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In this paper we present the application of the homotopy analysis method for determining the free vibrations of the simply supported beam. The linear and nonlinear cases are considered. The homotopy analysis gives the possibility to search for the solution of a wide range of problems described by means of the operator equations. The numerical examples are presented to confirm the exactness and fast convergence of the introduced method. The presented computational examples confirm the precision and the fast convergence of investigated method. In the linear case we knew the exact solutions, so we could compare with them the solutions obtained with the aid of homotopy analysis method. Differences between the obtained solutions were slight. However, the advantage of the examined method is that we receive here the approximate solution in the form of continuous function which can be used then in a further analysis or to perform various simulations.
W artykule przedstawiono homotopijną metodę analizy równań opisujących drgania swobodne układu o jednym stopniu swobody opisane równaniami liniowymi oraz nieliniowymi. Homotopijna metoda analizy daje możliwość poszukiwania rozwiązania szerokiego zakresu problemów opisanych za pomocą równań operatorowych. W artykule przedstawiono przykłady liczbowe w celu potwierdzenia dokładności i szybkiej zbieżności metody. W przypadku liniowym znane było dokładne rozwiązanie, dzięki czemu można było porównać je z rozwiązaniami uzyskanymi za pomocą analizy homotopijnej. Różnice między otrzymanymi roztworami były niewielkie. Zaletą badanej metody jest jednak to, że otrzymano przybliżone rozwiązania w postaci ciągłych funkcji, które można wykorzystać w dalszych analizach, w tym do interpretacji wyników badań doświadczalnych.
Czasopismo
Rocznik
Tom
Strony
61--71
Opis fizyczny
Bibliogr. 31 poz.
Twórcy
autor
- Faculty of Civil Engineering, The Silesian University of Technology, Akademicka 5, 44-100 Gliwice, Poland
autor
- Faculty of Applied Mathematics, The Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland
Bibliografia
- [1] S. Abbasbandy and E. Shivanian, (2011). A new analytical technique to solve Fredholm’s integral equations, Numer. Algor. 56, 27-43.
- [2] R. Brociek, E. Hetmaniok, J. Matlak, and D. Słota, (2016). Application of the homotopy analysis method for solving the systems of linear and nonlinear integral equations, Math. Model. Anal. 21, 350-370.
- [3] D.S. Chauhan, R. Agrawal, and P. Rastogi, (2012). Magnetohydrodynamic slip flow and heat transfer in a porous medium over a stretching cylinder: homotopy analysis method, Numer. Heat Transfer A 62, 136-157.
- [4] T. Fan and X. You, (2013). Optimal homotopy analysis method for nonlinear differential equations in the boundary leyer, Numer. Algor. 62, 337-354.
- [5] H. Gliński, R. Grzymkowski, A. Kapusta, and D. Słota, (2012) Mathematica 8, WPKJS, Gliwice (in Polish).
- [6] K. Gromysz, (2013). Examination of the Reinforced Concrete Plates Loaded Temporarily, Periodicaly and Kinematically, Monograph no 452, Silesian University of Technology Press, Gliwice (in Polish).
- [7] E. Hetmaniok, I. Nowak, D. Słota, and R. Wituła, (2015). Convergence and error estimation of homotopy analysis method for some type of nonlinear and linear integral equations, J. Numer. Math. 23, 331-344.
- [8] E. Hetmaniok, D. Słota, T. Trawiński, and R. Wituła, (2014). Usage of the homotopy analysis method for solving the nonlinear and linear integral equations of the second kind, Numer. Algor. 67, 163-185.
- [9] E. Hetmaniok, D. Słota, R. Wituła, and A. Zielonka, (2015). An analytical method for solving the twophase inverse Stefan problem, Bull. Pol. Ac.: Tech 63(3), 583-590.
- [10] E. Hetmaniok, D. Słota, R. Wituła, and A. Zielonka, (2015). Solution of the one-phase inverse Stefan problem by using the homotopy analysis method, Appl. Math. Modelling 39, 6793-6805.
- [11] T. Kaczorek, (2016). A new approach to the realization problem for fractional discrete-time linear systems, Bull. Pol. Ac.: Tech 64(1), 9-14.
- [12] Y. Khan, M. Fardi, (2015). A new efficient multiparametric homotopy approach for two-dimensional Fredholm integral equations of the second kind, Hacettepe J. Math. Stat. 44, 93-99.
- [13] Y. Khan, K. Sayevand, M. Fardi, M. Ghasemi, (2014). A novel computing multi-parametric homotopy approach for system of linear and nonlinear Fredholm integral equations, Appl. Math. Comput. 249, 229-236.
- [14] R. Lewandowski, (2006). Dynamics of the Building Structures, Poznań University of Technology Press, Poznań 2006 (in Polish).
- [15] S. Liao, (1998). Homotopy analysis method: a new analytic method for nonlinear problems, Appl. Math. Mech. - Engl. Ed. 19, 957-962.
- [16] S. Liao, (2003). Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman and Hall-CRC Press, Boca Raton.
- [17] S. Liao, (2009). Notes on the homotopy analysis method: some definitions and theorems, Commun. Nonlinear Sci. Numer. Simulat. 14, 983-997.
- [18] S. Liao, (2010). An optimal homotopy-analysis approach for strongly nonlinear differential equations, Commun. Nonlinear Sci. Numer. Simulat. 15, 2003-2015.
- [19] S. Liao, (2012). Homotopy Analysis Method in Nonlinear Differential Equations, Springer/Higher Education Press, Berlin/Beijing.
- [20] Z.M. Odibat, (2010). A study on the convergence of homotopy analysis method, Appl. Math. Comput. 217, 782-789.
- [21] P. Ostalczyk, (2015). On simplified forms of the fractional- order backward difference and related fractional- order linear discrete-time system description, Bull. Pol. Ac.: Tech 63(2), 423-433.
- [22] M. Russo and R.A. Van Gorder, (2013). Control of error in the homotopy analysis of nonlinear Klein-Gordon initial value problems, Appl. Math. Comput. 219, 6494-6509.
- [23] D. Słota, (2011). Homotopy Analysis Method and the Examples of Its Applications, Wyd. Pol. Śl., Gliwice (in Polish).
- [24] W. Sumelka, (2016). Fractional calculus for continuum mechanics - anisotropic non-locality, Bull. Pol. Ac.: Tech 64 (2), 361-372.
- [25] M. Turkyilmazoglu, (2010). A note on the homotopy analysis method, Appl. Math. Lett. 23, 1226-1230.
- [26] M. Turkyilmazoglu, (2010). Convergence of the homotopy analysis method, arXiv, 1006.4460v1.
- [27] K. Vajravelu, R.A. Van Gorder, (2012). Nonlinear Flow Phenomena and Homotopy Analysis. Fluid Flow and Heat Transfer, Springer/Higher Education Press, Berlin/Beijing.
- [28] R.A. Van Gorder, (2012). Control of error in the homotopy analysis of semi-linear elliptic boundary value problems, Numer. Algor. 61, 613-629.
- [29] R.A. Van Gorder and K. Vajravelu, (2009). On the selection of auxiliary functions, operators, and convergence control parameters in the application of the homotopy analysis method to nonlinear differential equations: a general approach, Commun. Nonlinear Sci. Numer. Simulat. 14, 4078-4089.
- [30] X. Zhang, B. Tang, and Y. He, (2011). Homotopy analysis method for higher-order fractional integrodifferential equations, Comput. Math. Appl. 62, 3194-3203.
- [31] M. Zurigat, S. Momani, Z. Odibat, and A. Alawneh, (2010). The homotopy analysis method for handling systems of fractional differential equations, Appl. Math. Modelling 34, 24-35.
Uwagi
PL
Opracowanie w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018)
Typ dokumentu
Bibliografia
Identyfikator YADDA
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