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  1. 16 Demonstratio Mathematica
  2. 11 Opuscula Mathematica
  3. 5 Fasciculi Mathematici
  4. 5 Journal of Applied Analysis
  5. 5 Journal of Theoretical and Applied Mechanics
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  1. 9 Jaroszewicz J.
  2. 3 Filipowska R.
  3. 3 Uzny S.
  4. 2 Benchohra M.
  5. 2 Cabański J.
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  1. 1 2024
  2. 2 2021
  3. 6 2020
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  5. 3 2018
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1
Content available Positive solutions to a third order nonlocal boundary value problem with a parameter
Szajnowska Gabriela, Zima Mirosława
Opuscula Mathematica
|
2024
|
Vol. 44, no. 2
267--283
EN
We present some sufficient conditions for the existence of positive solutions to a third order differential equation subject to nonlocal boundary conditions. Our approach is based on the Krasnosel’skiĭ-Guo fixed point theorem in cones and the properties of the Green’s function corresponding to the BVP under study. The main results are illustrated by suitable examples.
2
Content available Matlab implementation of direct and indirect shooting methods to solve an optimal control problem with state constraints
Karbowski Andrzej
Journal of Automation Mobile Robotics and Intelligent Systems
|
2021
|
Vol. 15, No. 1
43--50
EN
The paper presents a general procedure to solve nume‐ rically optimal control problems with state constraints. It is used in the case, when the simple time discretization of the state equations and expressing the optimal cont‐ rol problem as a nonlinear mathematical programming problem is too coarse. It is based on using in turn two multiple shooting BVP approaches: direct and indirect. The paper is supplementary to the earlier author’s paper on direct and indirect shooting methods, presenting the theory underlying both approaches. The same example is considered here and brought to an end, that is two full listings of two MATLAB codes are shown.
3
L1-solutions of the boundary value problem for implicit fractional order differentia equations
Telli Benoumran, Souid Mohammed Said
Journal of Applied Analysis
|
2021
|
Vol. 27, nr 1
121--128
EN
The aim of this paper is to present new results on the existence of solutions for a class of the boundary value problem for fractional order implicit differential equations involving the Caputo fractional derivative. Our results are based on Schauder’s fixed point theorem and the Banach contraction principle fixed point theorem.
4
Content available remote Approximate solutions and numerical analysis of a spring-mass running model
Wróblewska Zofia
Mathematica Applicanda
|
2020
|
Vol. 48, no. 1
25--48
EN
The paper refers to the classic spring-mass model of running, which was created on the basis of an inverted elastic pendulum. A new approximate solution of the boundary value problem relayed to the governing system based on two nonlinear ordinary differential equations is introduced, which we get in this model in a natural way. We give theoretical support by deriving asymptotic behaviour of obtained approximations. Simulations show that new solutions turn out very well. Our results are illustrated with some practical examples.
PL
W pracy rozważamy klasyczny model masy sprężynowej dla biegania oparty na odwróconym elastycznym wahadle. Przedstawiamy nowe przybliżone rozwiązanie interesującego zagadnienia brzegowego dla układu dwóch nieliniowych równań różniczkowych, które w naturalny sposób uzyskujemy w tym modelu. Badamy asymptotyczne zachowanie uzyskanych aproksymacji i podajemy asymptotyczną postać współczynnika spężystości nogi dla małych kątów ataku. Symulacje pokazują, że nowe rozwiązanie wypadło bardzo dobrze i wykazało dużą zgodność przybliżenia z rozwiązaniem dokładnym. Nasze wyniki zostały zilustrowane kilkoma praktycznymi przykładami pokazując, że pomiary parametrów biegu lekkoatletów są bliskie wartościom uzyskanym z modelu.
5
Content available The finite difference method on adaptive mesh for singularly perturbed nonlinear 1D reaction diffusion boundary value problems
Duru Hakkı, Güneş Baransel
Journal of Applied Mathematics and Computational Mechanics
|
2020
|
Vol. 19, nr 4
45--56
EN
In this paper, we study singularly perturbed nonlinear reaction-diffusion equations. The asymptotic behavior of the solution is examined. The difference scheme which is accomplished by the method of integral identities with using of interpolation quadrature rules with weight functions and remainder term integral form is established on adaptive mesh. Uniform convergence and stability of the difference method are discussed in the discrete maximum norm. The discrete scheme shows that orders of convergent rates are close to 2. An algorithm is presented, and some problems are solved to validate the theoretical results.
6
Content available An intial-value technique for self-adjoint singularly perturbed two-point boundary value problems
Padmaja P., Aparna P., Gorla R. S. R.
International Journal of Applied Mechanics and Engineering
|
2020
|
Vol. 25, no. 1
106--126
EN
In this paper, we present an initial value technique for solving self-adjoint singularly perturbed linearvboundary value problems. The original problem is reduced to its normal form and the reduced problem is converted to first order initial value problems. This replacement is significant from the computational point of view. The classical fourth order Runge-Kutta method is used to solve these initial value problems. This approach to solve singularly perturbed boundary-value problems is numerically very appealing. To demonstrate the applicability of this method, we have applied it on several linear examples with left-end boundary layer and rightend layer. From the numerical results, the method seems accurate and solutions to problems with extremely thin boundary layers are obtained.
7
Content available remote Existence and uniqueness of mild solutions for a fractional differential equation under Sturm-Liouville boundary conditions when the data function is of Lipschitzian type
Harjani Jackie, López Belen, Sadarangani Kishin
Demonstratio Mathematica
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2020
|
Vol. 53, nr 1
167--173
EN
In this article, we present a sufficient condition about the length of the interval for the existence and uniqueness of mild solutions to a fractional boundary value problem with Sturm-Liouville boundary conditions when the data function is of Lipschitzian type. Moreover, we present an application of our result to the eigenvalues problem and its connection with a Lyapunov-type inequality.
8
Content available Differential quadrature methodfor some diffusion-reaction problems
Szukiewicz Mirosław K.
Chemical and Process Engineering
|
2020
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Vol. 41, nr 1
3--11
EN
In the paper, differential quadrature method (DQM) is used to find numerical solutions of reaction-diffusion equations with different boundary conditions. The DQM-method changes the reaction-diffusion equation (ordinary differential equation) into a system of algebraic equations. The obtainedsystem is solved using built-in procedures of Maple®(Computer Algebra System-type program).Calculations were performed with Maple®program. The test problems include reaction-diffusionequation applied in heterogeneous catalysis. The method can be employed even in relatively hard tasks(e.g. ill-conditioned, free boundary problems).
9
Content available Theoretical Analysis and Experimental Verification of Top Orthogonal to Bottom Arrays of Conducting Strips on Piezoelectric Slab
Tasinkevych Yuriy, Trots Ihor, Tymkiewicz Ryszard
Archives of Acoustics
|
2020
|
Vol. 45, No. 3
433--444
EN
The purpose of this work is to present a theoretical analysis of top orthogonal to bottom arrays of conducting electrodes of infinitesimal thickness (conducting strips) residing on the opposite surfaces of piezoelectric slab. The components of electric field are expanded into double periodic Bloch series with corresponding amplitudes represented by Legendre polynomials, in the proposed semi-analytical model of the considered two-dimensional (2D) array of strips. The boundary and edge conditions are satisfied directly by field representation, as a result. The method results in a small system of linear equations for unknown expansion coefficients to be solved numerically. A simple numerical example is given to illustrate the method. Also a test transducer was designed and a pilot experiment was carried out to illustrate the acoustic-wave generating capabilities of the proposed arrangement of top orthogonal to bottom arrays of conducting strips.
10
Content available Existence and uniqueness of solutions for nonlinear Katugampola fractional differential equations
Basti Bilal, Arioua Yacine, Benhamidouche Nouredine
Journal of Mathematics and Applications
|
2019
|
Vol. 42
35--61
EN
The present paper deals with the existence and uniqueness of solutions for a boundary value problem of nonlinear fractional differential equations with Katugampola fractional derivative. The main results are proved by means of Guo-Krasnoselskii and Banach fixed point theorems. For applications purposes, some examples are provided to demonstrate the usefulness of our main results.
11
Content available Analysis of natural frequency of flexural vibrations of a single-span beam with the consideration of Timoshenko effect
Jaroszewicz J., Łukaszewicz K.
Technical Sciences / University of Warmia and Mazury in Olsztyn
|
2018
|
nr 21(3)
215--232
EN
This paper presents general solution of boundary value problem for constant cross-section Timoshenko beams with four typical boundary conditions. The authors have taken into consideration rotational inertia and shear strain by using the theory of influence by Cauchy function and characteristic series. The boundary value problem of transverse vibration has been formulated and solved. The characteristic equations considering the exact bending theory have been obtained for four cases: the clamped boundary conditions; a simply supported beam and clamped on the other side; a simply supported beam; a cantilever beam. The obtained estimators of fundamental natural frequency take into account mass and elastic characteristics of beams and Timoshenko effect. The results of calculations prove high convergence of the estimators to the exact values which were calculated by Timoshenko who used Bessel functions. Characteristic series having an alternating sign power series show good convergence. As it is shown in the paper, the error lower than 5% was obtained after taking into account only two first significant terms of the series. It was proved that neglecting the Timoshenko effect in case of short beams of rectangular section with the ratio of their length to their height equal 6 leads to the errors of calculated natural frequency: 5%÷12%.
12
Content available remote Transport of temperature fluctuations across a two-phased laminate conductor
Wodzyński Ł., Kula D., Wierzbicki E.
Mechanics and Mechanical Engineering
|
2018
|
Vol. 22, nr 3
775--787
EN
In the periodic composite materials temperature or displacement fluctuations suppressed in directions perpendicular to the periodicity surfaces should expect a damping reaction from the composite. This phenomenon, known as the boundary effect behavior has been investigated only in the framework of approximated models. In this paper extended tolerance model of heat transfer in periodic composites is used as a tool allows analytical investigations of highly oscillating boundary thermal loadings. It has been shown that mentioned reaction is dual - different for even and odd fluctuations.
13
Content available Zastosowanie wybranej metody Trefftza do rozwiązania dwuwymiarowego zagadnienia Poissona z uwzględnieniem własności materiałowych ośrodka
Borkowska D.
Autobusy : technika, eksploatacja, systemy transportowe
|
2018
|
R. 19, nr 6
363--367, CD
PL
Celem pracy jest analiza teoretyczna oraz numeryczna jednej z wersji nieosobliwych metod Trefftza na przykładzie zagadnienia dwuwymiarowego opisanego równaniem Poissona. Przekształcając klasyczne sformułowanie zagadnienia brzegowego za pomocą metody residuów ważonych do sformułowań wariacyjnych otrzymuje się równanie obszarowo-brzegowe opisujące dane zagadnienie. W pracy rozpatruje się silne sformułowanie wariacyjne. Przyjmując rozwiązanie w postaci szeregu funkcji Trefftza spełniających jednorodne równanie Laplace’a oraz zakładając również funkcje Trefftza jako funkcje wagowe uzyskuje się równanie metody Trefftza w wersji Galerkina o symbolicznej nazwie O-S;T-T. Artykuł zawiera teoretyczną analizę metody O-S;T- T na przykładzie zagadnienia spełniającego równanie Poissona z uwzględnieniem parametru materiałowego ośrodka.
EN
The aim of this paper is theoretical and numerical analysis of one of the nonsingular Trefftz method. Two-dimensional boundary value problem governed by Poisson’s equation is taken as the example. Domain boundary equation is obtained by transformation of classical formulation of the boundary problem with the use of weighted residual method. In this paper the original variation formulation is considered. The solution of the problem is assumed as the superposition of Trefftz functions, which satisfy Laplace’s equation. Taking the same functions as the weighting functions one obtains equations of the Galerkin version of the Trefftz method with symbolic name O-S;T-T. The paper contains the theoretical analysis of the O-S;T-T method which is confirmed with numerical example.
14
Content available remote Homotopy perturbation method for solving fourth – order boundary value problems with additional boundary condition
Filipowska R.
Technical Transactions
|
2017
|
Vol. 9(114)
173--179
EN
This paper presents the homotopy perturbation method for solving linear and non–linear two–point boundary value problems in the form of a fourth–order differential equation and five boundary conditions. Three initial and two final conditions were taken into account. The solution of this problem is possible only when the considered equation includes an unknown parameter. The presented method has been illustrated with a numerical example.
PL
W artykule przedstawiono homotopijną metodę perturbacyjną zastosowaną do rozwiązywania zarówno liniowego, jak i nieliniowego dwupunktowego zagadnienia brzegowego składającego się z równania różniczkowego czwartego rzędu oraz pięciu warunków brzegowych. Pod uwagę wzięto trzy początkowe i dwa końcowe warunki brzegowe. Rozwiązanie tak postawionego problemu jest możliwe tylko wtedy, gdy rozpatrywane równanie zawiera nieznany parametr. Prezentowaną metodę zilustrowano przykładem obliczeniowym.
15
Content available The effect of influence of conservative and tangential axial forces on transverse vibrations of tapered vertical columns
Jaroszewicz J., Radziszewski L., Dragun Ł.
Technical Sciences / University of Warmia and Mazury in Olsztyn
|
2017
|
nr 20(4)
333--342
EN
The Cauchy function and characteristic series were applied to solve the boundary value problem of free transverse vibrations of vertically mounted, elastically supported tapered cantilever columns. The columns can be subjected to universal axial point loads which considerate – conservative and follower /tangential/ forces, and to distributed loads along the cantilever length. The general form of characteristic equation was obtained taking into account the shape of tapered cantilever for attached and elastically secured. Bernstein-Kieropian double and higher estimators of natural frequency and critical loads were calculated based on the first few coefficients of the characteristic series. Good agreement was obtained between the calculated natural frequency and the exact values available in the literature.
16
Content available On nonexistence of global in time solution for a mixed problem for a nonlinear evolution equation with memory generalizing the Voigt-Kelvin rheological model
Pukach P., Il'kiv V., Nytrebych Z., Vovk M.
Opuscula Mathematica
|
2017
|
Vol. 37, no. 5
735--753
EN
The paper deals with investigating of the first mixed problem for a fifth-order nonlinear evolutional equation which generalizes well known equation of the vibrations theory. We obtain sufficient conditions of nonexistence of a global solution in time variable.
17
Content available The general formula for calculation of fundamental frequency of axisymmetric vibrations of circular plates with linearly variable thickness
Jaroszewicz J., Radziszewski L.
Technical Sciences / University of Warmia and Mazury in Olsztyn
|
2016
|
nr 19(4)
401--410
EN
This work has derived the general formula, sufficiently precise for engineering calculations of base frequency of axisymmetric free vibrations of uniform, circular diaphragm type plates clamped at the edge with linearly variable thickness. To solve the boundary problem, the Cauchy’s function method and characteristic series have been applied. The above formula has been derived on the basis of Dunkerley’s formula which is based on the first major term of the characteristic series and results in the simplest, lower bound estimator. The analysis of the formula shows that the base frequency coefficient for diaphragm plates clamped at the edge depends to only a small extent on the Poisson’s ratio, and therefore it may be averaged in the case of construction materials. Comparison of the calculations results of the simplest lower bound estimators for the base frequency obtained by using proposed method, with the results known from the literature as precise solutions, including Conway method, confirmed the high accuracy of the proposed method.
18
A boundary-value problem for the equation of mixed type with generalized operators of fractional differentiation in the boundary conditions
Repin O. A., Kumykova S. K.
Journal of Applied Analysis
|
2016
|
Vol. 22, nr 1
27--36
EN
The paper is devoted to the study of a boundary-value problem for an equation of mixed type with generalized operators of fractional differentiation in boundary conditions. We prove uniqueness of solutions under some restrictions on the known functions and on the different orders of the operators of generalized fractional differentiation appearing in the boundary conditions. Existence of solutions is proved by reduction to a Fredholm equation of the second kind, for which solvability follows from the uniqueness of the solution of our original problem.
19
Content available remote Variational iteration technique for solving higher order boundary value problem with additional boundary condition
Filipowska R.
Czasopismo Techniczne. Mechanika
|
2016
|
Y. 113, iss. 4-M
15--20
EN
This paper treats a variational iteration technique, which is based on variational iteration method, for solving linear and non – linear two – point boundary value problems in the form of a fourth – order differential equation and five boundary conditions. The solution of this problem is possible only when the considered equation includes an unknown parameter. The presented method has been illustrated with a numerical example.
PL
W artykule przedstawiono iteracyjną technikę wariacyjną opartą na iteracyjnej metodzie wariacyjnej, zastosowaną do rozwiązywania zarówno liniowego, jak i nieliniowego dwupunktowego zagadnienia brzegowego składającego się z równania różniczkowego czwartego rzędu oraz pięciu warunków brzegowych. Rozwiązanie tak postawionego problemu jest możliwe tylko wtedy, gdy rozpatrywane równanie zawiera nieznany parametr. Prezentowaną metodę zilustrowano przykładem obliczeniowym.
20
Content available About sign-constancy of Green's functions for impulsive second order delay equations
Domoshnitsky A., Landsman G., Yanetz S.
Opuscula Mathematica
|
2014
|
Vol. 34, no. 2
339--362
EN
We consider the following second order differential equation with delay [formula] In this paper we find necessary and sufficient conditions of positivity of Green's functions for this impulsive equation coupled with one or two-point boundary conditions in the form of theorems about differential inequalities. By choosing the test function in these theorems, we obtain simple sufficient conditions. For example, the inequality [formula] is a basic one, implying negativity of Green's function of two-point problem for this impulsive equation in the case 0<γi≤1, 0<δi≤1 for i=1,…,p.
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