and prove the existence and uniqueness of the solution. The proofs are based on the theory of evolution equations with monotone operators, a classical existence and uniqueness result for parabolic inequalities, and fixed point arguments.

9

Enhancement to Lohner's Enclosures for Verified Trajectory Tracing in Dynamical Systems

Kleiner A.

Schedae Informaticae

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2006

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Vol. 15

9-28

EN

The paper presents some improvements of the Lohner's algorithm for the rigorous enclosure of trajectories of an ODE.

10

Rozwiązywanie równań i układów równań różniczkowych zwyczajnych pierwszego rzędu

Ufnalski W.

LAB Laboratoria, Aparatura, Badania

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2004

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R. 9, nr 2

39-47

11

Pewne nielokalne zagadnienie dla silnie zaburzonego równania różniczkowego zwyczajnego

Flyud V., Tsymbal V.

Zeszyty Naukowe. Matematyka / Politechnika Opolska

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2002

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z. 18

27-35

PL

W artykule skonstruowano i uzasadniono asymptotyczne rozwinięcie nielokalnego zagadnienia dla równania różniczkowego zwyczajnego z dwoma małymi parametrami.

EN

Singularly perturbed problem of the previous article is considered. The asymptotic expansions of solutions in some cases are obtained.

12

O pewnym nielokalnym zagadnieniu dla równania różniczkowego zwyczajnego

Flyud V., Tsymbal V.

Zeszyty Naukowe. Matematyka / Politechnika Opolska

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2002

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z. 18

21-26

PL

W niniejszym artykule udowodniono istnienie dokładnie jednego klasycznego rozwiązania nielokalnego zagadnienia dla równania różniczkowego zwyczajnego rzędu drugiego.

13

A comparison of center-manifold reduction of nonlinear integro-differential flutter equation and approximate set of ordinary differential equations

Grzędziński J.

Research Bulletin / Warsaw University of Technology, Institute of Aeronautics and Applied Mechanics

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1999

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nr 9

7-14

EN

A nonlinear integro-differential flutter equation of a thin airfoil placed in an incompressible flow is solved by two different methods. The first method involves the center-manifold reduction and gives the asymptotic limit cycle amplitude and frequency in terms of power series expansions. The second method replaces the integro-differential equation by an approximate set of first-order ordinary differential equations which are solved by using bifurcation and continuation software package. A comparison of these two methods shows that the domain of a good agreement between them varies significantly depending on the parameters of the problem.