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1

Discussion on angular asymmetry in the solutions of SLIP model

Kowalczyk Piotr, Płociniczak Łukasz, Wróblewska Zofia

Mathematica Applicanda

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2023

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Vol. 51, no. 2

223--237

EN

We consider a spring-mass model of human running which is built upon an inverted elastic pendulum. The model itself consists of two sets of differential equations - one set describes the motion of the centre of mass of a runner in contact with the ground (support phase), and the second set describes the phase of no contact with the ground (flight phase). In our previous approach, we assumed that periodic solutions in the support phase are symmetrical with respect to the touch-down and take-off angles for the large spring constant (or small angle of attack). Based on proposed solutions, we introduce analytical approximations of an asymmetrical boundary value problem, which brings our model closer to real running. By appropriately concatenating asymptotic solutions for the two gait phases, we are able to reduce the dynamics to a one-dimensional apex to apex return map and then to investigate the existence and stability of periodic solutions. Unlike in the symmetrical version, we could not find sufficient conditions for this map to have a unique stable fixed point. Extending the model with the possibility of taking off with the angle other than during landing, the aforementioned asymmetry, is necessary in the context of real run considerations. Thanks to this, our work could be enriched by experimental results. In this paper, we will present the possible reasons for the instability of asymmetric solutions in conjunction with conclusions from the observation of real runs.

PL

W pracy rozważamy model biegu, w którym człowiek sprowadzony jest do punktu masy na nieważkiej sprężynie, a momencie kontaktu z podłożem staje się odwróconym sprężystym wahadłem. Sam model składa się z dwóch zestawów równań różniczkowych - jedno opisuje ruch środka masy biegacza podczas kontaktu stopy z podłożem (faza podparcia), a drugi fazę lotu. W naszym poprzednim podejściu zakładaliśmy, że rozwiązania okresowe w fazie podparcia są symetryczne względem kątów lądowania i odbicia dla dużej wartości sztywności nogi (lub małego kąta ataku). Na podstawie proponowanych rozwiązań wprowadzamy analityczne przybliżenia asymetrycznego problemu brzegowego, co zbliża nasz model do rzeczywistego biegu. Odpowiednio łącząc asymptotyczne rozwiązania dla obu faz biegu, jesteśmy w stanie zredukować dynamikę do jednego wymiaru i utworzyć odwzorowanie powrotu od wierzchołka do kolejnego wierzchołka praboli lotu, a następnie badać istnienie i stabilność rozwiązań okresowych. W odróżnieniu od wersji symetrycznej, nie mogliśmy znaleźć wystarczających warunków, aby to odwzorowanie miało jednoznacznie określony stabilny punkt stały. Rozszerzenie modelu o możliwość odbicia pod innym kątem, niż podczas lądowania (asymetria), jest konieczne w kontekście rozważań nad rzeczywistym biegiem. Dzięki temu nasza praca mogła zostać wzbogacona o wyniki eksperymentalne. W tym artykule przedstawimy możliwe przyczyny niestabilności asymetrycznych rozwiązań w połączeniu z wnioskami z obserwacji rzeczywistych biegów.

2

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

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2023

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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.

3

Numerical scheme methods for solving nonlinear pseudo-hyperbolic partial differential equations

Abdulazeez Sadeq Taha, Modanli Mahmut, Husien Ahmad Muhamad

Journal of Applied Mathematics and Computational Mechanics

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2022

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Vol. 21, nr 4

5--15

EN

The numerical solutions to the nonlinear pseudo-hyperbolic partial differentia equation with nonlocal conditions are presented in this study. This equation is solved using the homotopy analysis technique (HAM) and the variational iteration method (VIM). Both strategies are compared and contrasted in terms of approximate and accurate solutions. The results show that the HAM technique is more appropriate, effective, and close to the exact solution than the VIM method. Finally, the graphical representations of the obtained results are given.

4

Existence and convergence results for Caputo fractional Volterra integro-differential equations

Hamoud A. A., Bani Issa M. Sh., Ghadle K. P., Abdulghani M.

Journal of Mathematics and Applications

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2018

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Vol. 41

109--121

EN

In this article, homotopy analysis method is successfully applied to find the approximate solution of Caputo fractional Volterra integro-differential equation. The reliability of the method and reduction in the size of the computational work give this method a wider applicability. Also, the behavior of the solution can be formally determined by analytical approximate. Moreover, we proved the existence and convergence of the solution. Finally, an example is included to demonstrate the validity and applicability of the proposed technique.

5

Dense Projection Tomography on the Triangular Tiling

Nagy B., Lukić T.

Fundamenta Informaticae

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2016

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Vol. 145, nr 2

125--141

EN

In this paper, we consider the binary tomography reconstruction problem. A new approach is proposed what exploits a possibility provided by the natural structure of the triangular grid, which is not available in the case of the classical square grid. In contrast to the square grid, in the case of the triangular grid information need for the reconstruction of the unknown image is increasing when not only one, but two projections are used by lanes. In this way, the number of Δ and ∇ shaped pixels per lane can be determined. We propose this type of projection approach and call it dense projections. The reconstruction is based on three projection directions by the lane directions of the grid (they are analogous to row and column directions on the square grid). Our algorithm is deterministic and uses energy minimization technique to find (near) optimal solution in a reasonable time. The experimental evaluation of the new method, using regular hexagon shaped test images, is given. Comparison with reconstructions based on the square grid is also considered.

6

Variational iteration technique for solving higher order boundary value problem with additional boundary condition

Filipowska R.

Czasopismo Techniczne. Mechanika

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2016

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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.

7

Rational operation of variable declining rate filters

Dąbrowski W.

Environment Protection Engineering

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2011

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Vol. 37, nr 4

35-53

EN

An approximate solution to the system of equations governing the flow distribution among variable declining rate (VDR) filters results in flow rates through filters being elements of a geometrical progression. Based on this approximation, it was deduced how to operate a plant in order to keep the same flow rates through VDR filters for various total head losses of flow. These principles of operation were carefully verified using the accurate di Bernardo mathematical model of VDR filter plants. It was deduced that the longest filter runs result from such an operation of a plant for which the ratio of the highest to the average flow rates through a filter and simultaneously the affordable total head loss of flow through the plant are the highest.

8

Electromagnetic scattering by a periodic array of thick-walled parallel plate waveguides

Tasinkevych Y.

Journal of Technical Physics

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2009

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Vol. 50, no 1

41-53

EN

Electromagnetic wave scattering by a periodic array of semi-infinite thick-walled parallel plate waveguides is studied in this paper. The cases of TE and TM polarization of an incident plane harmonic wave are considered separately. The scattered field above the waveguides is sought in the form of a series of spatial harmonics in accordance with the Floquet's theorem, whereas in the waveguide regions it is sought in the form of parallel plate waveguide modes. To satisfy the boundary and edge conditions by field components in the free space above the array, the Fourier expansion for spatial harmonics amplitudes with corresponding coefficients, being properly chosen Legendre functions, is exploited. The unknown coefficients are the solutions of certain doubly infinite systems of linear equations. The approximate solution is found numerically.

9

Application of Chebyshew and trigonometric polynomials to the approximation of a solution of a singular integral equation with a multiplicative Cauchy kernel in the half-plane

Karczmarek P.

Opuscula Mathematica

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2008

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Vol. 28, no. 2

129-136

EN

In this article Chebyshev and trigonometric polynomials are used to construct an approximate solution of a singular integral equation with a multiplicative Cauchy kernel in the half-plane.

10

The Shortest Common Superstring Problem and Viral Genome Compression

Ilie L., Popescu C.

Fundamenta Informaticae

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2006

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Vol. 73, nr 1-2

153-164

EN

Viruses compress their genome to reduce space. One of the main techniques is overlapping genes. We model this process by the shortest common superstring problem. We give an algorithm for computing optimal solutions which is slow in the number of strings but fast (linear) in their total length. This algorithm is used for a number of viruses with relatively few genes. When the number of genes is larger, we compute approximate solutions using the greedy algorithm which gives an upper bound for the optimal solution. We give also a lower bound for the shortest common superstring problem. The results obtained are then compared with what happens in nature. Remarkably, the compression obtained by viruses is very close to the one achieved by modern computers.